Fisher College of Business
Working Paper Series
Charles A. Dice Center for
Research in Financial Economics
Changing Times: The Pricing Problem
in Non-Linear Models
Robert L. Kimmel, Department of Finance, The Ohio State University
Dice Center WP 2008-24
Fisher College of Business WP 2008-03-022
December 17, 2008
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Changing Times: The Pricing Problem in Non-Linear Models∗
Robert L. Kimmel
Fisher College of Business
The Ohio State University†
This Version: 17 December 2008
Abstract
Finding conditional moments and derivative prices is a common application in continuous-time financial
economics, but these quantities are known in closed-form only for a few specific models. Recent research
identifies a large class of models for which solutions to such problems have convergent power series, allowing
approximation even when not known in closed-form. However, such power series may converge slowly or not
at all for long time horizons, limiting their practical use. We develop the method of time transformation,
in which the variable representing time is replaced by a non-linear function of itself. With appropriate
choice of the time transformation, power series often converge for much longer time horizons, and also
much faster, sometimes uniformly for all time horizons. For applications such as bond pricing, in which
the time-to-maturity may be many years, rapid convergence is very important for practical application.
The ability to approximate solutions accurately and in closed-form simplifies the estimation of non-affine
continuous-time term structure models, since the bond pricing problem must be solved for many different
parameter vectors during a typical estimation procedure. We show through several examples that the
series are easy to derive, and, using term structure models for which bond prices are known explicitly,
also show that the series are extremely accurate over a wide range of interest rate levels for arbitrarily
long maturities; in some cases, they are many orders of magnitude more accurate than series constructed
without time transformations. Other potential applications include pricing of callable bonds and credit
derivatives.
∗ I would like to thank seminar participants at INFORMS, CIREQ/CIRANO, the Econometric Society North American Winter
Meetings, Princeton University, the FERM annual meeting, Quantitative Methods in Finance, the Hong Kong University of Science
and Technology, Singapore Management University, The Ohio State University, the Atlanta Federal Reserve Bank, the Bachelier
Society Fourth World Congress, and the China International Conference in Finance, Jin Duan, Mark Fisher, Rüdiger Frey, Kewei
Hou, Nengjiu Ju, Yue-Kuen Kwok, Haitao Li, and Jun Yu, for many helpful comments and suggestions. Any remaining errors are
solely the responsibility of the author. Much of the material in this paper was originally part of Complex Times: Asset Pricing
and Conditional Moments under Non-Affine Diffusions.
† Columbus, OH 43210-1144. Phone: +1 614 292 1875. E-mail: kimmel.42@osu.edu.
1.
Introduction
Finding conditional moments or derivative prices in an economy driven by a continuous-time process is a
common problem in financial economics. However, for all but a few models, numeric methods are needed
to solve such problems. For some applications (typically estimation by a maximum likelihood or minimum
distance search), the pricing or conditional moment problem must be solved many times; if it must be solved
numerically each time, the intense computational burden severely limits the class of models that may be used
in practice.
Non-affine models of, for example, the short interest rate process, are frequently found in the literature;
for just a few examples, see Chan, Karolyi, Longstaff, and Sanders (1992), Aı̈t-Sahalia (1996), and Andersen,
Benzoni, and Lund (2004). However, such studies usually examine the behavior of the observed short interest
rate process, without taking bond pricing implications into account. When the term structure of interest rates
as a whole is examined, however, use of affine models is much more common; see, for example, Duffie and Kan
(1996), Dai and Singleton (2000), Duffee (2002), Dai and Singleton (2002), Duarte (2004), Thompson (2004),
Mosburger and Schneider (2005), Aı̈t-Sahalia and Kimmel (2008), and Cheridito, Filipović, and Kimmel
(2007). Egorov, Li, and Ng (2008) use an affine model of the simultaneous evolution of yield curves in two
different currencies. The widespread use of affine models in this literature is largely due to the difficulty
of calculating bond prices for non-affine models. Although non-linear models with explicit term structure
implications have appeared in the literature, they are drawn from a few very specific classes that allow closedform derivation of bond prices; see, for example, Ahn and Gao (1999) and Ahn, Dittmar, and Gallant (2002).1
Kimmel (2008) develops methods for approximation, by power series, of conditional moments and contingent claim prices in a large class of non-affine diffusion models. However, the convergence properties of
such power series may be poor for long time horizons. For some applications, this is not much of a problem;
for example, if conditional moments are needed for a method of moments estimation, and data are observed
frequently, then approximations need only be accurate at short time intervals, such as daily or weekly. However, for other applications, accuracy at much longer time horizons is needed. For example, in term structure
applications, one may need to calculate bond prices or swap rates with maturities of many years. A power
series representation of bond prices may converge for maturities up to some value, and then diverge for longer
maturities; for example, in the Cox, Ingersoll, and Ross (1985) model, inspection of the expression for bond
prices reveals that it has singularities (i. e., a point where it is not complex differentiable) at complex values
of maturity. Although such complex maturities have no meaning in a bond pricing application, nonetheless,
the singularities at these points prevent convergence of the power series of the bond price beyond positive
maturities of the same absolute magnitude. Even when the power series representation of a bond price converges for all maturities, as is the case for the model of Vasicek (1977), the convergence may be so slow as to
render the use of the series worthless for practical applications. In these two models, bond prices are known
in closed-form, so approximation is unnecessary; however, the same issues arise in other models in which bond
1 The
non-affine model of Ahn, Dittmar, and Gallant (2002) can be embedded within an affine model with the introduction
of additional state variables. These extra state variables do not have independent variation; that is, they are instantaneously
perfectly correlated with the original state variables.
1
prices are not known explicitly.
We therefore develop the method of time transformations, which often both extends the interval and
improves the rate of convergence of power series. This method works by replacing the time variable in the
partial differential equation representation of a pricing or conditional moment problem with a non-linear
function of itself. We show that proper choice of a non-linear transformation of time never decreases the range
of maturities for which the power series converges, and often increases it. A well-chosen time transformation can
extend the range of convergence to all positive maturities in many cases, even if the interval of convergence is
very short without the time transformation. Furthermore, appropriate use of a time transformation sometimes
establishes uniform convergence for all positive time horizons; that is, the asset price or conditional moment
can be approximated twenty, fifty, or one hundred years into the future just as accurately as one week ahead.
Without the use of time transformations, such uniform convergence can only be achieved in a few very special
cases.
The rest of this paper is organized as follows. In Section 2, we review the method of series solution of
conditional moment and contingent claim pricing problems, and discuss some of the problems of this approach
at long time horizons. In Section 3, we consider a family of non-affine transformations of the time variable,
and use these transformations to develop power series with improved convergence properties. In Section 4,
we apply time transformation methods to the two large families of models studied by Kimmel (2008), and
show that both the range and the rate of convergence of power series solutions to conditional moments and
bond prices can often be improved dramatically. Section 5 explores several specific models in detail. For
the term structure models of Ahn, Dittmar, and Gallant (2002) and Cox, Ingersoll, and Ross (1985), bond
prices are known in closed-form; these models therefore serve as illustrative examples to show the accuracy
of series approximations using our method. We find that approximations are both easy to derive and very
accurate (even with only a few terms) over a very wide range of initial interest rates, for both short and very
long maturities. By contrast, approximations based on power series constructed without time transformations
perform very poorly for maturities beyond a year or two; they are often many orders of magnitude less accurate
than approximations based on our method. We also consider other models, such as the callable bond pricing
model of Jarrow, Li, Liu, and Wu (2006), in which prices are not known in closed-form, and show how to
approximate them using our technique. Finally, Section 6 concludes. Proofs of all results are found in the
appendix.
2.
Series Solutions at Long Horizons
Continuous-time asset pricing or conditional moment problems are often based on N -dimensional diffusion
processes:
dXt = µ (Xt ) dt + σ (Xt ) dWt
with an initial condition Xt0 = x, where Xt is a vector of N state variables, µ (x) is an N -vector valued
function of x, σ (x) is an N × N -matrix valued function of x, and Wt is an N -dimensional standard Brownian
2
motion. Criteria for existence of a solution are found in, for example, Karatzas and Shreve (1991), Stroock
and Varadhan (1979), or Liptser and Shiryaev (2001). The conditional moment of ψ (x) is defined as:
¯
£
¤
f (∆, x) = E ψ (Xt+∆ ) ¯Xt = x
(2.1)
The price of derivative security that pays a given function ψ (x) of the state vector at maturity satisfies:
h R t+∆
i
¯
f (∆, x) = E e− t r(Xu )du ψ (Xt+∆ ) ¯Xt = x
(2.2)
where r (x) is the short interest rate. Applications from the early days of continuous-time finance were typically
option pricing problems, in which Xt would be the price process of a stock or other underlying asset, and
ψ (Xt ) would be the payoff of the option as a function of the underlying asset price at maturity. However,
typical bond pricing problems are also structured this way; the final condition is then ψ (Xt ) = 1, which is
the (fixed) final payoff of a zero-coupon bond at maturity. The expectation in (2.1) is normally taken under
true probabilities, whereas the expectation in (2.2) is taken under risk-neutral probabilities. Both types of
problems can be formulated as a partial differential equation problem:
N
N
N
X
∂2f
∂f
∂f
1 XX 2
σij (x)
(∆, x) =
µi (x)
(∆, x) +
(∆, x) − r (x) f (∆, x)
∂∆
∂xi
2 i=1 j=1
∂xi ∂xj
i=1
(2.3)
(2.4)
f (0, x) = ψ (x)
with r (x) = 0 for a conditional moment problem. The equivalence of the probabilistic problem of (2.1) or
(2.2), and the PDE problem of (2.3) and (2.4), is subject to technical conditions; see Levendorskii (2004a)
and Levendorskii (2004b) for a recent discussion of this issue for affine models.
Techniques for approximating conditional probability densities of non-linear diffusions are derived in Aı̈tSahalia (2002) (for a single state variable) and Aı̈t-Sahalia (2008) (for multiple state variables); see Aı̈t-Sahalia
(1999) for examples, and Egorov, Li, and Xu (2001) for an extension to time-inhomogeneous diffusions. With
some modification, such techniques could be used to find state price densities for the case of r (x) 6= 0. It
might seem that conditional moment and asset pricing problems could be solved by first approximating the
conditional probability density or the state price density, and then integrating the ψ (x) function over this
density. However, this integral may be difficult or impossible to find explicitly, and convergence of the density
approximations to the true density does not guarantee convergence of conditional moment (or asset price)
approximations derived in this way to the true conditional moment (or asset price). It is possible, even in
cases where the integral of the final condition over the true density exists, that the integral of the final condition
over an approximate density does not exist. Noting these problems, Kimmel (2008) skips the intermediate
step of finding a density, and instead writes the solution to (2.3) with the specific final condition (2.4) directly
as a power series in ∆, centered at zero:
f (∆, x) = a0 (x) +
∞
X
n=1
3
an (x)
∆n
n!
(2.5)
and derives a recursive relation for the coefficients:
a0 (x) = ψ (x)
an (x) =
N
X
(2.6)
N
µi (x)
i=1
N
∂an−1
1 XX 2
∂ 2 an−1
(x) +
σij (x)
(x) − r (x) an−1 (x)
∂xi
2 i=1 j=1
∂xi ∂xj
(2.7)
2
If the µi (x), σij
(x), and r (x) coefficients, as well as the final condition ψ (x), are all infinitely differentiable,2
then derivation of the power series coefficients ai (x), as in (2.6) and (2.7), is straightforward. Note the the
time-homogeneity of the coefficients of (2.3) is important; if these coefficients were instead explicit functions
of time, calculating the power series coefficients by a method similar to (2.6) and (2.7) would be much more
difficult or perhaps impossible.
Even in the time-homogeneous case, though, establishing convergence of the power series is a much trickier
issue than establishing existence of the coefficients; if the conditional moment or derivative price, f (∆, x),
extended to complex values of ∆, has a singularity at some ∆0 6= 0, then the power series does not converge
for any |∆| > |∆0 |; furthermore, if f (∆, x) has a singularity at ∆0 = 0, the power series does not converge
anywhere (except trivially at ∆ = 0). The region of convergence of the power series thus depends critically
on the nature of f (∆, x), not just for the positive real values of ∆ that are of interest for applications, but for
all complex ∆. However, f (∆, x) in most cases is known only implicitly as the solution to the PDE, so it is
not possible to determine by direct inspection whether f (∆, x) has the necessary properties for a convergent
power series (and, furthermore, there would be little reason to find the power series if f (∆, x) were known
explicitly). Rather, it must be determined from the coefficients and final condition of the problem (2.3) and
(2.4), whether f (∆, x) can be represented by a power series.
Kimmel (2008) shows that, for any specification of the coefficients of the scalar version of the PDE, there
exists an infinite-dimensional class of final conditions ψ (x) such that the corresponding solution f (∆, x) is
analytic in some neighborhood of ∆ = 0; its power series then converges within any circle |∆| < r contained
within that region.3 Although he provides a full characterization of all solutions to (2.3) and (2.4) that have
convergent power series representations, in practice, this characterization is somewhat tricky to use; it is
difficult to determine whether a given ψ (x) lies within the class of final conditions that generate solutions
with convergent power series. He therefore goes on to describe two particular families of scalar PDEs for which
it is possible to express the class of solutions with convergent power series in a more useful form. Using the
change of variable techniques of Colton (1979), he expresses the solution to (2.3) and (2.4) in the scalar case
as:
f (∆, x) = η (x) h (∆, y (x))
2 As
noted in Kimmel (2008), infinite differentiability of these functions is a sufficient, but not necessary condition for existence
of the power series coefficients; it is neither necessary nor sufficient for the convergence of the series.
3 The equivalence of the PDE problem and the probabilistic problem must still be established.
4
For appropriate choice of η (x) and y (x), the function h (∆, y) satisfies the transformed problem:
∂h
1 ∂2h
(∆, y) − rh (y) h (∆, y)
(∆, y) =
∂∆
2 ∂y 2
h (0, y) = g (y)
(2.8)
(2.9)
Since any scalar PDE problem can be converted to this form by change of variables, there is no loss of generality
by considering only this apparently restrictive form. In particular, note that the changes in variables do not
involve the time variable, so that if, for some x, f (∆, x) is an analytic function in some region of ∆, then
h (∆, y (x)) is also an analytic function of ∆ in the same region. Kimmel (2008) refers to PDEs expressed in
this form as “canonical,” and explicitly characterizes the analytic solutions to (2.8) and (2.9) for two particular
specifications of rh (y):4
b2
2
(y − a) + d
2
a
b2
rh (y) = 2 + y 2 + d
y
2
rh (y) =
(2.10)
(2.11)
These two specifications include many of the models that are commonly used in the literature. For example,
the term structure models of Vasicek (1977), Cox, Ingersoll, and Ross (1985), Ahn, Dittmar, and Gallant
(2002), and Ahn and Gao (1999) are all covered by one of these two cases, as is the callable bond pricing
problem of Jarrow, Li, Liu, and Wu (2006). However, many models that have never previously appeared in
the literature can also be expressed in one of these two forms, including many in which the state variable
process is strongly non-linear (see the discussion in Kimmel (2008) and in Section 5). For both specifications,
the region of analyticity (i. e., complex differentiability) of h (∆, y) (and therefore the interval of convergence
for its power series) is determined by two attributes of the final condition g (y). Specifically, for the power
series to converge at all, the final condition must be everywhere analytic. Note that this requirement applies
to all complex values of y, even if only real values are meaningful in the real-world problem. If this smoothness
requirement is satisfied, then the size of the region of analyticity (and therefore the region of convergence of
the power series) is determined by a growth condition on g (y). For (2.10), this condition is:
¯ b
¯
kyk2
¯ − 2 (y−a)2
¯
g (y)¯ ≤ ke 2
¯e
for some k > 0 and some norm (over the reals) kyk.5 A special case is obtained by taking the norm kyk ≡
√
|y| / k0 for some k0 > 0, in which case the growth condition is:
¯ b
¯
|y|2
¯ − 2 (y−a)2
¯
g (y)¯ ≤ ke 2k0
¯e
Note that, like the smoothness condition, the growth condition must be satisfied for all complex values of y,
even if only real values are meaningful for the application. The solution h (∆, y) is then analytic in a region
4 Kimmel
(2008) also considers the case where rh (y) is linear, but for our purposes, analysis of this case is not very different
from (2.10) with b = 0.
√
5 See Kimmel (2008) for a brief discussion of such norms. Multiples of the absolute value function kyk ≡ |y| / k are valid
0
q
norms, but so are elliptical norms such as kyk ≡ (Re y)2 /k1 + (Im y)2 /k2 . Many other types of norms are also possible.
5
that is circular in the special case b = 0, but is in general an elongated ellipse-like shape.6 This region includes
the circle |∆| < [ln (1 + k0 |2b|)] / |2b|, and the power series of h (∆, y) converges within this circle. If the
growth condition is not satisfied for this type of norm with any k0 , then the power series does not converge
anywhere (except trivially at ∆ = 0); if the growth condition is satisfied by this type of norm for any k0 , then
the power series converges everywhere. Within the region of analyticity in ∆, the solution h (∆, y) is also
everywhere analytic in y, although this property is not particularly important, since we, like Kimmel (2008),
consider power series only in the time variable. Analyticity of the final condition g (y) is needed to establish
analyticity of h (∆, y) in ∆, and therefore convergence of the power series, but analyticity of h (∆, y) in y has
no relevance for convergence, since we do not consider power series in y.
The situation for (2.11) is more complicated, but in many ways similar to that of (2.10). For the solution
h (∆, y) to be analytic, the final condition must be of the form:7

√
√

g1 (y) y 1− 21+8a + g2 (y) y 1+ 21+8a
g (y) =
√
√

g1 (y) y 1− 21+8a + g2 (y) y 1+ 21+8a ln y
√
√
1+8a
2
∈
/N
1+8a
2
∈N
(2.12)
where g1 (y) and g2 (y) are both even, everywhere analytic functions that satisfy the growth bounds:
¯ b 2
¯
kyk2
¯ −2y
¯
g2 (y)¯ ≤ ke 2
¯e
¯ b 2
¯
kyk2
¯ −2y
¯
g1 (y)¯ ≤ ke 2
¯e
for some k > 0 and some norm (over the reals) kyk. Analogously with (2.10), g1 (y) and g2 (y) must be smooth
(analytic) and satisfy the growth bounds for all complex values of y, even though the conditional moment or
pricing problem only has meaning for real values of y. Again taking a norm of the form kyk ≡ |y| /k0 , the
solution h (∆, y) is then analytic in a region of ∆ that includes the circle |∆| < [ln (1 + k0 |2b|)] / |2b|, and
its power series converges everywhere within that circle. Except for specific values of a, the solution is not
everywhere analytic in y, but can be expressed in terms of two functions that are analytic in both ∆ and y;
see Kimmel (2008). However, as in the previous case, analyticity of h (∆, y) in y is not particularly important.
The results of Kimmel (2008) therefore allow us to determine when, for certain versions of the canonical
PDE, solutions are analytic in the time variable, and can therefore be approximated by the first few terms of
a convergent power series. However, for many applications, we would like to approximate solutions for very
large values of ∆, and this presents some practical difficulties. First, if there is a singularity anywhere in the
complex plane, then convergence of the power series is limited to a circle whose radius is the distance from
the origin to the singularity. Therefore, if a power series is to converge for arbitrarily large time horizons,
the solution h (∆, y) must have an analytic extension to all complex values of ∆, since a singularity anywhere
in the complex plane limits the range of convergence for positive ∆. Second, even if h (∆, y) is everywhere
analytic, and its power series converges everywhere, such convergence is, in general, not uniform. Consequently,
6 Although
this shape superficially resembles an ellipse, in that it is a smooth, closed curve that is longer in one dimension
than the other, its exact shape is different.
√
7 We follow Kimmel (2008) here, but note that the first case listed in (2.12) may be used whether or not
1 + 8a/2 is an
integer. If it is, the presence of g2 (y) is redundant; simply adding y
generates exactly the same final condition.
6
√
1+8a g
2
(y) to g1 (y), and setting g2 (y) equal to zero,
convergence for large values of ∆ may be very slow, so that many terms of the power series must be calculated
for a good approximation; use of a series is then not practical.
Table 1 describes the convergence properties of series representations of bond prices in several term structure models. In all of these models, bond prices are known in closed-form.8 In the model of Vasicek (1977), the
bond pricing problem reduces to the canonical problem with rh (y) specified by (2.10), and the final condition
satisfies the corresponding growth condition for all values of ∆. Consequently, bond prices are everywhere
analytic, and a power series converges for all maturities. However, this convergence is not uniform, and can be
very slow for long maturities. The situation for the model of Cox, Ingersoll, and Ross (1985) is quite different.
Here, the bond pricing problem reduces to the canonical problem (2.8) and (2.9), with rh (y) specified by
(2.11). The model of Ahn, Dittmar, and Gallant (2002) is similar to that of Cox, Ingersoll, and Ross (1985),
but with rh (y) specified by (2.10) instead. However, in both cases, the applicable growth condition is satisfied
for some k0 and not for others, so that analyticity of bond prices cannot be established for all ∆. In fact,
in both models, bond prices have singularities at complex values of ∆. Although such complex values are
meaningless for applications, the singularities nonetheless prevent convergence of the power series for positive
values. As such, bond prices in these models can be represented by a power series up to a certain maturity;
beyond that maturity, the series diverges. The situation is worse still for the model of Ahn and Gao (1999),
in which bond prices have a singularity at ∆ = 0. The power series for bond prices in this model does not
converge for any positive maturity.
In all four of the models described in Table 1, bond prices are known in closed-form, so discussion of the
convergence properties of their power series is for illustrative purposes only. However, the results of Kimmel
(2008) apply to many other models, in which bond prices (or other derivative prices or conditional moments)
are not known in closed-form, and similar issues arise in those models. For example, Jarrow, Li, Liu, and Wu
(2006) consider a callable bond pricing problem, in which bond prices satisfy:9
³
∂f
∂f
σ2 x ∂ 2 f
c2 ´
(∆, x) = κ (θ − x)
(∆, x) +
(∆, x) − c1 x +
f (∆, x)
2
∂∆
∂x
2 ∂x
x
f (0, x) = 0
This problem can, by change of variables, be converted to the canonical form (2.8) and (2.9), with rh (y) given
by (2.11). With some parameter restrictions, the final condition is sufficiently smooth, and also satisfies the
growth condition for some k0 , but not for other values. Consequently, the solution to their problem can be
represented by a power series that converges for short maturities, but diverges for long maturities. In this
problem, the bond price has a singularity for complex values of ∆; although these values have no meaning
in the context of the real-world problem, such singularities prevent convergence of a power series for positive
values of ∆.
We propose the method of time transformations to remedy these problems. Specifically, we apply a change
8 For
one of the models, that of Ahn and Gao (1999), the definition of “closed-form” must include confluent hypergeometric
functions, which are usually defined as the solution to an ordinary differential equation. With a stricter definition of “closed-form,”
bond prices in this model are not known explicitly.
9 Our notation differs somewhat from theirs.
7
of variables to f (∆, x), the solution to (2.3) and (2.4), replacing the time variable, ∆, with a non-linear function
of itself, τ (∆). The solution, expressed in terms of τ instead of ∆, satisfies a modified general PDE, with the
same final condition. By expressing the pricing PDE in terms of τ instead of ∆, the range of convergence of
the power series solution can often be improved. In particular, if the solution h (∆, y) is well-behaved for all
positive values of ∆ (which are the values of interest in real-world applications), a well-chosen transformation
of the time variable extends the convergence of a power series to all ∆ ∈ [0, +∞). In other words, even if there
are singularities for negative or complex values of ∆, so that a power series directly in ∆ converges only for a
limited range of positive values, the method of time transformation can extend this range to any arbitrarily
long range. If certain additional conditions are met, the time transformation also makes such convergence
uniform; in this case, the power series representation of h (∆, y) converges just as quickly for ∆ equal to 30
or 50 years as for ∆ equal to one week. Furthermore, the time transformation we use preserves the ease
of calculation of power series coefficients through a recursive relation that is a slightly modified version of
(2.7). In the next section, we study the method of time transformations in detail, and examine its effect on
convergence of the power series representations of solutions to (2.3) and (2.4).
3.
Time Transformations
As noted in the preceding section and in Kimmel (2008), a singularity at a negative or complex value of ∆
(i. e., values not of interest for typical applications) nonetheless causes the power series representation of a
derivative price or conditional moment f (∆, x) to fail to converge on the interval ∆ ∈ [0, +∞) (i. e., the values
normally of interest in applications). However, provided any singularities in f (∆, x) are not near the positive
real axis (which is usually the case in applications), it is nonetheless possible, as we now show, to construct
power series that converge for all values of ∆ ∈ [0, +∞). In some cases, convergence on this interval can even
be uniform, so that the series converges just as quickly for a time period of ten, twenty, or one hundred years
into the future, as it does for one week ahead. This goal can be realized by performing a change of the time
variable, replacing ∆ with some function τ (∆), and constructing a power series in τ (∆) instead of one in
∆ directly. A power series converges within a circle in the complex plane whose radius is the distance from
the point of expansion to the nearest singularity. However, if the function τ (∆) is not affine in ∆, then the
transformation effectively distorts time, stretching it in some directions and compressing it in others, so that a
circle in the complex plane of τ (∆) is a non-circular shape in the complex plane of ∆. By appropriate choice
of τ (∆), singularities in f (∆, x) can be moved away from the origin, whereas the interval ∆ ∈ [0, +∞) can
be compressed, so that each point moves closer to the origin, and falls within the circle of convergence.
Throughout, we use the same time transformation (or more precisely, since it depends on a parameter,
family of time transformations), described in Section 3.1. However, we use this transformation for two distinct
purposes, which are described in Sections 3.2 and 3.3, respectively. The first use increases the range of
convergence of the power series of the conditional moment or asset price sought; that is, the time transformation
allows a series to converge for a longer range of time horizons than a series constructed without the time
transformation. We refer to this type of convergence as “small circle convergence,” for reasons discussed
8
below. If the price or conditional moment is well behaved for all positive time horizons, then its power series
can be made to converge for all values out to +∞ through use of the time transformation. Even in this case,
though, this convergence is in general not uniform; that is, it is slower for longer time horizons. Then only a
small number of terms in the power series may be required to approximate the solution very accurately for
small ∆, but the number of terms needed increases with ∆, so that practical use of the power series is limited
to smaller values of ∆.
The second use of the time transformation, which we refer to as “large circle convergence,” establishes
uniform convergence out to +∞. If convergence is uniform, then the solution to (2.3) and (2.4) can be
approximated by a given number of terms in the power series, and the accuracy of the approximation does
not depend on ∆; solutions at a one hundred year time horizon are as easy to approximate as solutions
at a one week time horizon, for example. Much stronger restrictions than in the small circle case must be
satisfied; specifically, f (∆, x) must have the right type of behavior as ∆ approaches +∞, and use of the time
transformation for large circle convergence often requires that the state variable be changed as well. In both
the small and large circle convergence cases, however, the computation of a power series in τ (instead of ∆)
proceeds by a simple recursive relation. In this section, we examine this family of transformations, and the
conditions needed for both small and large circle convergence. Throughout, we use the term structure model
of Cox, Ingersoll, and Ross (1985) (for which bond prices are known explicitly) as an illustrative example, but
the issues that arise for this model also arise for other models in which prices are not known explicitly.
3.1.
The Basic Time Transformation
The basic time transformation used throughout is:
τ = τk (∆) ≡ 1 − exp (−k∆)
(3.1)
for some k 6= 0. In typical applications, it is most useful to choose k to be a real positive number, but the
results here are valid for any complex (but non-zero) value of k. The inverse transformation is:
∆ = ∆k (τ ) = −
ln (1 − τ )
k
(3.2)
Note that every number (except zero) has infinitely many complex logarithms; we choose the principal branch
of the logarithm function −π < Im [ln (1 − τ )] ≤ +π. With this choice, ∆ is an analytic (i. e., complex
differentiable) function of τ for all values of τ except real values τ ≥ 1. We then express the solution to (2.3)
and (2.4), f (∆, x), as:
f (∆, x) = w (τ, x) = w (τk (∆) , x)
(3.3)
w (τ, x) = f (∆, x) = f (∆k (τ ) , x)
(3.4)
with the inverse relation:
9
Note that τ (0) = 0. Note also that f (∆, x), without additional restrictions, can be expressed in the form
(3.3) only for −π < Im [−k∆] ≤ +π; however, for real k, this range includes all real ∆, which are the values
that matter for applications. If f (∆, x) is an analytic function around ∆ = 0, then w (τ, x) is also an analytic
function around τ = 0; the reverse implication also holds. Therefore, either both have a convergent power
series (around ∆ = 0 and τ = 0, respectively), or neither does. The power series for w (τ, x) can be found
by essentially the same method used to find the power series for f (∆, x). By substituting the expression for
f (∆, x) from (3.3) into (2.3) and (2.4), the PDE can then be rewritten in terms of τ and w (τ, x) instead of
∆ and f (∆, x):
N
k (1 − τ )
N
N
X
1 XX 2
∂w
∂w
∂2w
(τ, x) =
(τ, x) +
(τ, x) − r (x) w (τ, x)
µi (x)
σij (x)
∂τ
∂xi
2 i=1 j=1
∂xi ∂xj
i=1
w (0, x) = ψ (x)
(3.5)
(3.6)
This PDE is of the same form as (2.3) and (2.4), except that the derivative with respect to the time variable
in (3.5) has an explicitly time-dependent coefficient that is not present in (2.3). However, because of the linear
form of this coefficient, it is still possible to derive a power series for w (τ, x) from a simple recursive relation.
We now write the power series representation of w (τ, x) in τ , rather than of f (∆, x) in ∆:
w (τ, x) = b0 (x) +
∞
X
bn (x)
n=1
τn
n!
(3.7)
The initial coefficient b0 (x) follows immediately from (3.6):
b0 (x) = ψ (x)
(3.8)
By plugging (3.7) into (3.5), and gathering terms of like order in τ , we find the subsequent coefficients bn (x)
for n ≥ 1 recursively:
·
¸
N
N N
1X
1 XX 2
r (x)
∂bn−1
∂ 2 bn−1
bn (x) =
(x) +
(x) −
+ 1 − n bn−1 (x)
µi (x)
σ (x)
k i=1
∂xi
2k i=1 j=1 ij
∂xi ∂xj
k
(3.9)
A power series in τ can therefore be calculated just as easily as a power series in ∆, using a very similar
recursive relation; the only differences are that the right-hand side has been divided through by k, and an
additional term appears in the coefficient of bn−1 (x) in (3.9) that does not appear in the coefficient of an−1 (x)
in (2.7). As in the case discussed in Section 3, time-homogeneity of the coefficients of the original pricing PDE
is important, but so is the particular form of the time transformation. Calculating the coefficients bn (x) would
be much more difficult, or perhaps impossible, if the coefficients of the original PDE were to depend explicitly
on time, or if some other type of time transformation were chosen.
The inverse time transformation ∆k (τ ) is an analytic function in a neighborhood of τ = 0. If f (∆, x) is
an analytic function around ∆ = 0, then it follows from (3.4) that w (τ, x) is also an analytic function around
τ = 0. If a solution f (∆, x) to (2.3) and (2.4) cannot be found explicitly, it can still be approximated by a
power series in two different ways, which we summarize in the following remark.
10
Remark 1. If the solution f (∆, x) to the conditional moment or asset pricing problem (2.3) and (2.4) can
be shown to be analytic in a neighborhood of ∆ = 0 (through the results of Kimmel (2008) or by some other
method), then it can be approximated through a truncated power series in two distinct ways:
1. The first few terms of the power series of f (∆, x) (the solution to the original problem, (2.3) and (2.4))
in ∆ can be calculated using (2.6) and (2.7).
2. The first few terms of the power series of w (τ, x) (the solution to the transformed problem, (3.5) and
(3.6)) in τ can be calculated using (3.8) and (3.9). An approximation of f (∆, x) (the solution to the
original problem (2.3) and (2.4)) can then be constructed from the approximation to w (τ, x) using (3.3).
With either of these methods, f (∆, x) can be approximated by the first few terms of a convergent power series
in a neighborhood of ∆ = 0. Since the power series converges in this neighborhood, the approximation becomes
more accurate when more terms are added. However, the size and shape of the neighborhood of ∆ = 0 in which
the series converges is, in general, not the same with the two methods.
The first method could even be considered a special case of the second. If we scale the time transformation
by k, then we see that τk (∆) approaches ∆ as k goes to zero:
lim
k→0
τk (∆)
=∆
k
Throughout, we sometimes speak loosely of the power series of w (τ, x) converging for certain values of ∆. A
power series in τ converges within |τ | < s for some s ≥ 0. Therefore, a statement that a power series in τ
converges in a given region of ∆ simply means that |τk (∆)| < s everywhere within that region of ∆. Since
the relation between τ and ∆ depends on a parameter k, the value of k must be stated explicitly, or be clear
from the context, for such statements to be meaningful.
If all that is needed is an approximation of f (∆, x) in some very small neighborhood of ∆ = 0, it usually
makes little difference which method is used; if f (∆, x) is analytic around ∆ = 0, then a power series in ∆
converges around this point, and the power series of w (τ, x) also converges around τ = 0, which corresponds
to a neighborhood of ∆ = 0. If f (∆, x) is not analytic at ∆ = 0, then neither power series converges for any
values except ∆ = 0 and τ = 0, respectively, at which f (0, x) = ψ (x) trivially.
However, as we will see in the next few sections, for large values of ∆, there are two significant advantages
to using the second method for many applications. First, the neighborhood of ∆ = 0 in which the series
converges can often be larger when the second method is used; this phenomenon is examined in Section 3.2.
Second, convergence within that neighborhood can often be faster with the second method; this phenomenon
is studied in Section 3.3.
3.2.
Small Circle Convergence
The radius of convergence of a power series is the distance from the point of expansion to the nearest singularity
in the function being represented. The power series representation of f (∆, x) around ∆ = 0 therefore converges
in the region |∆| < s for some s ≥ 0, with s = +∞ possible. (For finite s, the power series may or may not
11
converge on |∆| = s.) Similarly, the power series representation of w (τ, x) around τ = 0 converges in the
region |τ | < r for some r > 0. For financial and economic applications, we typically care only about positive
values of ∆, so the relevant values of ∆ for which the first power series converges are ∆ ∈ [0, s). We now show
that, for appropriate choice of k, use of the basic time transformation results in a power series representation
of w (τ, x) in τ that converges on at least ∆ ∈ [0, s), and often for larger values of ∆. That is, appropriate
use of the time transformation never decreases the interval of ∆ for which a power series converges, and often
increases it.
As discussed earlier, the circle |τ | < r within which the power series representation of w (τ, x) converges
implicitly specifies a region of ∆. Since the basic time transformation is non-affine, a circle in the complex
plane of τ corresponds to a non-circular shape in the complex plane of ∆. We now consider the requirements
for convergence of the power series within a circle in |τ | < r with radius 0 < r ≤ 1, and the corresponding
region of ∆. The implications of convergence in a circle with radius r > 1 are quite different, and are discussed
in the Section 3.3. Within |τ | < r for r ≤ 1, the inverse time transformation is analytic in τ . Consequently,
the function w (τ, x) = f (∆, x) is analytic in τ at all points ∆ = ∆k (τ ), |τ | < r, where f (∆, x) is analytic
in ∆ (with x representing a vector of state variables). The following theorem characterizes, for 0 < r < 1, a
region of ∆ whose analyticity guarantees the convergence of the power series in τ ; the case of r = 1 is treated
separately. By convention, the arccosine function takes values in [0, π].
Theorem 1. Let k 6= 0, and let 0 < r < 1. Holding x fixed, if f (∆, x) is defined and analytic in ∆ in the
region where:
p
p
1 − r2 <Im (k∆) < + arccos 1 − r2




cos [Im (k∆)]
cos [Im (k∆)]
 <Re (k∆) < − ln  p

− ln  p
− cos2 [Im (k∆)] − (1 − r2 )
+ cos2 [Im (k∆)] − (1 − r2 )
− arccos
(3.10)
(3.11)
then the function w (τ, x) = f (∆, x), with ∆ ≡ ∆k (τ ) defined by (3.2), is analytic within the circle |τ | < r.
Conversely, if w (τ, x) is defined and analytic in the circle |τ | < r, then f (∆, x) = w (τ, x), with τ ≡ τk (∆)
defined by (3.1), is analytic in ∆ in the region indicated by (3.10) and (3.11).
Proof: See appendix.
Figure 1 shows the shape of the region described by (3.10) and (3.11) for k = 0.1; these shapes are bounded
and elongated regions, extending farther in the real positive direction than in other directions. If w (τ, x) is
analytic within |τ | < r for one of the values of r shown (with k = 0.1), then its power series converges within
that circle, and f (∆, x) can be recovered from this power series, using (3.3), within the corresponding region.
Not surprisingly, circles in τ with larger radius correspond to larger regions of ∆; as the radius r approaches
1, all of the positive real axis is eventually included in the corresponding region of ∆. However, this region is
bounded in other directions; as r approaches 1 (holding k fixed at 0.1), the region of ∆ described by (3.10)
and (3.11) never has real part smaller than −10 ln 2, and never has imaginary part larger than +5π or smaller
than −5π. The figure also shows the locations of the singularities of the bond price function for the model of
Cox, Ingersoll, and Ross (1985), with parameter values κ = 0.5 and σ = 0.15. (See Table 1 for a description of
12
the risk-neutral interest rate process in this model. We do not specify the value of the θ parameter, because
it does not affect the location of the singularities.) The closest singularities to the origin have modulus of
approximately 8.237, so the power series (in ∆) representation of bond prices converges only for maturities of
less than this amount. (Note that the interval of convergence depends on the values of the κ and σ parameters,
as per Table 1.) However, by applying the time transformation with k = 0.1, and constructing a power series
representation in τ instead of ∆, the interval of convergence can be much larger. As shown, these singularities
all lie outside the r = 0.99 circle, which is the largest circle shown; in fact, the singularities lie outside of
any circle with r < 1, as can be confirmed by checking (3.10) and (3.11). Since the r = 0.99 circle includes
maturities of more than 50 years, a power series in τ representation of bond prices (with k = 0.1) converges for
maturities at least this large. Use of the time transformation in this example therefore increases the interval
of the power series convergence by a factor of more than six. (As we see later, the interval of convergence can
be improved further still with larger values of k.)
Figure 2 illustrates the effects of the time transformation in a different way. One of the shapes shown is
a circle in ∆, with radius equal to 50; the other shape, contained within the circle, is a circle in τ projected
onto the complex plane of ∆, as described by (3.10) and (3.11) for k = 0.1, with r approximately equal to
0.9866 (i. e., similar to the shapes shown in Figure 1, but with a different radius). With this combination of
r and k, the elongated shape reaches a value of +50 on the positive real axis, touching the circle in ∆ at this
point. A power series representation of f (∆, x), the solution to (2.3) and (2.4), can be calculated using (2.6)
and (2.7). However, if f (∆, x) has a singularity anywhere inside the circle, the radius of convergence of the
power series extends only as far as this singularity, and the series diverges for values of ∆ approaching +50.
Similarly, a power series representation of w (τ, x), the solution to (3.5) and (3.6), can be calculated using (3.8)
and (3.9); f (∆, x) can then be recovered using (3.3). If there is a singularity in f (∆, x) within the elongated
shape shown in Figure 2, then the power series representation of w (τ, x) does not converge for all |τ | < 0.9866,
and it is not possible to approximate values of f (∆, x) for ∆ approaching +50 this way either. However, if
there are singularities inside the circle in Figure 2, but outside the elongated shape, then the power series
representation of f (∆, x) does not converge for all |∆| < 50, but the power series representation of w (τ, x)
does converge for all |τ | < 0.9866, which effectively includes ∆ ∈ [0, +50). In this case, it is not possible to
approximate the solution f (∆, x) for a larger range of ∆ by finding a power series directly, but it is possible
by approximating w (τ, x) with a truncated power series instead, and then applying (3.3). In fact, for positive
real k, a power series in τ always converges for positive real values of ∆ at least as large as those for which a
power series directly in ∆ converges. Therefore, if positive real values of ∆ are of interest (as is usually the
case in applications), the use of the time transformation (with positive real values of k) never decreases the
range of convergence, and may well increase it. The figure also shows the points of singularity of the bond
price function in the model of Cox, Ingersoll, and Ross (1985), with κ = 0.5 and σ = 0.15. As shown, there
are eight singularities within the circle in ∆, preventing convergence of a power series for maturities up to
∆ = 50. However, none of these singularities are within the circle in τ , so that use of the time transformation
(with k = 0.1) extends convergence at least to ∆ = 50.
Both Figures 1 and 2 show circles in τ mapped to ∆ when the parameter of the time transformation is
13
k = 0.1. For the bond pricing example with the parameters chosen, the value k = 0.1 is sufficient to construct
convergent power series for very long maturities. However, the location of the singularities in the bond pricing
function depends on the parameters, and could be closer to the origin; in this case, the time transformation
with k = 0.1 would result in a convergent power series for a shorter range of ∆. For example, continuing
with the model of Cox, Ingersoll, and Ross (1985), but with parameter values κ = 1 (instead of κ = 0.5)
and σ = 0.15, then there would be singularities inside all of the circles shown in Figure 1, except the r = 0.4
circle. The interval of convergence using k = 0.1 would then be considerably smaller than when κ = 0.5,
including maturities up to some value that is considerably less than 20 years. However, when the parameter
of transformation k is larger, the elongation of the circles in τ (when mapped to ∆) is more extreme. Figure 3
shows circles in τ mapped to ∆ when the parameter of the time transformation is k = 0.2, with the axes
drawn to the same scale as Figure 1. Note, for example, that circles in τ that extend to ∆ = 50 in the positive
real direction include substantially smaller regions of ∆ in other directions than when k = 0.1. For example,
with k = 0.2, the circle with r = 0.99999 (i. e., the largest circle shown in Figure 3) extends to approximately
∆ = +61.030 in the positive direction, but to ∆ = −3.466 in the negative direction. With k = 0.1, the circle
with r = 0.99 (i. e., the largest circle shown in Figure 1) extends not quite as far in the positive direction, to
approximately ∆ = +52.958, but extends almost twice as far as the k = 0.2 circle in the negative direction, to
∆ = −6.906. The r = 0.99999 circle in Figure 3 also extends only to about ±7.84 along the imaginary axis,
whereas the r = 0.99 circle in Figure 1 extends to about ±14.69 along the imaginary axis. Larger positive
values of k mean smaller regions of ∆ are needed to extend the same distance in the positive real direction.
Figure 3 also shows the points of singularity for bond prices in the Cox, Ingersoll, and Ross (1985) model with
κ = 1; as shown, they are all outside all of the circles shown, guaranteeing convergence for maturities up to
more than 50 years. By contrast, with κ = 0.5, convergence cannot be guaranteed even to 20 years, as the
singularities appear inside all of the circles (except the r = 0.4 circle) in Figure 1.
Figure 4 shows the effect of varying values of k directly, with circles in τ mapped back to ∆ for various
combinations of k and r. The combinations are chosen so that the largest positive value of ∆ falling within
the circle is always the same (approximately +30.40). As shown, the shapes for large values of k are contained
entirely within the shapes for smaller values of k, but still extend the same distance in the positive real
direction. The significance of this phenomenon is that higher values of k can cause the power series for w (τ, x)
to converge for larger (real positive) values of ∆. For example, if there are no singularities in f (∆, x) within
the k = 0.25 shape, but there is a singularity within the k = 0.05 shape, then the power series for w (τ, x)
converges for at least |τ | < 0.999 when k = 0.25, which includes the values ∆ ∈ [0, 30.40). However, for
k = 0.05, the radius of convergence is less than 0.6105, and does not include the same interval in ∆. Larger
values of k never decrease the range of ∆ for which a power series converges, and may well increase it.
The qualitative implications of Theorem 1 may therefore be summarized as follows:
Remark 2. For applications in which real positive values of ∆ are of interest, if the solution f (∆, x) to a
conditional moment or derivative pricing problem (2.3) and (2.4) has a power series in ∆ that converges for
some ∆ ∈ [0, s), then:
1. The range of convergence using the time transformation with k > 0 is never smaller than the range
14
without the time transformation, and may well be larger.
2. Increasing the value of k > 0 in the time transformation never decreases the range of convergence, and
may well increase it.
3. If the solution has no singularity for any real ∆ ≥ 0, then the range of convergence can be made arbitrarily
large by using the time transformation with a sufficiently large value of k > 0.
Theorem 1 describes the region in ∆ that corresponds to |τ | < r for 0 < r < 1. If w (τ, x) (for a given value
of k) can be shown to be analytic for |τ | < 1, then it is possible to approximate f (∆, x) for all ∆ ∈ [0, +∞) even
when a power series directly in ∆ diverges for large values, by first approximating w (τ, x) with a truncated
power series in τ , and then applying (3.3). The power series then converges for values of τ that correspond to
all ∆ ∈ [0, +∞). The following corollary addresses this case:
Corollary 1. Let k 6= 0 be a constant. Holding x fixed, if f (∆, x) is defined and analytic in ∆ in the region
where:
π
π
< Im (k∆) < +
2
2
(3.12)
Re (k∆) > − ln (2 cos [Im (k∆)])
(3.13)
−
then the function w (τ, x) = f (∆, x), where ∆ = ∆k (τ ) is defined by (3.2), is analytic in τ in the region
|τ | < 1. Conversely, if w (τ, x) is defined and analytic in τ in the region |τ | < 1, then f (∆, x) = w (τ, x),
where τ = τk (∆) is defined by (3.1), is analytic in ∆ in the region indicated by (3.12) and (3.13).
Proof: See appendix.
The shape described by (3.12) and (3.13) is simply the union of the shapes described by (3.10) and (3.11)
for all r < 1. As shown in Figure 5, the unit circle |τ | = 1 maps to an open shape in ∆; if the parameter
of the time transformation k is positive, then the opening is toward the right. For larger values of k, the
circle (mapped to ∆) follows the positive real axis in ∆ more closely; provided f (∆, x) has no singularities
in the neighborhood of the positive real axis, convergence for arbitrarily large positive real values of ∆ can
be established by choosing a sufficiently large value of k. For example, suppose f (∆, x) has a singularity at
∆ = −5, and nowhere else. A power series directly in ∆ diverges for |∆| > 5. However, the power series
representation of w (τ, x) converges for all |τ | < 1, provided k > (ln 2) /5. This value can be determined
approximately from the graph, or exactly from (3.13). The interval of convergence is thus extended from
∆ ∈ [0, 5) to ∆ ∈ [0, +∞) through use of the time transformation with sufficiently large k. Similarly, for the
example of the term structure model of Cox, Ingersoll, and Ross (1985), a value of k = 0.1 suffices to establish
convergence on ∆ ∈ [0, +∞) when κ = 0.5 and σ = 0.15. For κ = 1 and σ = 0.15, a value of k = 0.15 suffices
for convergence on ∆ ∈ [0, +∞).
The qualitative implications of the corollary may thus be summarized as:
Remark 3. For applications in which real positive values of ∆ are of interest, if the solution f (∆, x) to
a conditional moment or contingent claim pricing problem (2.3) and (2.4) has no singularities near a real
positive value of ∆, then:
15
1. The range of convergence of the power series representation of the solution with the time transformation
with k > 0 is never smaller than the range without the time transformation, and may well be larger.
2. Increasing the value of k > 0 in the time transformation never decreases the range of convergence of the
power series representation of the solution, and may well increase it.
3. For sufficiently large k > 0, the range of convergence of the power series solution includes ∆ ∈ [0, +∞).
Although use of the basic time transformation establishes convergence of a power series for arbitrarily large
values of ∆ when there are no singularities near the positive real axis, such convergence is, in general, not
uniform on ∆ ∈ [0, +∞). For example, in the model of Vasicek (1977), bond prices are everywhere analytic, so
that their power series representation converges everywhere, but this convergence is not uniform for all ∆.10
For the model of Cox, Ingersoll, and Ross (1985), bond prices are not everywhere analytic, but the range of
convergence of a power series can be made to include ∆ ∈ [0, +∞) by use of the time transformation with
sufficiently large k > 0. However, in this case also, the convergence is, in general, not uniform. In such cases,
for very large ∆, the power series still converges, but the rate of convergence may be very slow. If it can
be established that a power series converges uniformly on ∆ ∈ [0, +∞), then convergence is guaranteed to
occur at a minimum rate no matter how large ∆ is. Uniform convergence on ∆ ∈ [0, +∞) can sometimes be
established for series solutions to asset pricing or conditional moment problems, but the conditions needed are
quite different than the conditions that establish uniform convergence on finite time intervals. We examine
this situation in detail in the next section.
3.3.
Large Circle Convergence
The previous section considers the problem of extending the range of convergence of a series solution to a
conditional moment or asset pricing problem; specifically, it examines convergence properties of power series
in τ for circles |τ | < r with 0 < r ≤ 1, and establishes sufficient conditions for convergence on intervals such
as ∆ ∈ [0, T ) with T > 0, or, in the case of r = 1, on ∆ ∈ [0, +∞). As discussed, if the solution to an asset
pricing or conditional moment problem is well-behaved on ∆ ∈ [0, +∞), then the solution to a transformed
problem, constructed using the time transformation, can be shown to be analytic within the circle |τ | < 1.
The power series of the solution to the transformed problem then converges within this circle in τ , which
effectively includes all positive values of ∆, even if a power series constructed directly in ∆ converges only for
a very short range of values. However, even when convergence can be extended to all positive values, such
convergence is, in general, only uniform on bounded sets of ∆. Thus, even if a power series converges for all
∆ ∈ [0, +∞), the convergence may be increasingly slow for larger and larger values of ∆. Calculation of a
large number of terms in the series may be needed before a good approximation to the solution f (∆, x) is
obtained when ∆ is large, and this may severely limit the practicality of the series method of solution. By
contrast, if a series converges uniformly on ∆ ∈ [0, +∞), then the rate of convergence does not depend on
the value of ∆, and f (∆, x) can be approximated for large and small values of ∆ equally well with the same
10 The
only functions for which power series converge uniformly for all ∆ are polynomials, and bond prices in the Vasicek (1977)
model are not polynomials in maturity.
16
number of terms. We now derive several results that can be helpful in deriving a series of approximations that
converge uniformly, for all time horizons, to an asset price or conditional moment.
3.3.1.
Large Circle Convergence with Change of Time
A power series representation of f (∆, x) in ∆ never converges uniformly for all ∆, except in the special cases
in which f (∆, x) is a polynomial in ∆. However, for k > 0, the basic time transformation maps the positive
real line, ∆ ∈ [0, +∞), to a finite interval, τ ∈ [0, 1). A power series converges uniformly on compact sets that
fall within the circle of convergence; thus, if the radius of convergence of a power series for w (τ, x) (the solution
to the transformed problem, (3.5) and (3.6)) in τ can be shown to be greater than 1, then this series converges
uniformly for |τ | ≤ 1, and the circle in τ includes all positive ∆. Uniform convergence on ∆ ∈ [0, +∞), possible
with a power series in ∆ only in very limited special cases, may therefore be possible with a power series in τ ,
since, for positive values of the parameter k of the time transformation, the unbounded interval ∆ ∈ [0, +∞)
maps to the bounded interval τ ∈ [0, 1).
Although uniform convergence on ∆ ∈ [0, +∞) is a powerful and useful result, it is much harder to establish
than simple convergence on the same interval. The implications of Theorem 1 and Corollary 1 are bidirectional.
If the solution f (∆, x) to the original problem, (2.3) and (2.4), can be shown to be analytic in ∆ within a
given region, then it follows that the solution w (τ, x) to the transformed problem, (3.5) and (3.6), is analytic
in a corresponding circle in τ . But the reverse implication also holds; if w (τ, x) is analytic in τ within a
circle, then f (∆, x) is analytic within the region indicated in the theorem or corollary statement. Establishing
analyticity of the solution to either problem within the given region is sufficient to establish convergence of the
power series representation of w (τ, x), and f (∆, x) can then be recovered using (3.3). However, for circles in τ
with radius greater than one, the implication breaks down in one direction. Specifically, although analyticity
of w (τ, x) in a circle in τ still implies analyticity of f (∆, x) within a corresponding region of ∆, the reverse
implication does not hold. The problem is that circles in τ with radius greater than one include the point
τ = 1, which does not correspond to any value of ∆. For k > 0, as ∆ approaches +∞, τk (∆) approaches
one, but never reaches it for finite values of ∆. Consequently, even if f (∆, x) is everywhere analytic in ∆,
the function w (τ, x) may fail to be analytic at τ = 1, and its power series then diverges for all |τ | > 1. The
following theorem is essentially analogous to Theorem 1 and Corollary 1, but for circles with r > 1. However,
with the earlier results, to establish convergence of a power series solution within a circle in τ , it suffices either
to show that w (τ, x) (the solution to the transformed problem) is analytic in τ within a circle, or that f (∆, x)
(the solution to the original problem) is analytic in the corresponding region of ∆. When r > 1, however,
only the first method may be used; convergence can only be shown by establishing analyticity of the solution
to the transformed problem.
Theorem 2. Suppose for some x, some k 6= 0, and some region of ∆, that f (∆, x) can be written as:
f (∆, x) = w (τk (∆) , x)
(3.14)
where w (τ, x) is analytic in the circle |τ | < r for some r > 1. Denote by wn (τ, x) the power series (in τ ) of
17
w (τ, x) including terms up to order n:
wn (τ, x) ≡ b0 (x) +
n
X
bi (x) τ i
i=1
where the bi (x) are the power series coefficients, and define:
fn (∆, x) ≡ wn (τk (∆) , x)
(3.15)
Then holding x fixed, fn (∆, x) converges to f (∆, x) for all complex ∆ such that:
³
´
p
Re (k∆) > − ln cos [Im (k∆)] + cos2 [Im (k∆)] + r2 − 1
(3.16)
Furthermore, for any 1 ≤ s < r, fn (∆, x) converges uniformly to f (∆, x) for all complex ∆ such that:
³
´
p
Re (k∆) ≥ − ln cos [Im (k∆)] + cos2 [Im (k∆)] + s2 − 1
Proof: See appendix.
Theorem 2 provides a means of constructing a series of approximations to f (∆, x) that converges uniformly
on ∆ ∈ [0, +∞). The solution to the asset pricing or conditional moment problem, f (∆, x), is first expressed
in terms of w (τ, x), using the time transformation. If w (τ, x) can be shown to be analytic in a circle that
includes τ = 1 (in Section 5, we use the results of Kimmel (2008), but other methods may be feasible as well),
then it is possible to construct approximations to f (∆, x) that converge uniformly for all positive ∆. Each
wn (τ, x) is the power series of w (τ, x), truncated after n + 1 terms. From these approximations to w (τ, x),
approximations to f (∆, x) can be constructed through (3.15). The theorem establishes that, given analyticity
of w (τ, x) in a sufficiently large circle of τ , the approximations to f (∆, x) converge uniformly in the indicated
region. For real ∆ and k > 0, this region simplifies to ∆ ∈ [− ln (1 + s) , +∞). Since s > 0, this interval
includes ∆ ∈ [0, +∞), so for most applications, it suffices to show that w (τ, x) is analytic in |τ | < r for any
r > 1; increasing the value of r does not result in improved convergence properties on ∆ ∈ [0, +∞). However,
as discussed below, it is impossible to apply Theorem 2 at all for some problems, and even when applicable,
there are strong restrictions on the choice of the parameter k in the time transformation.
The region described by (3.16) is quite different from that described in either Theorem 1 or Corollary 1;
Figure 6 shows this region for circles of various radii with k = 0.15. All three circles (in τ ) shown have radius
greater than one, but for the goal of establishing uniform convergence on ∆ ∈ [0, +∞), it suffices to show
analyticity of w (τ, x) in τ within any circle with r > 1; clearly, the smaller this radius is (while still being
greater than one), the easier this task tends to be. Every point to the right of the curves shown in Figure 6
falls within the circle |τ | < r for the corresponding value of r, and every point to the left falls outside the
same circle. However, as previously noted, the point τ = 1, which falls within any circle with radius greater
than one, does not correspond to any value of ∆, regardless of the value of the parameter k. Consequently,
analyticity of f (∆, x) everywhere to the right of the curves shown in Figure 6, while necessary for analyticity
of w (τ, x) within the corresponding circle in τ , is not sufficient; w (τ, x) may still fail to be analytic at τ = 1.
The figure also shows the points of singularity of bond prices in the model of Cox, Ingersoll, and Ross (1985);
18
as shown, all such singularities are outside the r = 1.01 circle. Since the r = 1 circle is included inside the
r = 1.01 circle, this establishes convergence of a power series (in τ ) on ∆ ∈ [0, +∞). However, inspection of
the bond price function (which is known explicitly) shows that there is no choice of k at all that makes this
function analytic at τ = 1, corresponding to the limiting value of τ as ∆ approaches +∞. It might seem that
a series approximation method that results in uniform convergence for all maturities is not feasible for this
model, but, as is shown in Section 4, a modified version of Theorem 2 (see Theorem 3 below) may be used to
establish the desired result.
Another way to see that analyticity of f (∆, x) in ∆, even everywhere, does not necessarily mean analyticity
within |τ | < r for r > 1, is to note that Theorem 2 produces f (∆, x) functions that are periodic in ∆. To find
a w (τ, x) that generates a given f (∆, x) using (3.14), the choice of k in the time transformation is severely
restricted by the periodicity of f (∆, x); furthermore, if f (∆, x) is not periodic, then there is no value of k at
all for which an appropriate w (τ, x) can be found. In the case of r ≤ 1, convergence of a power series (in τ )
follows from the analyticity of the function being represented within the region of ∆ corresponding to |τ | < r.
In the r > 1 case, however, uniform convergence does not follow from analyticity of the function at all points
to the right of the circles (in τ ) shown in Figure 6, because the point τ = 1 does not correspond to any value
of ∆.
3.3.2.
Examples
Consider the example of an Ornstein-Uhlenbeck process. The expected value of such a process satisfies (2.1)
with coefficients µ (x) = −κx, σ (x) = 1, and r (x) = 0, and with final condition ψ (x) = x. If κ > 0, the
process is stationary, and its expected value converges to a limit as the time horizon approaches positive
infinity. The solution is known explicitly, and is f (∆, x) = x exp (−κ∆), which is everywhere analytic in ∆.
It can be expressed in terms of τ instead of ∆:
κ
w (τ, x) = f (∆k (τ ) , x) = x (1 − τ ) k
This solution is analytic in τ within a circle |τ | < r for some r > 1 (in fact, for any r > 1) provided κ is a
positive integer multiple of k. A power series in τ then converges uniformly for all |τ | ≤ 1, which includes the
interval ∆ ∈ [0, +∞) (provided κ, and therefore k, are positive). Note, however, that the function w (τ, x) is
not analytic in τ at τ = 1 for any other values of k. For such values, a power series in τ converges within the
circle |τ | < 1, corresponding to ∆ ∈ [0, +∞), but this convergence is not uniform. A series representation of
the solution that converges for all ∆ ∈ [0, +∞) is therefore possible, as long as κ > 0. However, note that
the situation is very different from the case of small circle convergence. Increasing the value of k in the time
transformation never decreases the range of convergence, although it may fail to increase it. Here, increasing
k can actually cause the convergence properties of the series to get worse; uniform convergence only happens
for specific values of k, specifically, those which are κ divided by a positive integer. The largest such value is
κ itself, and increasing k to larger positive values causes the convergence of a series on ∆ ∈ [0, +∞) to cease
to be uniform. In the large circle case, k which is too large can be just as bad as k which is too small.
¡
¢
Now, consider the same general PDE, but with final condition ψ (x) = exp −cx2 . The solution (which is
19
the conditional expectation of the final condition) is given by:
³
´
2
exp(−2κ∆)
exp − 1+cx
c
(1−exp(−2κ∆))
κ
f (∆, x) = p
1 + κc (1 − exp (−2κ∆))
If we choose k = 2κ/n for any integer n > 0, then this solution can be expressed as:
³
´
cx2 (1−τ )n
exp
−
n
c
¡
¢
1+ κ [1−(1−τ ) ]
w (τ, x) = f ∆2κ/n (τ ) , x = q
n
c
1 + κ [1 − (1 − τ ) ]
For the specific case n = 1, the above expression simplifies to:
³
´
2
)
exp − cx1+(1−τ
c
κτ
p
w (τ, x) = f (∆2κ (τ ) , x) =
1 + κc τ
This function has a singularity at τ = −κ/c. If |c| < |κ|, then the singularity lies outside the unit circle in τ ,
and the conditions of Theorem 2 are satisfied for some r > 1. For c > 0, the series of approximations to f (∆, x)
constructed using (3.15) converges uniformly on ∆ ∈ [0, +∞), provided the tails of the final condition do not
go to zero too quickly (i. e., provided |c| < |κ|). However, in the case |c| > |κ|, the singularity at τ = −κ/c lies
within the unit circle in τ , and the conditions of Theorem 2 are not satisfied for k = 2κ. Choosing k = 2κ/n
for some n > 1 does not improve the situation. Consequently, for c > 0, if the tails of the final condition go
to zero too quickly, the conditions of Theorem 2 cannot be satisfied for any value of k at all, and the basic
time transformation cannot establish uniform convergence on ∆ ∈ [0, +∞) (although the weaker conditions
of Corollary 1 do establish non-uniform convergence on this interval). In the small circle case, provided there
is no singularity near the positive real line, larger values of k always increase the range of convergence, until
that range includes ∆ ∈ [0, +∞). Here, the situation is very different; a choice of k which is too large is just
as bad as one which is too small, in that it fails to establish uniform convergence, and even when there are no
singularities near the positive real line, it may be impossible to establish uniform convergence with any choice
of k at all.
The qualitative aspects of Theorem 2 may be summarized as:
Remark 4. The same time transformation (3.1) that can extend the range of convergence of a power series
solution to (2.3) and (2.4), can also sometimes establish uniform convergence of the power series on ∆ ∈
[0, +∞), in which case the solution to the pricing or conditional moment problem can be approximated equally
accurately for all positive ∆. However:
1. Analyticity of the solution f (∆, x), even for all ∆, is not sufficient for uniform convergence of the power
series on ∆ ∈ [0, +∞), for any value of the parameter k.
2. If the solution w (τ, x) to the transformed problem, (3.5) and (3.6), is analytic in τ within a circle |τ | < r
for any combination of k > 0 and r > 1, then the power series of the solution to the transformed problem
converges uniformly on |τ | ≤ 1. The solution f (∆, x) to the original problem, (2.3) and (2.4), can then be
recovered from w (τ, x) using (3.3), and the power series effectively converges uniformly on ∆ ∈ [0, +∞).
20
3. The transformed problem can only be analytic in a circle |τ | < r with r > 1 for specific values of k > 0;
it may be that no value of k at all suffices. Analyticity for all such |τ | < 1 is always preserved when k is
increased to a larger positive value, but in general, analyticity at τ = 1 is then lost.
Increasing the value of k > 0 in the time transformation never decreases the range of convergence of a power
series, but can decrease the rate of convergence; since the series converges uniformly only for specific values
of k, increasing k can cause uniform convergence to become non-uniform.
3.3.3.
Large Circle Convergence with Change of Time and State
To apply Theorem 2, analyticity of the solution to the transformed problem, (3.5) and (3.6), must be established. However, this task is difficult; for example, the results of Kimmel (2008) do not apply (at least not
directly) to such time-inhomogeneous problems. We therefore often find it more convenient to work with a
modified version of Theorem 2, which is based on a change of both time and state variables, as well as the dependent variable; it is sometimes possible to use the change of state variable to eliminate the time-dependency
introduced by the change of time variable. We find it convenient, in this case, to focus only on scalar problems.
We also assume the PDE is already in the canonical form (2.8) and (2.9); as previously discussed, this results
in no loss of generality. We then express the solution of the original problem as:
h (∆, y) = ν (∆, y) w (τ (∆) , z (∆, y))
(3.17)
If w (τ, z) is analytic in τ , Theorem 2 can still be applied. However, it establishes uniform convergence in ∆
of approximations to w (τ (∆) , z) holding z fixed, rather than what is desired, which is uniform convergence
of h (∆, y) holding y fixed; furthermore, it is harder to establish the conditions of the theorem, because of the
time-homogeneity of the coefficients of the PDE solved by w (τ, z). We therefore consider a modified version
of Theorem 2 that considers convergence of a series of approximations to w (τ, z), with y held fixed instead of
z. The particular transformations used are:
ν (∆, y) = eλ∆ ξ (y)
i
√ h
k∆
z (∆, y) = k θ + e 2 (y − θ)
for arbitrary numbers λ and θ, and an arbitrary function ξ (y). By substituting the expression for h (∆, y)
from (3.17) into the canonical PDE, a different PDE and final condition, satisfied by w (τ, z), can be derived.
In general, this alternate PDE does not admit straightforward analysis. However, as we see in Section 4, in
the specific cases where rh (y) is specified by either (2.10) or (2.11), it is possible to choose k, λ, θ, and ξ (y) so
that the PDE satisfied by w (τ, z) is quite simple, and is covered by the results of Kimmel (2008). This may
not seem particularly noteworthy, since, for these two specifications of rh (y), the original problems are also
covered by the results of Kimmel (2008). However, there is an important difference between the two situations.
For the original problem, these results apply to h (∆, y) with time variable ∆ and state variable y; if h (∆, y) is
everywhere analytic, then a power series converges on the interval ∆ ∈ [0, +∞), but convergence is in general
not uniform. However, in the transformed problem, the results of Kimmel (2008) apply to w (τ, z), with τ as
21
the time variable and z as the state variable. Provided k > 0, the interval ∆ ∈ [0, +∞) maps to τ ∈ [0, 1),
and uniform convergence of a power series in τ on this interval is often possible.
The following theorem states that if w (τ, z) is analytic in a sufficiently large region, then, with some
restrictions on k and λ, it is possible to construct a series of approximations to h (∆, y), the solution to the
original problem (2.8) and (2.9), that converge uniformly for all ∆. The problem of establishing that w (τ, z)
has the sufficient properties to allow application of the theorem is examined in Section 4.
Theorem 3. Suppose the solution h (∆, y) to a conditional moment or asset pricing can be written as:
³
√ £
¡
¢
¤´
h (∆, y) = eλ∆ ξ (y) w τk (∆) , k θ + 1 − τk/2 (∆) (y − θ)
for some k 6= 0, arbitrary numbers λ and θ, and an arbitrary function ξ (y), where w (τ, z) is an analytic
function of both variables for all z and for all |τ | < r for some r > 1. Denote by wn (τ, z) the power series
approximation (in τ ) to w (τ, z) including terms up to order n in τ :
wn (τ, z) ≡ b0 (z) +
n
X
bi (z) τ i
i=1
where the bi (z) are the power series coefficients. Define:
³
√ £
¡
¢
¤´
hn (∆, y) ≡ eλ∆ ξ (y) wn τk (∆) , k θ + 1 − τk/2 (∆) (y − θ)
Then for a fixed value of y, hn (∆, y) converges to h (∆, y) for all complex ∆ such that:
³
´
p
Re (k∆) > − ln cos [Im (k∆)] + cos2 [Im (k∆)] + r2 − 1
(3.18)
Furthermore, for any 1 ≤ s < r and for any real c, hn (∆, y) converges uniformly to h (∆, y) for all complex
∆ such that:
³
´
p
Re (k∆) ≥ − ln cos [Im (k∆)] + cos2 [Im (k∆)] + s2 − 1
and
Re (λ∆) ≤ c
(3.19)
Proof: See appendix.
The region of ∆ for which hn (∆, y) converges, for a given w (τ, z), is the same as in Theorem 2. However,
this theorem may fail to establish uniform convergence in regions where Theorem 2 does. For example,
suppose k is positive. If w (τ, z) is analytic in a circle |τ | < r for some r > 1, then Theorem 2 establishes
uniform convergence on ∆ ∈ [0, +∞). However, Theorem 3 does not establish uniform convergence for
these values if λ is positive (or more generally, if the real part of λ is positive); in this cases, the uniformly
convergent approximations wn (τ, z) are premultiplied by a quantity that grows without bond as ∆ approaches
positive infinity. The theorem still establishes uniform convergence on ∆ ∈ [0, T ] for any T > 0, but uniform
convergence on the entire positive real axis remains elusive. However, if the real part of λ is negative (or zero),
uniform convergence on ∆ ∈ [0, +∞) follows by choosing any c ≥ 0.
Theorem 3 is particularly useful for several of the cases considered in Section 4. In those cases, a change
of time variables introduces time-inhomogeneity to the coefficients of the pricing partial differential equation.
22
But, a corresponding change of the state vector restores time-homogeneity once again. The results of Kimmel
(2008) can be used to establish analyticity of the solution to the transformed problem, and Theorem 3 can then
be applied to establish uniform convergence of a power series in τ on an interval that includes ∆ ∈ [0, +∞),
taking into account the change of both the time and the state variables. Note that this result can be used
even in some non-stationary models; for example, uniform convergence of series for bond prices for all positive
maturities can be established in the model of Ahn, Dittmar, and Gallant (2002), even when the driving
diffusion process is non-stationary. In this case, bond prices are already known explicitly, but the results of
this section also apply to a large class of non-linear models for which bond prices are not known explicitly.
The next section characterizes in detail two large families of conditional moment and contingent claim pricing
problems to which the results of this section can easily be applied.
4.
Applying the Time Transformation
The previous section develops tools to improve the convergence properties of power series representations of
solutions to general diffusion problems. Specifically, it shows how to improve both the range of convergence
of a series solution to a conditional moment or asset pricing problem, and also how to extend the rate
of convergence, by making it uniform for all positive time horizons, when appropriate conditions are met.
However, to apply these results in practice, it is necessary to show that the solution to either the original
problem (2.8) and (2.9), or a transformed problem that arises from change of the time variable (and possibly
also the state and dependent variables) has an analytic solution; furthermore, since the solution is unknown
(which is the whole reason for constructing a series representation), this determination must be made with
knowledge only of the coefficients of the PDE problem and the final condition. In this section, we consider
the two broad families of such problems discussed in Kimmel (2008), and show how the results of the previous
section may be applied to these problems. Both of these families of problems involve scalar diffusions. We
assume these problems have already been transformed into the canonical form (2.8) and (2.9); as previously
discussed, this results in no loss of generality.
4.1.
Brownian Motion
The first case we consider is the one Kimmel (2008) refers to as the “Brownian motion” case. This terminology
is used because the canonical form of the PDE to be solved is one that arises when a state variable follows a
Brownian motion. Note, however, that many non-linear problems also give rise to the same canonical PDE,
so this terminology should not be interpreted to mean that the results apply only to problems based on a
Brownian motion. This PDE is (2.8) and (2.9) with rh (y) specified by (2.10):
· 2
¸
∂h
1 ∂2h
b
2
(∆, y) =
(∆, y) −
(y − a) + d h (∆, y)
∂∆
2 ∂y 2
2
h (0, y) = g (y)
(4.1)
(4.2)
23
Kimmel (2008) expresses the region of analyticity of h (∆, y), given smoothness and growth conditions on
g (y). However, as discussed in the previous section, even if h (∆, y) were to be everywhere analytic, this
still would not guarantee uniform convergence of a power series on ∆ ∈ [0, +∞). We therefore consider the
problem of establishing analyticity of w (τ, z), the function in Theorem 3 from which h (∆, y) is constructed,
for an appropriate choice of k > 0, λ, θ, and ξ (y). If analyticity within |τ | < r can be established for some
r > 1 (with the additional restriction on λ), then Theorem 3 applies, and uniform convergence for all positive
∆ can be established. The following theorem provides conditions under which such analyticity in w (τ, z) can
be established.
Theorem 4. Let b 6= 0, a, and d be arbitrary numbers, and for
√
2b, choose either square root. Let g (y) be
analytic for all complex y, and let there exist some c > 0 and some norm (over the reals) kyk such that g (y)
satisfies the bound:
¯
µ
¶¯
¯ y2
¯
kyk2
¯e 4 g √y
¯ ≤ ce 2
¯
¯
2b
√
Then there exists a w (τ, z), defined and analytic for all complex z and k τ k < 1, that satisfies the partial
differential equation with final condition:
∂w
1 ∂2w
(τ, z) =
(τ, z)
∂τ
2 ∂z 2
µ
¶
√
(z−a 2b)2
z
4
g √
w (0, z) = e
2b
Furthermore, h (∆, y), defined by:
b
2
h (∆, y) ≡ e− 2 (y−a)
−( 2b +d)∆
³
√ £
¤´
w τ2b (∆) , 2b a + e−b∆ (y − a)
°p
°
°
°
satisfies (4.1) and (4.2) for all complex y and ∆ such that ° τ2b (∆)° < 1.
Proof: See appendix.
Theorem 3 takes as given a w (τ, z) that is analytic in τ , and shows how to use the power series of this
function to construct a series of convergent approximations to another function, h (∆, y). Theorem 4 does
three things. First, given smoothness and growth conditions on g (y), it shows the existence of a w (τ, z) that
satisfies the conditions of Theorem 3, with:
k = 2b
b
λ=− −d
2
θ=a
b
2
ξ (y) = e− 2 (y−a)
Second, it establishes that w (τ, z) solves a PDE with final condition, so that, since w (τ, z) is analytic for
√
k τ k < 1, its power series can be found by the recursive method of (2.6) and (2.7) (with the appropriate
changes in notation). Finally, it establishes that the h (∆, y) constructed in Theorem 3 is the solution to
√
the problem (4.1) and (4.2). If b is positive and the positive square root 2b is chosen, and the region
of analyticity of w (τ, z) established by Theorem 4 includes |τ | ≤ 1, then Theorem 3 applies and, with an
additional restriction on b and d, establishes uniform convergence of a series on ∆ ∈ [0, +∞). Since the power
series of w (τ, z) converges within a circle, Theorem 4 is best applied with a norm of the type kyk = |y| /k0 .
24
However, the result is no harder to derive with a general norm, and the more general result can sometimes be
useful.11 For this reason, we express the result in terms of any norm over the reals, although in all examples
considered in Section 5, a circular norm suffices.
We further note that the PDE (4.1) and (4.2) is unchanged if b is replaced by −b. The final condition
for the PDE satisfied by the solution w (τ, z) to the transformed problem is then potentially complex, even
when the final condition in the original problem is a real function. However, this presents no problems; the
change of variables that construct h (∆, y) from w (τ, z) ensure that the latter is a real function (provided the
final condition is a real function), for either choice. The theorem therefore establishes two different ways to
construct a solution to (4.1) and (4.2). Although the two solutions coincide (where they are both defined),
they are based on different functions w (τ, z), which may have different regions of analyticity (in τ ). Even if
the region of analyticity in τ were the same, these regions would map to different regions of ∆, because the
parameter of the basic time transformation is k = 2b for one and k = −2b for the other. Since we are usually
interested in problems in which the coefficients of the PDE are real, b will be either real or imaginary; if it is
real, then the preferred choice of k is normally the positive value, which helps to establish uniform convergence
for ∆ ∈ [0, +∞). The negative choice can help to establish uniform convergence on ∆ ∈ (−∞, 0], but this is
of no interest for practical applications.
Many problems involving conditional moments or bond pricing with a mean-reverting process are covered
by Theorem 4 (after a change of variables), although other cases are covered as well. Conditional moments
and bond prices under the term structure models of Vasicek (1977) and Ahn, Dittmar, and Gallant (2002),
after a change of variables that puts the appropriate PDE in the canonical form, are covered by this case.
For both models, conditional moments of the state variable are everywhere analytic in the time horizon; this
follows by applying the results of Kimmel (2008) to the conditional moment sought. The final conditions
are sufficiently smooth, and grow at a polynomial rate, more slowly than any exponential function, which
is sufficient to guarantee existence of an everywhere analytic moment. If the state variable process is also
stationary, Theorem 4 establishes uniform convergence for all ∆ ∈ [0, +∞). Bond prices in the Vasicek (1977)
model are everywhere analytic, so a series in ∆ converges for all maturities, whether or not the interest rate
process is stationary. However, this convergence is not uniform. Provided the interest rate process is stationary,
Theorem 4 establishes uniform convergence of series representations of bond prices for all maturities.12 For
the scalar version of the model of Ahn, Dittmar, and Gallant (2002), considered in detail in Section 5, series
representations of bond prices directly in maturity converge only for a limited range of maturities. However,
Theorem 4 establishes that a series in τ instead of ∆ converges uniformly for all maturities, even if the state
variable process is not stationary. Of course, the value of Theorem 4 is not to establish a method of finding
convergent series for bond prices in models for which they are already known explicitly, but to find bond prices,
11 If
w (τ, z) has singularities within |τ | < 1, then its power series does not converge within any circle that includes τ = 1, and
consequently, the approximations to h (∆, y) do not converge on ∆ ∈ [0, +∞). However, provided the singularities are away from
the positive real axis in τ , the time transformation can be applied again, and the results of Section 3.2 can extend the region of
convergence to τ [0, 1]. This uniformly convergent series of approximations to w (τ, z) could then be used to construct a uniformly
convergent series of approximations to h (∆, y) on ∆ ∈ [0, +∞), with a slightly modified version of Theorem 3. However, we do
not formally state and prove such a result.
12 These results are not shown, but can be found in a manner similar to the analysis of other models in Section 5.
25
other contingent claim prices, or conditional moments in the wide variety of problems, many non-linear, whose
canonical form is given by (4.1) and (4.2). For many such problems, the solutions are not known in closed-form,
but can be approximated accurately by our methods.
We finally note that Theorem 4 can be extended; if the conditions hold for any norm, then the PDE
solution is analytic for all values of ∆. However, uniform convergence of a power series could still only be
established in certain directions in the complex plane. We do not formally state and prove such a result.
4.2.
General Affine
The other case we consider is (2.8) and (2.9) with rh (y) specified by (2.11):
µ
∂h
1 ∂2h
(∆, y) =
(∆, y) −
∂∆
2 ∂y 2
¶
a
b2 2
+
y
+
d
h (∆, y)
y2
2
(4.3)
h (0, y) = g (y)
(4.4)
If a = 0, this PDE is a special case of the PDE (4.1). As in that case, the convergence properties of the
power series for h (∆, y) can be considerably improved if a time transformation is applied first. Several term
structure models that have appeared in the literature reduce to the general affine case after changes of variables.
Conditional moments and bond prices in an affine model (specifically, one in which the state variable follows
the square-root process of Feller (1951), as in Cox, Ingersoll, and Ross (1985)) are covered by this case; for
this reason, Kimmel (2008) refers to this version of the canonical PDE as the “general affine” case. However,
many non-affine pricing or conditional moment problems also reduce (when expressed in the canonical form)
to this case, so the results of this section are not limited to affine problems.
As with the Brownian motion case, Kimmel (2008) expresses the region of analyticity of h (∆, y) in terms
of the smoothness and growth properties of g (y), although the class of final conditions that generate analytic
solutions is more complicated to characterize. Specifically, the final condition must be of the form:
g (y) = g1 (y) y
√
1− 1+8a
2
+ g2 (y) y
1+
√
1+8a
2
(4.5)
where g1 (y) and g2 (y) are everywhere analytic and even functions. In the case where
√
1 + 8a/2 ∈ N, the
alternate final condition:
g (y) = g1 (y) y
may be used instead; in this case,
√
√
1− 1+8a
2
+ g2 (y) y
1+
√
1+8a
2
ln y
(4.6)
1 + 8a should be interpreted as the positive square root.13 But also as
with the Brownian motion case, analyticity of h (∆, y), even everywhere in ∆, does not establish uniform
convergence on ∆ ∈ [0, +∞). The following theorem establishes analyticity of w (τ, z), the auxiliary function
from Theorem 3, used to construct the solution h (∆, y):
13 As
noted in Section 2, Kimmel (2008) requires the alternate final condition in the case where
√
1 + 8a/2 is an integer.
However, the final condition (4.5) can still be used, as the g2 (y) part of the final condition is redundant in this case, and could
be incorporated into the g1 (y) part of the final condition instead.
26
Theorem 5. Let b 6= 0, a, and d be arbitrary numbers, and for
√
2b, choose either square root. Let g1 (y)
and g2 (y) be even functions that are analytic for all complex y, and let there exist some c > 0 and some norm
(over the reals) kyk such that g1 (y) and g2 (y) satisfy the bounds:
¯
¯
µ
¶¯
µ
¶¯
¯ y2
¯
¯
¯ y2
kyk2
kyk2
¯e 4 g1 √y
¯ ≤ ce 2
¯e 4 g2 √y
¯ ≤ ce 2
¯
¯
¯
¯
2b
2b
√
Then there exist w1 (τ, z) and w2 (τ, z), analytic for all complex z and k τ k < 1, that satisfy the partial
differential equations with final conditions:
√
∂w1
1 − 1 + 8a ∂w1
1 ∂ 2 w1
(τ, z) =
(τ, z) +
(τ, z)
∂τ
∂z
2 ∂z 2
√2z
1 + 1 + 8a ∂w2
1 ∂ 2 w2
∂w2
(τ, z) =
(τ, z) +
(τ, z)
∂τ
2z
2 ∂z 2
µ
¶∂z
µ
¶
z2
z2
z
z
w1 (0, z) = e 4 g1 √
w2 (0, z) = e 4 g2 √
2b
2b
Furthermore, h (∆, y), defined by:
h (∆, y) ≡ e
(
− 2b y 2 −
b
2 +d


µ
¶ 1−√21+8a
µ
¶ 1+√21+8a
z
z
)∆  √
w1 (τ, z) + √
w2 (τ, z)
2b
2b
√ −b∆
2be
y, satisfies (4.3) and (4.4), with g (y) specified by (4.5), for all complex y
°p
°
°
°
and ∆ such that y =
6 0 and ° τ2b (∆)° < 1.
where τ = τ2b (∆) and z =
Proof: See appendix.
This theorem is in many ways similar to Theorem 4, but is also somewhat more complicated, in that the
final condition is expressed in terms of two everywhere analytic functions, and the growth condition is applied
to each. Like Theorem 4, it establishes existence of analytic (in τ ) auxiliary functions, and specifies the PDE
(with final condition) satisfied by these functions; their power series can therefore be found by the recursive
method of (2.6) and (2.7). It allows application of Theorem 3, with:
µ
k = 2b
λ = −b 1 ±
√
1 + 8a
2
¶
−d
θ=0
b
2
ξ (y) = e− 2 y y
1±
√
1+8a
2
To apply Theorem 3 to the w1 (τ, z) part of the solution, the minus signs should be chosen in λ and ξ (y);
to apply the theorem to the w2 (τ, z) part of the solution, the plus signs should be chosen. The theorem
further shows that the h (∆, y) constructed in Theorem 3 is the solution to the problem (4.3) and (4.4),
with g (y) specified by (4.5). If b is positive, and the conditions of the theorem are satisfied for the norm
√
kyk = |y| /k0 for any k0 > 1 (choosing the positive square root 2b), then the power series for w (τ, z)
converges uniformly (holding z fixed) for the values of τ that include ∆ ∈ [0, +∞). With some additional
parameter restrictions, Theorem 3 establishes uniform convergence of the series of approximations of h (∆, y),
constructed by truncating the power series of w1 (τ, z) and w2 (τ, z) and applying the definition of h (∆, y)
in the theorem statement, on ∆ ∈ [0, +∞), holding y fixed. As with Theorem 4, circular norms, of the
form kyk = |y| /k0 , are the most useful, since the power series of w1 (τ, z) and w2 (τ, z) converge within a
27
circle. However, in some cases, non-circular norms could still be useful, since a second application of the
time transform (with a slight modification of Theorem 3) would allow uniform approximation of w1 (τ, z) and
√
w2 (τ, z) on τ ∈ [0, 1] whenever k1k < 1, even if the region k τ k < 1 does not include |τ | ≤ 1. (See the brief
discussion in the previous section.) We do not formally state and prove such a result.
As with Theorem 4, it is possible to replace b with −b; the PDE satisfied by h (∆, y), (4.3) and (4.4), is
then unaltered. The final condition satisfied by the transformed problem may well then be complex, even if
the original final condition is a real function. However, this is not a problem since, if the final condition is
a real function, the changes of variables that construct h (∆, y) from w1 (τ, z) and w2 (τ, z) ensure that the
former is a real function, even if the latter are not. Provided the quadratic coefficient in (4.3) (i. e., b2 /2)
is positive, then b is real, and can be chosen to be either positive or negative. The positive choice is most
useful for establishing uniform convergence on ∆ ∈ [0, +∞); the negative choice may help to establish uniform
convergence on ∆ ∈ (−∞, 0], but this is not a particularly useful result for most real world problems.
√
We also note that it is possible to modify Theorem 5 to use the alternate final condition (4.6) in the case
1 + 8a/2 ∈ N. This requires a slight modification to the construction of h (∆, y) from w1 (τ, z) and w2 (τ, z),
but also to the partial differential equation solved by w1 (τ, z); this equation is no longer homogeneous, but
includes a term on w2 (τ, z) and a term on its first derivative. The recursive procedure to calculate the
coefficients is still feasible, but most be modified for this inhomogeneous case. We do not consider this
possibility further.
As with Theorem 4, many conditional moment or bond pricing problems for models that have appeared
in the literature, including (but not limited to) those with a mean-reverting state process, are covered by
Theorem 5. In Section 5, we apply this theorem and Theorem 3 to bond prices under the model of Cox,
Ingersoll, and Ross (1985), and find that series using the time transformation converge uniformly and rapidly
for all positive maturities. We also apply these results to the callable bond pricing model of Jarrow, Li, Liu,
and Wu (2006) (for which prices are not known in closed-form), and find that the series representation of bond
prices in this model also converges uniformly for all positive maturities.
5.
Examples
We now consider several examples, and show how to construct approximations of solutions to several asset
pricing problems, using time transformations to extend the range and increase the rate of convergence. For
two of the problems considered, in Sections 5.1 and 5.2, the quantity sought is already known explicitly, so
these cases serve only as illustrative examples, to test the accuracy of approximations based on power series,
and to see how much the accuracy is improved by the use of time transformations. For both of these cases,
which are bond pricing models, we show that naı̈ve power series (i. e., those constructed directly in the time
variable ∆, without using a time transformation) diverge for maturities beyond a certain value; however, time
transformation methods extend the range of convergence to include all maturities. Furthermore, we show that
approximations based on power series using time transformations are much more accurate than those based
on naı̈ve power series, for an extremely wide range of maturities and values of the state variable, virtually
28
certain to include any values needed for a real world application.
For the other examples considered, including the callable bond pricing problem of Jarrow, Li, Liu, and Wu
(2006), the solution sought is not known in closed-form, so we do not compare approximate solutions based
on power series to the true solution. For these cases, we show how to derive a series representation of the
solution, and to determine its convergence properties.
5.1.
ADG Model
We now construct approximations to bond prices under the interest rate model of Ahn, Dittmar, and Gallant
(2002). Since bond prices in this model are known in closed-form, this model allows us to determine the
accuracy of approximations constructed using our methods.
The risk-neutral interest rate process under the scalar version of the model of Ahn, Dittmar, and Gallant
(2002) is given by:14
dxt = κ (θ − xt ) dt + σdWt
rt = x2 + φ
where Wt is a Brownian motion. Zero-coupon bond prices f (∆, r) then satisfy the PDE with final condition:
¡
¢
∂f
∂f
σ2 ∂ 2 f
(∆, x) = κ (θ − x)
(∆, x) +
(∆, x) − x2 + φ f (∆, x)
∂∆
∂x
2 ∂x2
f (0, x) = 1
This PDE can be converted to the canonical form by the change of dependent and independent variables
discussed in Section 2. With the specific coefficients given by (5.3), these changes of variables are:
y (x) =
x−θ
σ
f (∆, r) = e
κ(x−θ)2
2σ 2
h (∆, y (x))
The canonical form PDE, with final condition, is then:
· 2
¸
∂h
1 ∂2h
b
2
(∆, y) =
(∆,
y)
−
(y
−
a)
+
d
h (∆, y)
∂∆
2 ∂y 2
2
κ
h (0, y) = e− 2 y
2
(5.1)
(5.2)
with:
a≡−
2θσ
b2
b≡
p
κ2 + 2σ 2
d≡
κ
κ2 θ 2
− +φ
b2
2
The quantity inside the square root in the definition of b is clearly positive, and b itself is assigned the positive
14 Our
notation differs somewhat from that used by these authors, and is also slightly less general. Ahn, Dittmar, and Gallant
(2002) allow linear as well as quadratic interest rate specifications, nesting the model of Vasicek (1977), which is precluded by
our parameterization. However, as our purpose here is to provide an illustrative example rather than to construct the most
comprehensive model, this less general parameterization suffices.
29
square root. We now apply Theorem 4, with:
r
2
−κ
2y
g (y) ≡ e
kyk ≡ |y|
1
κ
−
2 2b
Note that the expression inside the square-root in the definition of the norm kyk is necessarily between 0
and 1/2, provided κ is not negative. If κ is negative, this expression could approach 1 arbitrarily closely,
depending on the relative magnitudes of κ and σ, but is always smaller than 1, as long as σ 6= 0. In all cases,
the square-root sign should be interpreted as the positive square root. Theorem 4 establishes the existence
√
of a w (τ, z) that is analytic for all k τ k < 1, and that satisfies the PDE in the theorem statement; since
√
k1k < 1, it follows that the region of analyticity of w (τ, z) established by Theorem 4, k τ k < 1, includes a
circle |τ | < r for some r > 1.
The solution w (τ, z) is known explicitly for this model:
Ã
w (τ, z) =
e
!2
√
z− a 2b
1− κ
a2 κ
b
!−
Ã
2 1− κ
1 −τ
4
b
2
1− κ
b
(
q
1−
τ
2
¡
1−
κ
b
)
¢
However, even if it were not known in closed-form, a power series could be calculated using the recursive
relation of (3.8) and (3.9), since w (τ, z) is known to solve the PDE (with final condition) in Theorem 4. The
first few terms are:


³
√ ´2 
a 2b
´
³
´
z
−
κ
1− b

 1 + τ 1 − κ 1 + 1 − κ





b
4
b
8








Ã
!


µ
¶
√
2
2
√
³
´
1− κ
2


2b
b
a 2b
3
κ
3
z− a
− a κκ 

4
1− κ
2(1− ) 


b
z
−
+
1
−
b
κ
w (τ, z) = e

´2  16 16
2 ³
b
1
−

τ
κ
b


 +
1−


Ã
!

√
4

2
b 
³
´

κ 2
a 2b
 1



+
1−
z−


64
b
1 − κb




+ ...

³
Since k1k < 1, this power series converges within a circle that includes τ ∈ [0, 1].
Theorem 4 also provides a construction of an h (∆, y) that is the solution to (5.1) and (5.2). This solution
can therefore be approximated by first approximating w (τ, z) with a few terms of its power series, and
then applying the construction in the theorem statement to the approximation. Convergence of a series of
approximations to h (∆, y) on ∆ ∈ [0, +∞) is then established by Theorem 3, with only mild restrictions on
the value of φ (φ ≥ 0, precluding negative interest rates, suffices). Since b is positive, the interval τ ∈ [0, 1]
maps to ∆ ∈ [0, +∞), and (given the restriction on φ) Theorem 3 then establishes uniform convergence on
this range. Note that stationarity of the state variable process is not required for this uniform convergence.
Table 2 shows bond yields based on approximations (in τ ) with terms up to order 2 and 4, and also power
series approximations in ∆ (i. e., without using any time transformation methods) with terms up to order
30
8, and compares these approximate yields to the true yields. The parameter values used are κ = 0.4995,
θ = 0.1860, σ = 0.0812 and φ = 0.0088, which are chosen so that the unconditional mean and variance of
the interest rate process are 5% and 0.001, respectively, and the kurtosis coefficient is 5. For these parameter
values, a naı̈ve power series representation of bond prices directly in maturity converges for maturities up to
approximately 8.7 years, and diverges for longer maturities (see Table 1). Since the relation between the state
variable and the interest rate is not one-to-one, Table 2 shows results for four values of the state variable: two
values corresponding to an initial interest rate of 6%, and two values corresponding to an initial interest rate
of 600%. For each initial value of the state variable, bond prices are calculated for an extremely wide range
of maturities, ranging from 1 to 10, 000 years. The accuracy of the approximations based on series in τ is
clear when they are compared to the exact bond prices, and to those based on a naı̈ve power series directly
in ∆. As shown, even the 2-term approximations in τ are extremely accurate over this huge range of interest
rates and maturities. Far and away the largest yield approximation error occurs with maturity of 1 year and
initial interest rate of r = 600% (corresponding to an initial state variable value of x = 2.44769); this error is
slightly more than one tenth of a basis point, when the exact yield is almost 400%. Most approximate yields
are off by less than one hundredth of a basis point, and often by much less than that. In relative terms, the
largest error is less than one part in one hundred thousand of the yield being approximated; for very long
maturities, the error is closer to one part in four hundred million. For the 4-term approximations in τ , the
errors are much smaller, generally tiny fractions of a basis point; in only one case is the relative error slightly
more than one part per billion. By contrast, approximations based on a power series in ∆, even though
they include terms up to order 8 in ∆, are much less accurate. For maturity of one year in the two cases in
which the initial interest rate is 6%, these approximations are accurate to less than a basis point, but are very
inaccurate in the two cases corresponding to r = 600%, producing large negative yields instead of the correct
large positive yields. For longer maturities, the accuracy is poor for all four initial values of the state variable
process; even at maturity of five years, which is within the interval of convergence, the approximation error
is so large as to render the approximations useless in three of the four cases. For longer maturities, series
constructed directly in ∆ diverge, and the table shows extreme inaccuracy in these cases. The method of time
transformation here has both extended the range of maturities for which power series converge, to positive
infinity, and also improved their accuracy, often by many orders of magnitude relative to series derived without
time transformations.
5.2.
CIR Model
We now construct approximations to bond prices under the interest rate model of Cox, Ingersoll, and Ross
(1985). Since bond prices are known in closed-form, this model, like that of the previous section, allows us to
evaluate the accuracy of approximations constructed using time transformations.
The risk-neutral interest rate process is given by:
√
drt = κ (θ − rt ) dt + σ rt dWt
where Wt is a Brownian motion. We require 2κθ ≥ σ 2 so that the interest rate process cannot achieve the
31
boundary value of rt = 0 with positive probability (see Feller (1951)). Zero-coupon bond prices f (∆, r) then
satisfy the PDE with final condition:
∂f
∂f
σ2 r ∂ 2 f
(∆, r) − rf (∆, r)
(∆, r) = κ (θ − r)
(∆, r) +
∂∆
∂r
2 ∂r2
(5.3)
(5.4)
f (0, r) = 1
This PDE can be converted to canonical form by the change of dependent and independent variables discussed
in Section 2 in Kimmel (2008). With the specific coefficients given by (5.3), these changes of variables are:
µ
√
2 r
y (r) =
σ
f (∆, r) =
4r
σ2
¶ 14 − θκ2
σ
rκ
e σ2 h (∆, y (r))
The canonical form PDE, with final condition, is then:
¸
·
∂h
1 ∂2h
a
b2 2
(∆, y) =
y
+
d
h (∆, y)
(∆,
y)
−
+
∂∆
2 ∂y 2
y2
2
κ
h (0, y) = y α e− 4 y
2
(5.5)
(5.6)
with:
2θ2 κ2
2θκ 3
a≡
− 2 +
σ4
σ
8
√
b≡
κ2 + 2σ 2
2
d≡−
θκ2
σ2
α≡
1+
√
1 + 8a
2θκ 1
= 2 −
2
σ
2
The boundary nonattainment condition is needed to establish equality of the two expressions for α. The
quantity inside the square root sign in the expression for b is clearly positive, and b is taken to be the positive
square root. We now apply Theorem 5, with:
"r
g1 (y) ≡ 0
2
−κ
4y
g2 (y) ≡ e
kyk ≡ |y|
#
1
κ
−
+²
2 4b
for some ² > 0. Note that the expression inside the square root sign in the definition of the norm kyk is
necessarily between 0 and 1/2 if κ is not negative; if κ is negative, this value is always less than one, but can
approach one arbitrarily closely if the magnitude of σ is small compared to the magnitude of κ. In either case,
by choosing a sufficiently small ², k1k < 1. Theorem 5 establishes the existence of w1 (τ, z) and w2 (τ, z) that
√
are analytic for all k τ k < 1, and that satisfy the PDEs in the theorem statement; it is easily verified that
w1 (τ, z) is everywhere zero. Since k1k < 1, it follows that the region of analyticity of w2 (τ, z) established by
√
Theorem 5, k τ k < 1, includes a circle |τ | < r for some r > 1.
The solution w2 (τ, z) for this problem is known explicitly:
w2 (τ, z) = ¡
1−
z2
4
−2τ
1− κ
2b
)
e(
¡
¢ ¢α+ 21
1
κ
2 1 − 2b τ
However, even if it were not known in closed-form, a power series could be calculated using the recursive relation
of (3.8) and (3.9), since w2 (τ, z) is known to solve the PDE (with final condition) stated in Theorem 5. The
32
first few terms are:

¶µ
¶
µ
¶2 #
1
κ
z2 1
κ
1
+α
−
+
−
1 + τ

2
2 4b
2 2 4b



 µ

¶µ
¶µ
¶2




1
3
1
κ
2

+α
+α
−
κ
z

2 
2
2
2
4b
τ
w2 (τ, z) = e 4 (1− 2b ) 


 +


µ
¶
µ
¶
µ
¶

3
4
4
2 



3
1
κ
z
1
κ
2


+z
+
α
−
+
−


2
2 4b
4 2 4b


+ ...

"µ
Since k1k < 1, the power series converges within a circle that includes τ ∈ [0, 1]. Theorem 5 also provides a
construction of an h (∆, y) that is the solution to (5.5) and (5.6). This solution can therefore be approximated
by first approximating w2 (τ, z) with a few terms of its power series, and then applying the construction in the
theorem statement to the approximation. Theorem 3 can then be applied; note that b is positive; the interval
τ ∈ [0, 1] then maps to ∆ ∈ [0, +∞). Furthermore, the λ coefficient in the theorem is necessarily negative,
so the theorem establishes uniform convergence of approximations to h (∆, y), based on the power series of
w2 (τ, z), for all positive ∆.
The accuracy of the approximation is clear when it is compared to the exact bond prices, and to those
obtained by a power series approximation of f (∆, r) directly in ∆. Table 3 shows bond yields based on
approximations (in τ ) with terms up to order 2 and 4, and also power series approximations in ∆ (i. e.,
without using any time transformation methods) with terms up to order 8, and compares these approximate
yields to the true yields. As shown, even the 2-term approximations in τ are extremely accurate over a huge
range of interest rates (r = 0.06%, r = 6%, and r = 600%) and maturities (from 1 to 10,000 years). The
largest relative error occurs with maturity of 1 year and initial interest rate of r = 0.06%, and is less than
four parts in one thousand; most errors are much smaller. For the 4-term approximations in τ , the errors are
typically measured in a few thousands of a basis point, or less, providing far greater accuracy then is likely to
be needed in any real application. By contrast, the approximations in ∆, even though they include terms up
to order 8 in ∆, are severely inaccurate for all but the shortest maturities. With the very high initial interest
rate of r = 600%, this approximation is not even accurate for a maturity of one year, the approximation error
being twice the size of the exact yield itself. For the two smaller initial interest rates of r = 6% and r = 0.06%,
the approximation in ∆ is quite accurate for a maturity of one year, but deviates from the true yield by about
40 basis points (in both cases) with a maturity of five years, despite the fact that the series converges (for
these parameter values) for maturities up to approximately 8.24 years. For maturies of 10 years or longer, the
approximations based on seres in ∆ are, for all initial interest rates shown, so inaccurate as to be useless for
any reasonable application, as might be expected, since the series diverges for these maturities. As with the
case of the Ahn, Dittmar, and Gallant (2002) model, the method of time transformation here has not only
extended the range of maturities for which the series converges, but also improved the accuracy of the series
by many orders of magnitude.
33
5.3.
Callable Bonds
Jarrow, Li, Liu, and Wu (2006) consider a model in which the prices of callable bonds satisfy:
³
c2 ´
∂f
∂f
σ2 x ∂ 2 f
(∆, x) = κ (θ − x)
(∆, x) +
(∆,
x)
−
c
x
+
f (∆, x)
1
∂∆
∂x
2 ∂x2
x
f (0, x) = 0
(5.7)
(5.8)
with c1 > 0 and 2θκ ≥ σ 2 . This PDE can be converted to the canonical form by the change of dependent and
independent variables as discussed in Section 2. With the specific coefficients given by (5.7), these changes of
variables are:
y (x) =
√
2 x
σ
µ
f (∆, x) =
4x
σ2
¶ 14 − θκ2
σ
xκ
e σ2 h (∆, y (x))
The canonical form PDE, with final condition, is then:
¸
·
∂h
1 ∂2h
a
b2 2
(∆, y) =
y
+
d
h (∆, y)
(∆,
y)
−
+
∂∆
2 ∂y 2
y2
2
κ
h (0, y) = y α e− 4 y
(5.9)
2
(5.10)
with:
2θ2 κ2
2θκ − 4c2
3
a≡
−
+
σ4
σ2
8
√
b≡
κ2 + 2c1 σ 2
2
d≡−
θκ2
σ2
α≡
2θκ 1
−
σ2
2
The quantity inside the square root sign in the expression for b is positive, and b takes the value of the positive
square root. The results of Kimmel (2008) apply to (5.9) and (5.10) (with a restriction on the values of a and
α), establishing that h (∆, y) is analytic in ∆. However, the region of analyticity (and therefore the interval
of convergence of a power series) is bounded; the power series of h (∆, y) diverges for large ∆. But, using our
results, the range of convergence can be extended to ∆ ∈ [0, +∞), and this convergence can be made uniform.
We now apply Theorem 5, with:
"r
g1 (y) ≡ e
2
−κ
4y
g2 (y) ≡ 0
kyk ≡ |y|
#
1
κ
−
+²
2 4b
for any ² > 0.15 Note that the expression inside the square root sign in the definition of the norm kyk is
necessarily between 0 and 1/2 if κ is not negative, and between 1/2 and 1 (but never equal to 1) if κ is
negative. Either way, with a sufficiently small choice of ² > 0, we have k1k < 1. Theorem 5 establishes the
√
existence of w1 (τ, z) and w2 (τ, z) that are analytic for all k τ k < 1, and that satisfy the PDEs (with final
conditions) in the theorem statement; consequently, their power series can be found using the recursive method
of (3.8) and (3.9). It is clear that w2 (τ, z) is everywhere zero; the region of analyticity of w1 (τ, z) established
√
by Theorem 5, k τ k < 1, includes a circle |τ | < r for some r > 1, because k1k < 1 and the norm is circular.
15 This
choice of g1 (y) and g2 (y) is appropriate for c2 ≥ 0. If c2 < 0, then the definitions of g1 (y) and g2 (y) can be reversed;
this allows application of Theorem 3 with a weaker restriction on the parameters. If the two parts of the final condition are
reversed, then the references to w1 (τ, z) and w2 (τ, z) in the subsequent text should also be reversed.
34
The power series of w1 (τ, z) therefore converges uniformly on a region that includes τ ∈ [0, 1]. The first few
terms of this series are:

µ
¶
¸
(α − γ) (α + γ − 1)
2α + 1 ³
κ ´ z2 ³
κ ´2
+
1−
+
1−
1 + τ

2z 2
4
2b
8
2b








(α − γ) (α − γ − 2) (α + γ − 1) (α + γ − 3)


4




4z



³
´




(2α
−
1)
(α
−
γ)
(α
+
γ
−
1)
κ
µ
¶α−γ

+

1−
z2
κ
z
2


2


1−
4z
2b
w1 (τ, z) = √
e 4 ( 2b )  + τ  µ

¶³
´

2


2b
(2α + 3) (2α + 1) (α − γ) (α + γ − 1)
κ 
2 


+
1−


+
16
8
2b 









κ ´3 z 4 ³
κ ´4
(2α + 3) z 2 ³


1
−
+
1
−
+


16
2b
64
2b


+ ...
·
where:
γ≡
1−
√
1 + 8a
2
Theorem 5 also provides a construction of an h (∆, y) that is the solution to (5.5) and (5.6). This solution
can therefore be approximated by first approximating w1 (τ, z) with a few terms of its power series, and
then applying the construction in the theorem statement to the approximation. Theorem 3, given sufficient
restrictions on α, then establishes uniform convergences of the approximations to h (∆, y) on ∆ ∈ [0, +∞).
Note that b is positive; the interval τ ∈ [0, 1] then maps to ∆ ∈ [0, +∞). As with the previous two models,
the method of time transformation here has not only extended the range of maturities for which the series
converges, but also established uniform convergence on the entire range.
5.4.
Other Models and Applications
For two of the three models considered in the preceding sections, the solution to the pricing problem is known
in closed-form. In all three models, the state variable follows an affine process. However, as noted in Kimmel
(2008), the same canonical form PDEs that arise in affine diffusion problems also arise in may non-linear
problems. Continuing with bond pricing as the motivating example (the task in two of the three preceding
sections), we note that non-linear term structure models in which bonds can be priced using our results are
rather easily constructed. This can be seen simply by reversing the change of variables (in particular, the
change of dependent variable) needed to put a pricing problem in the canonical form. We begin with:
1 ∂2h
∂h
(∆, y) =
(∆, y) − rh (y) h (∆, y)
∂∆
2 ∂y 2
h (0, y) = g (y)
where rh (y) is specified by either (2.10) or (2.11). Suppose that the final condition g (y) satisfies the conditions
of Theorem 4 (when rh (y) is given by (2.10)) or Theorem 5 (when rh (y) is given by (2.11)) for the norm
kyk = |y| (1 − ²) for some 0 < ² < 1, and that b2 > 0. Then by either Theorem 4 or 5, w (τ, z) is analytic for all
35
2
|τ | < 1/ (1 − ²) , and its power series converges uniformly on |τ | < 1. It follows that approximations to h (∆, y)
that converge uniformly on ∆ ∈ [0, +∞) can be constructed, using either Theorem 4 or 5 (depending on the
specification of rh (y)), and also Theorem 3. In the case of rh (y) given by (2.10), g (y) must be everywhere
analytic and satisfy:
¯ b 2
¯
2
2
¯ 2z
¯
¯e g (z)¯ ≤ ceb(1−²) |z|
In the case of rh (y) given by (2.11), it is g1 (y) and g2 (y) that must be everywhere analytic and satisfy the
same growth condition. Provided the additional constraint on the sign of λ (from Theorem 3) is satisfied, the
result is uniform convergence of approximations to h (∆, y) for all ∆.
The class of functions g (y) (or g1 (y) and g2 (y)) that satisfy these conditions is extremely broad. Polynomials, exponential functions, products of the two, linear combinations of the products, etc., all qualify, as do
¡
¢
functions of the form exp ky 2 for k < 1, such exponentials multiplied by polynomials, linear combinations
thereof, etc. Periodic functions such as sin (ky) and cos (ky), these functions multiplied by polynomials, linear
combinations of such functions, etc., also qualify. Some functions that may even seem to be non-analytic
¡ √ ¢
¡ √ ¢
at first glance can also qualify; Kimmel (2008) cites functions of the form exp k y + exp −k y , which,
¡√ ¢ √
despite the appearance of the square root, is analytic in y; functions such as sin y / y are also analytic in
y. Such functions can be multiplied by polynomials, multiplied by each other, added together, etc., to form
final conditions that satisfy the conditions of Theorem 4 or 5. Given such a final condition, the power series
of the corresponding PDE solution can be found, and the series converges for values of τ corresponding to all
∆ ∈ [0, +∞).
For such final conditions, a corresponding term structure model can be reverse engineered, by reversing
the change of dependent variable. Take:
f (∆, y) = h (∆, y) /g (y)
Then f (∆, y):
·
¸
∂f
g 0 (y) ∂f
1 ∂2f
g 00 (y)
(∆, y) =
(∆, y) +
(∆,
y)
−
r
(y)
−
f (∆, y)
h
∂∆
g (y) ∂y
2 ∂y 2
2g (y)
f (0, y) = 1
so that f (∆, y) may be interpreted as the price of a zero-coupon bond in a model driven by a state variable
36
and interest rate process:16
g 0 (Yt )
dt + dWt
g (Yt )
g 00 (Yt )
rt = rh (Yt ) −
2g (Yt )
dYt =
Bond prices in the implied model can be approximated by truncated power series, and the series converge for
all maturities ∆ ∈ [0, +∞). There are two types of choices of g (y) for which the implied state variable process
is affine, or can be converted to an affine process by change of independent variable; those two choices are:
g (y) = ec0 +c1 y+c2 y
2
g (y) = c0 ec1 y y c2
Any other choice of final condition generates a non-affine term structure model, but bond prices can still be
approximated uniformly using our methods in such models.
The above analysis assumes a g (y) (or g1 (y) and g2 (y)) that satisfies the growth condition for a norm of
the form kyk = |y| (1 − ²). However, it is possible to construct term structure models for which bond prices
can be approximated uniformly for all maturities, from an even broader class of final conditions. For a very
¡
¢
simple example, take g (y) = exp −3bc0 y 2 /2 for some c0 ≥ 1. This final condition does not satisfy the growth
condition for any norm of the form kyk = |y| (1 − ²). However, it does satisfy the final condition with other
types of norms, and for some of these norms, k1k < 1. For example, take:
r
kyk ≡
1
4 + ²2
2
2
(Re y) +
(Im y)
1 + ²1
3c0 − 1
where ²1 > 1 and ²2 > 0. Theorems 5 and 3 still apply with this norm, but establish the analyticity of w (τ, z)
√
(the solution to the transformed problem) only within k τ k < 1; this region does not include the circle |τ | ≤ 1,
needed for uniform convergence for all ∆ ∈ [0, +∞). However, the “small circle” results can be applied to the
transformed problem, resulting in a compound time transform. First, the time and state variable are changed
(as per Theorems 5 and 3), so that the solution w (τ, z) can be shown to be analytic in a region that includes
τ = 1; next, a time transformation is applied to the transformed problem, extending the range of convergence,
as discussed in Section 3.2. A power series in the second transformed time variable, then converges on an
interval that includes ∆ ∈ [0, +∞).
Most of the examples discussed so far are the pricing of default-free zero-coupon bonds in term structure
models, although we have also considered the callable bond pricing problem of Jarrow, Li, Liu, and Wu (2006).
Another potential application of time transformation methods is the pricing of credit derivatives, such as credit
default swaps. When calculating prices of such instruments, it is often necessary to calculate quantities such
16 It
is still necessary to establish that the coefficients of the PDE and the final condition are sufficiently regular so that the
probabilistic problem (i. e., finding the risk-neutral expectation of the discounted payoff of the bond) is equivalent to the partial
differential equation problem. These conditions can be verified on a case-by-case basis, using well-known results in the extant
literature.
37
as:
h R t+∆
i
f (∆, x) = E e− t ru +λu du
where the short interest rate rt and the default intensity λt are both functions of the state variable process Xt .
For appropriate rt and λt processes, our methods provide a way of approximating quantities such as f (∆, x),
possibly uniformly for all time horizons, even when the closed-form solution is unknown.
Finally, we note that although some of our methods apply only to single-factor models, many multi-factor
problems can be decomposed into a system of single-factor problems. For example, if multiple state variables
follow independent processes, and enter the interest rate in an additively separable way, then the problem
of pricing a zero-coupon bond in the multi-factor setting is equivalent to the problem of pricing zero-coupon
bonds in several single-factor models. If our methods apply to each of the single-factor models, then the
price of the bond in the multi-factor model can be approximated, perhaps uniformly in all maturities, even if
closed-form bond prices are not known.
6.
Conclusion
We have developed the method of time transformations to improve both the range and the rate of convergence
of power series representations of solutions to asset pricing or conditional moment problems that arise in a
continuous-time setting. In some cases, our methods allow accurate approximation of prices (or conditional
moments) for arbitrarily long time horizons. These methods make feasible the rapid calculation of bond prices
for many models in which such calculation would otherwise not be practical, and therefore make feasible
estimation techniques for non-affine models based on likelihood or minimum distance searches. We use the
term structure models of Ahn, Dittmar, and Gallant (2002) and of Cox, Ingersoll, and Ross (1985), for which
bond prices are already known in closed-form, to evaluate our methods, and find that, although naı̈ve power
series expansion of the bond price function performs very poorly for long maturities, use of time transformations
dramatically improves the accuracy of the series, resulting in uniform converge for all maturities; comparison
to true bond yields shows extreme accuracy with only a few terms in the power series over a very wide range of
initial interest rates and bond maturities. We also use the callable bond pricing model of Jarrow, Li, Liu, and
Wu (2006), in which prices are not known in closed-form, and show that our methods allow approximations
of bond prices in this model also that converge uniformly for all maturities. We also consider other potential
applications, such as pricing of credit derivatives.
Possible future work includes extension of our method to multivariate diffusions. Some multivariate diffusion problems can be broken into independent scalar diffusion problems; for this class of multiple diffusion
problems, no additional work is needed to apply our methods. However, in general, this is not the case; one
state variable may appear in the drift or diffusion coefficient of another state variable, or multiple state variables may not enter the final condition or interest rate function additively. It is possible to construct a large
class of models for which the pricing or conditional moment PDE is the same as the pricing PDE for multiple
affine diffusions. Since expectations of polynomials of affine diffusions are analytic in the time horizon, at least
38
a partial characterization of the final conditions with analytic moments is possible; as in the univariate case,
each final condition with analytic moments corresponds to a term structure model with analytic bond prices.
Such methods remain to be explored in full detail, however.
39
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41
A.
Appendix: Proofs
This appendix includes proofs of all theorems and corollaries in the main text.
A.1.
Proof of Theorem 1
We first show that, for a given value of k 6= 0 and 0 < r < 1, ∆ satisfies (3.11) if and only if:
|τk (∆)| < r
(A.1)
Note that (3.11) is equivalent to:
¡
¢
2
(exp [− Re (k∆)] − cos [Im (k∆)]) < cos2 [Im (k∆)] − 1 − r2
This follows by multiplying (3.11) through by negative one (which reverses the inequalities), exponentiating
and subtracting cos [Im (k∆)] from all three expressions (which preserves the inequalities), and squaring the
results (which replaces two inequalities by one). After some rearrangement, this inequality can be written as:
exp [−2 Re (k∆)] − 2 exp [− Re (k∆)] cos [Im (k∆)] + 1 < r2
But note that:
|τk (∆)| =
p
(A.2)
1 − 2 exp [− Re (k∆)] cos [Im (k∆)] + exp [−2 Re (k∆)]
(A.3)
The quantity inside the square root sign is the left-hand side of (A.2). Therefore, if ∆ satisfies (3.11), it also
satisfies (A.1). Since all the steps can be reversed, it follows that if ∆ satisfies (A.1), it must also satisfy
(3.11).
These results hold in particular for ∆ = ∆k (τ ) for some τ 6= 1. But for any τ 6= 1, τk (∆k (τ )) = τ , so it
follows from the above results that if |τ | < r, then ∆ = ∆k (τ ) satisfies (3.11). Satisfaction of (3.10) follows
from the definition of the inverse time transformation:

Re (1 − τ )
− ln |1 − τ | + ı arccos

|1 − τ |
k∆k (τ ) = − ln (1 − τ ) = 

Re (1 − τ )
− ln |1 − τ | − ı arccos
|1 − τ |

Im τ ≥ 0
Im τ < 0



The argument to the arccos function is always positive, so it takes values in [0, π/2). Holding |τ | fixed,
2
2
the smallest possible argument is 1 − |τ | , which results when Re τ = |τ | ; the arccos function thus takes its
√
√
maximum and minimum values ± 1 − r2 when τ = r2 ±ır 1 − r2 . Satisfaction of (3.10) follows immediately.
We now note that τk (∆) is everywhere analytic in ∆. The inverse transformation ∆k (τ ) has a singularity
at τ = 1, and, since we choose the logarithm such that −π < Im ln (z) ≤ +π, it also has a branch cut
discontinuity on the portion of the positive real axis τ ∈ [1, +∞). But these points lie outside of |τ | < r, so
in this region, ∆k (τ ) is analytic in τ . The analyticity of f (∆, x) in ∆ given the analyticity of h (τ, x) in τ , as
well as the converse, then follow immediately from the composition of two analytic functions.
42
A.2.
Proof of Corollary 1
The restrictions on ∆ given by (3.12) and (3.13) are simply those of (3.10) and (3.11) with the value of r
approaching 1 from below (note that the upper limit on Re (k∆) approaches +∞ in this case). Thus, if |τ | < 1,
∆k (τ ) satisfies (3.12) and (3.13), and conversely, if ∆ satisfies (3.12) and (3.13), then |τk (∆)| < r for some
r < 1. Therefore, given analyticity of h (τ, x) for some |τ | < 1, analyticity of f (∆, x) at ∆ = ∆k (τ ) follows
from Theorem 1. The converse follows analogously.
A.3.
Proof of Theorem 2
The condition |τk (∆)| < r is equivalent to:
2
(exp [− Re (k∆)] − cos [Im (k∆)]) < r2 − 1 + cos2 [Im (k∆)]
This follows by substituting in the right-hand side of (A.3) for |τk (∆)|, squaring both sides (which preserves
the inequality), and rearranging. However, for r > 1, the right-hand side is strictly positive for any value of
Im (k∆). Consequently, the only requirement for |τk (∆)| < r with r > 1 is:
p
p
− r2 − 1 + cos2 [Im (k∆)] < exp [− Re (k∆)] − cos [Im (k∆)] < + r2 − 1 + cos2 [Im (k∆)]
Note that the left inequality is always satisfied. The right inequality is equivalent to:
³
´
p
Re (k∆) > − ln cos [Im (k∆)] + r2 − 1 + cos2 [Im (k∆)]
(A.4)
So if ∆ satisfies (A.4), then |τk (∆)| < r. Since w (τ, x) is analytic for all |τ | < r, its power series converges
in this region, and converges uniformly in |τ | ≤ s for any 0 < s < r. The series approximations fn (∆, x) are
simply wn (τ, x) evaluated at τ = τk (∆), so fn (∆, x) converges in the region of ∆ corresponding to |τ | < r,
and uniformly in the region of ∆ corresponding to |τ | ≤ s for any 0 < s < r.
A.4.
Proof of Theorem 3
The function w (τ, z) also satisfies the conditions of Theorem 2 (with x replaced by z). So wn (τk (∆) , z) converges to w (τk (∆) , z) for all ∆ that satisfy (3.18) and for all z. But hn (∆, y) and h (∆, y) are simply wn (τ, z)
√ £
¡
¢
¤
and w (τ, z) premultiplied by exp (λ) ξ (y) with τ = τk (∆) and z = z (∆, y) ≡ k θ + 1 − τk/2 (∆) (y − θ)
substituted in. Therefore, for any y and for any ∆ that satisfies (3.18), hn (∆, y) converges to h (∆, y).
£
¤2
All that remains is to establish uniform convergence on (3.19). To do this, we note that 1 − τk/2 (∆) =
p
¯
¯
1 − τk (∆), from which it follows that ¯τk/2 (∆)¯ ≤ 1 + 1 + |τk (∆)|. Therefore, if |τk (∆)| ≤ s, we have
¯
¯
√
¯τk/2 (∆)¯ ≤ 1 + 1 + s, so τk/2 (∆) belongs to a compact set. Since z (∆, y) is continuous, z also belongs to
a compact set. It follows that wn (τ, z) converges uniformly to w (τ, z). Furthermore, from the restriction on
λ, the value of exp (λ∆) ξ (y) is bounded. Consequently, hn (∆, y) converges uniformly to h (∆, y).
A.5.
Proof of Theorem 4
The theorem establishes the existence (and analyticity) of a solution w (τ, z) to one PDE with final condition,
and another solution h (∆, y) to a different PDE with final condition. The first result, the existence and
43
analyticity of w (τ, z), is established by application of Theorem 2 in Kimmel (2008). The function:
µ
¶
√
(z−a 2b)2
z
4
gw (z) ≡ e
g √
2b
satisfies the conditions of that theorem, so there exists a w (τ, z), defined and analytic for all complex z and
√
τ such that k τ k < 1, that satisfies:
∂w
1 ∂2w
(τ, z) =
(τ, z)
∂τ
2 ∂z 2
(A.5)
(A.6)
w (0, z) = gw (z)
°p
°
°
°
Then h (∆, y) is defined and analytic for all complex y and ∆ such that ° τ2b (∆)° < 1, and:
· µ
¶
¸
√
2
b
b
∂h
b
∂w
∂w
(∆, y) = e− 2 (y−a) −( 2 +d)∆ −
+ d w (τ, z) + 2be−2b∆
(τ, z) − 2bbe−b∆ (y − a)
(τ, z)
∂∆
2
∂τ
∂z
·h
¸
i
2
2
3
∂ h
2
− 2b (y−a)2 −( 2b +d)∆
2
−2b∆ ∂ w
−b∆ ∂w
2
(∆, y) = e
b (y − a) − b w (τ, z) + 2be
(τ, z) − (2b) (y − a) e
(τ, z)
∂y 2
∂z 2
∂z
√ £
¤
where τ = τ2b (∆) and z = 2b a + e−b∆ (y − a) . Substituting these into the PDE in the theorem statement,
and taking advantage of the fact that w (τ, z) is a solution of (A.5) and (A.6), it can be seen that h (∆, y) is
indeed a solution. Furthermore, h (∆, y) satisfies the final condition, since:
³
³ √ ´
³√ ´
√ ´
2
2
2
b
b
b
h (0, y) = e− 2 (y−a) w τ (0) , 2by = e− 2 (y−a) w 0, 2by = e− 2 (y−a) gw
2by = g (y)
So h (∆, y) satisfies both the general PDE and the final condition.
A.6.
Proof of Theorem 5
The proof is similar to the proof of Theorem 4, but uses Lemma 5 instead of Theorem 2 from Kimmel (2008).
The final conditions:
gw1 (z) ≡ e
z2
4
µ
g1
z
√
2b
¶
gw2 (z) ≡ e
z2
4
µ
g2
z
√
2b
¶
√
satisfy the conditions of Lemma 5, so there exist w1 (τ, z) and w2 (τ, z), analytic for all z and all k τ k < 1,
that satisfy:
√
∂w1
1 − 1 + 8a ∂w1
(τ, z) =
(τ, z) +
∂τ
∂z
√2z
∂w2
1 + 1 + 8a ∂w2
(τ, z) =
(τ, z) +
∂τ
2z
∂z
1 ∂ 2 w1
(τ, z)
2 ∂z 2
1 ∂ 2 w2
(τ, z)
2 ∂z 2
(A.7)
(A.8)
w1 (0, z) = gw1 (z)
(A.9)
w2 (0, z) = gw2 (z)
(A.10)
44
°p
°
°
°
Then h (∆, y) is defined and analytic for all complex y 6= 0 and ∆ such that ° τ2b (∆)° < 1, and:

b 2
b
∂h
(∆, y) =e− 2 y −( 2 +d)∆
∂∆
µ
µ
(
− 2b y 2 −
+e
)∆
b
2 +d
b
b 2
∂2h
(∆, y) =e− 2 y −( 2 +d)∆
2
∂y
µ
µ
(
− 2b y 2 −
+e
where τ = τ2b (∆) and z =
√
)∆
b
2 +d
z
√
2b
z
√
2b
z
√
2b
z
√
2b
¶ 1−√21+8a
¶ 1+√21+8a
¶ 1−√21+8a
¶ 1+√21+8a

∂w1
∂w1
2be−2b∆
(τ, z) − bz
(τ, z)


∂τ
∂z
 · µ

√
¶
¸


1 + 8a
− b 1−
+ d w1 (τ, z)
2


∂w2
∂w2
(τ, z) − bz
(τ, z)
2be−2b∆


∂τ
∂z
 · µ

√
¶
¸


1 + 8a
− b 1+
+ d w2 (τ, z)
2
√


µ
¶
1 + 8a
∂w1
−2b∆ 1 −
2b
e
−
z
(τ,
z)


2
∂z






∂ 2 w1
−2b∆
 +2be

(τ,
z)
2


∂z
 µ

√
µ
¶
¶


2a
1
+
8a
2 2
+ b y − 2b 1 −
+ 2 w1 (τ, z)
2
y
√


¶
µ
1
+
1
+
8a
∂w
2
−2b∆
−z
(τ, z)
 2b e

2
∂z




2


∂
w
2
 +2be−2b∆

(τ,
z)
2


∂z
 µ

√
µ
¶
¶


1 + 8a
2a
2 2
+ b y − 2b 1 +
+ 2 w2 (τ, z)
2
y
2be−b∆ y. Substituting these into the PDE and taking advantage of the fact that
w1 (τ, z) and w2 (τ, z) are the solution of (A.7) through (A.10), it can be seen that h (∆, y) is a solution to the
general PDE in the theorem statement. Furthermore, h (∆, y) satisfies the final condition, since:
³
√
³
√ ´
√ ´i
1+ 1+8a
w1 τ (0) , 2by + y 2 w2 τ (0) , 2by
√
³ √ ´i
h 1−√1−8a
1+ 1−8a
b 2
2
= e− 2 y y
w1 + y 2
w2 0, 2by
√
³√ ´
³√ ´i
h 1−√1−8a
1+ 1−8a
b 2
2
gw1
2by + y 2
gw2
2by
= e− 2 y y
b
h (0, y) = e− 2 y
2
h
y
1−
√
1+8a
2
= g (y)
So h (∆, y) satisfies both the general PDE and the final condition.
45
Projection of Circles _W_=r onto 'for Basic Time Transformation (k=0.1)
20
15
10
r=0.40
Im(')
5
r=0.70
r=0.90
0
-10
0
10
20
30
40
50
60
r=0.97
r=0.99
-5
CIR
Singularities
-10
-15
-20
Re(')
Figure 1: This figure shows the position of circles |τ | = r with radii 0 < r < 1 projected onto
the plane of ∆, with the parameter of transformation in (3.1) given by k = 0.1. As shown, the
projections of the circles in τ onto ∆ are elongated shapes which extend farther in the positive
real direction than in any other direction. In general, the direction of elongation is the same as
the direction of k in the complex plane. If a function is analytic in ∆ at every point inside one
of the circles in τ , then the power series representation to that function in τ converges within
that circle. The figure also shows the singularity points for bond prices in the model of Cox,
Ingersoll, and Ross (1985), with κ = 0.5 and σ = 0.15. (The value of θ in this model does not
affect the location of the singularities.) For these parameter values, the singularities nearest
the origin are located at ∆ = −5.865 ± 5.784, and, as a result, the power series representation
of the bond price in ∆ converges only for maturities of up to 8.237 years. However, these
singularities lie outside all of the circles shown; therefore, the power series representation of
the bond price in τ = τk (∆) (for k = 0.1) converges in a region at least as large as the region
enclosed by the largest circle shown, which includes maturities of more than 50 years.
46
Required Region of Analyticity with and without Time Transformation (k=0.1)
60
40
20
Im(')
Without
With k=0.1
0
-80
-60
-40
-20
0
20
40
60
80
CIR
Singularities
-20
-40
-60
Re(')
Figure 2: This figure shows a circle in ∆ with r = 50 (i. e., a circle without the time
transformation), and a circle in τ with an approximate radius r = 0.9866 (i. e., a circle with
the time transformation, using k = 0.1), projected onto the plane of ∆. As shown, the circle
in τ is completely contained within and makes up only a small portion of the circle in ∆;
however, both include ∆ ∈ [0, 50). The use of the time transformation therefore effectively
extends the interval of convergence of a power series. If the function being represented has a
singularity inside the circle in ∆, but outside the circle in τ , then a power series in ∆ does not
converge on the interval ∆ ∈ [0, +50). However, a power series in τ converges on the interval
τ ∈ [0, 0.9866) (the ending point being approximate), which corresponds to ∆ ∈ [0, +50); the
value of f (∆, x) on this interval can then be found from w (τ, x) by applying (3.3). The figure
also shows the points of singularity of the bond price function for the model of Cox, Ingersoll,
and Ross (1985), with κ = 0.5 and σ = 0.15. As shown, there are no singularities within the
circle in τ , so that convergence of the series representation (in τ ) of bond prices is guaranteed
for maturities of up to at least 50 years; however, there are eight singularities within the circle
of ∆, so that a series (in ∆) converges only for a much smaller range of maturities. (Specifically,
series in ∆ converge on ∆ ∈ [0, +8.237).
47
Projection of Circles _W_=r onto 'for Basic Time Transformation (k=0.2)
20
15
10
r=0.70000
5
Im(')
r=0.90000
r=0.99000
0
-10
0
10
20
30
40
50
60
70
r=0.99900
r=0.99999
-5
CIR
Singularities
-10
-15
-20
Re(')
Figure 3: This figure shows the position of circles |τ | = r with radii 0 < r < 1 projected onto
the plane of ∆, with the parameter of transformation in (3.1) given by k = 0.2. Note that the
circles in τ are more elongated and follow the positive real axis more closely than in the k = 0.1
case, shown in Figure 1. A power series in τ therefore always converges for at least as large
a range of positive real values of ∆ in the k = 0.2 case as in the k = 0.1 case. Increasing the
(positive real) value of k never decreases the interval of convergence for positive real ∆, and
often increases it. The figure also shows the points of singularity of the bond price function
for the model of Cox, Ingersoll, and Ross (1985), with κ = 0.5 and σ = 0.15. Note that all
of the singularities lie outside all of the circles shown, so that, with k = 0.2, convergence for
maturities of at least 60 years can be established. As can be seen from Figure 1, the interval
of convergence for a series based on k = 0.1 is substantially smaller.
48
Projection of Circles _W_=r onto 'for Basic Time Transformation
20
15
10
Im(')
5
0
-15
-10
-5
0
5
10
15
20
25
30
35
k=0.05
k=0.10
k=0.15
k=0.20
k=0.25
-5
-10
-15
-20
Re(')
Figure 4: This figure shows circles in τ projected onto the plane of ∆, for various combinations
of r and k. For each choice of k shown in the legend, the corresponding value of r was chosen
so that all circles reach the same maximum value on the positive real axis, of approximately
∆ = 30.4. The radii are approximately r = 0.6105, r = 0.9066, r = 0.9792, r = 0.9954, and
r = 0.9990 for k = 0.05 through k = 0.25, respectively. As shown, the regions enclosed by
the circles are smaller for larger values of k, even though all circles reach the same maximum
positive real value ∆ = 30.4. Use of larger values of k can therefore effectively increase the
range of convergence of a power series. If there are singularities inside the k = 0.05 circle but
outside the k = 0.25 circle, then use of the basic time transformation ensures convergence of
a power series on ∆ ∈ [0, +∞) if k = 0.25, but only for a smaller range of ∆ if k = 0.05.
49
Projection of Circle _W_=1 onto 'for Basic Time Transformation
40
30
20
Im(')
10
0
-20
-10
0
10
20
30
40
50
k=0.05
k=0.10
k=0.15
k=0.20
k=0.25
-10
-20
-30
-40
Re(')
Figure 5: This figure shows the position of the circles |τ | = 1 projected onto the plane of
∆, for different values of the parameter of transformation of (3.1). As shown, the unit circle
projections onto ∆ are elongated shapes opening up toward the right (i. e., toward large
positive real values) in the complex plane of ∆. In general, the direction of the opening is the
same as the direction of k in the complex plane. If a function is analytic in ∆ at every point
inside one of the circles in τ (i. e., to the right of the the corresponding shapes in the figure),
then the power series of τ converges within that region. For positive values of k, all circles in
τ with radius equal to one include ∆ ∈ [0, +∞).
50
Projection of Circle _W_=r onto 'for Basic Time Transformation (k=0.15)
80
60
40
Im(')
20
r= 1.01
r= 5.00
0
-30
-20
-10
0
10
20
30
40
r=25.00
CIR
Singularities
-20
-40
-60
-80
Re(')
Figure 6: This figure shows the position of circles |τ | = r projected onto the plane of ∆,
where the parameter of the transformation of (3.1) is k = 0.15. As shown, the projections of
the circles in τ onto ∆ are periodic functions of the imaginary part of ∆. Points to the right
of the curves are inside the corresponding circles in τ , and points to the left are outside the
same circles. If a function is analytic within the indicated circle in τ , then it is also analytic
as a function of ∆ in the indicated region. Note that all the circles shown include the interval
∆ ∈ [0, +∞). However, analyticity of a function to the right of one of the curves shown is not
sufficient to ensure analyticity within the indicated circle in τ , and therefore also not sufficient
to establish uniform convergence of a power series on ∆ ∈ [0, +∞); analyticity at τ = 1 is also
required, but this point does not correspond to any value of ∆.
51
52
3
2
dr = κr (θ − r) dt + σr dW
r = x2 + d
dx = κ (θ − x) dt + σdW
dr = κ (θ − r) dt + σ rdW
³
³
2κ2
σ4
2 2
+ κ 8θ y 2
1
y2
2κ+4
2
³σ
3
8
´
´
κ2 θ
σ2
+
− κθ +
+
´
´2
2 2
3
+ 2θσ4κ − 2θκ
8
σ2
2
2
θκ2
−
+y 2 κ +2σ
8
σ2
³
κ2 +2σ 2
y + κ22θσ
2
+2σ 2
κ2 θ 2
+ κ2 +2σ2 − κ2 + d
1
y2
κ2
2
¢2
¡
y + κσ2
κ
σ2
+θ − 2 − 2κ2
2
2
2
σ
¡ yσ ¢ 3 + 2κ2
κ
σ
κ
2
2
y 2 κθ
4
e− 4 y
2
e−
e− 2 y
2
¡ yσ ¢ 1 − 2θκ
2
κ
e− 2 y
Canonical PDE
Coefficient
Final Condition
∆=
∆=
σ
√
σ
2
κ2 +2σ 2
∆=0
√
³
´i
h
√
ln 1+ κ2 κ− κ2 +2σ 2 +(2n+1)πı
κ2 +2σ 2
³
´i
h
√
ln 1+ κ2 κ− κ2 +2σ 2 +(2n+1)πı
None
Singularities
, n∈N
, n∈N
∆ ∈ 0,

∆ ∈ 0,

σ
κ2 +2σ 2
κ2 +2σ 2
∆=0
2
√
√
³
´i2
h
√
ln 1+ κ2 κ− κ2 +2σ 2
+π 2
r
σ
´i2
h
³
√
+π 2
ln 1+ κ2 κ− κ2 +2σ 2
r
∆ ∈ [0, +∞)
Range of Power
Series Convergence




This table shows information relevant to the convergence properties of the power series representations of bond prices in four term structure
models. In all four models, bond prices are known in closed form. The first column specifies the model; “Vas” denotes the model of Vasicek
(1977), “CIR” refers to Cox, Ingersoll, and Ross (1985), “ADG” denotes the model of Ahn, Dittmar, and Gallant (2002), and “AG” refers
to Ahn and Gao (1999). The second column specifies the interest rate processes in the respective models. The next two columns show
the coefficients and final conditions of the pricing PDE for each model, when expressed in the canonical form of (2.8) and (2.9). The next
column shows the location of singularities in the bond price, as a function of ∆. The last column shows the interval of convergence of
the power series for positive values of ∆. Note that in the Vas model, there is no singularity anywhere in the bond price function; the
power series representation of bond prices therefore convergences for all maturities. In the AG model, there is a singularity at ∆ = 0,
and consequently, a power series representation of bond prices does not converge for any non-zero maturity. For the CIR and ADG
models, there are singularities at complex values of the bond price function, which prevent convergence of a series representation for large
positive values of ∆. The range of convergence for these two models can be extended by the use of time transformation methods; uniform
convergence in maturity can also be established under these two models and under the Vas model using time transformations.
AG
ADG
CIR
dr = κ (θ − r) dt + σdW
Vas
√
Interest Rate
Process (Risk-Neutral)
Model
Table 1: Convergence Properties of Bond Prices in Several Models
Table 2: Bond Yield Approximations in ADG Model
Maturity
(years)
Exact
Zero-coupon Bond Yields
Order 2 Approx. in τ
Order 4 Approx. in τ
Yield
Rel. Error
Yield
Rel. Error
Order 8 approx. in ∆
Yield
Rel. Error
Typical initial interest rate—r = 6%, x = +0.226223
1
5
10
20
50
100
1000
10000
5.850026%
5.354871%
5.112256%
4.967059%
4.878684%
4.849224%
4.822709%
4.820058%
5.850060%
5.354894%
5.112268%
4.967064%
4.878686%
4.849225%
4.822709%
4.820058%
1
5
10
20
50
100
1000
10000
3.255580%
2.593916%
3.510528%
4.156640%
4.554494%
4.687128%
4.806500%
4.818437%
3.255598%
2.593937%
3.510540%
4.156646%
4.554496%
4.687130%
4.806500%
4.818437%
1
5
10
20
50
100
1000
10000
393.2333%
135.6025%
71.67386%
38.29582%
18.21030%
11.51503%
5.489290%
4.886716%
393.2344%
135.6026%
71.67387%
38.29583%
18.21030%
11.51503%
5.489290%
4.886716%
5.75 × 10−6
4.37 × 10−6
2.31 × 10−6
1.19 × 10−6
4.83 × 10−7
2.43 × 10−7
2.44 × 10−8
2.44 × 10−9
5.850026%
5.354871%
5.112257%
4.967059%
4.878684%
4.849224%
4.822709%
4.820058%
4.25 × 10−10
7.53 × 10−10
4.01 × 10−10
2.06 × 10−10
8.39 × 10−11
4.22 × 10−11
4.25 × 10−12
4.25 × 10−13
5.850024%
5.366424%
N/A
N/A
N/A
N/A
N/A
N/A
−3.31 × 10−7
2.16 × 10−3
N/A
N/A
N/A
N/A
N/A
N/A
3.254145%
−54.71145%
−84.53861%
−70.68393%
−43.11839%
−27.13691%
−4.558774%
−0.640115%
−4.41 × 10−4
−2.21 × 101
−2.51 × 101
−1.80 × 101
−1.05 × 101
−6.79 × 100
−1.95 × 100
−1.13 × 100
−596.6695%
−385.3111%
−248.7733%
−152.2866%
−75.61808%
−43.36144%
−6.178866%
−0.802100%
−2.52 × 100
−3.84 × 100
−4.47 × 100
−4.98 × 100
−5.15 × 100
−4.77 × 100
−2.13 × 100
−1.16 × 100
−627.5459%
−391.1627%
−251.6678%
−153.7253%
−76.19144%
−43.64775%
−6.207464%
−0.804959%
−2.72 × 100
−4.70 × 100
−5.63 × 100
−6.21 × 100
−6.18 × 100
−5.47 × 100
−2.17 × 100
−1.17 × 100
Typical initial interest rate—r = 6%, x = −0.226223
5.64 × 10−6
8.21 × 10−6
3.33 × 10−6
1.42 × 10−6
5.17 × 10−7
2.51 × 10−7
2.45 × 10−8
2.44 × 10−9
3.255580%
2.593916%
3.510528%
4.156640%
4.554494%
4.687128%
4.806500%
4.818437%
3.10 × 10−10
1.36 × 10−9
5.78 × 10−10
2.46 × 10−10
8.99 × 10−11
4.37 × 10−11
4.26 × 10−12
4.25 × 10−13
High initial interest rate—r = 600%, x = +2.44769
2.80 × 10−6
2.82 × 10−7
1.71 × 10−7
1.54 × 10−7
1.29 × 10−7
1.02 × 10−7
2.15 × 10−8
2.41 × 10−9
393.2333%
135.6025%
71.67386%
38.29582%
18.21030%
11.51503%
5.489290%
4.886716%
7.53 × 10−10
5.86 × 10−11
3.01 × 10−11
2.67 × 10−11
2.25 × 10−11
1.78 × 10−11
3.73 × 10−12
4.19 × 10−13
High initial interest rate—r = 600%, x = −2.44769
1
5
10
20
50
100
1000
10000
365.1619%
105.7295%
54.34346%
29.52725%
14.70262%
9.761194%
5.313906%
4.869178%
365.1622%
105.7295%
54.34348%
29.52725%
14.70263%
9.761195%
5.313906%
4.869178%
9.19 × 10−7
1.37 × 10−7
2.08 × 10−7
1.99 × 10−7
1.60 × 10−7
1.21 × 10−7
2.22 × 10−8
2.42 × 10−9
365.1619%
105.7295%
54.34346%
29.52725%
14.70262%
9.761194%
5.313906%
4.869178%
1.55 × 10−10
1.88 × 10−11
3.55 × 10−11
3.47 × 10−11
2.79 × 10−11
2.10 × 10−11
3.85 × 10−12
4.21 × 10−13
This table shows bond yields for the model of Ahn, Dittmar, and Gallant (2002), calculated exactly, by
series approximation in ∆, and by series approximation in τ . For series in τ , approximations including
terms up to order τ 2 and τ 4 are included; for series in ∆, terms up to order ∆8 are included. The
parameters used are κ = 0.4995, θ = 0.1860, σ = 0.0812, and φ = 0.0088. For these parameter values,
the unconditional mean and variance of the interest rate process are 5% and 0.001, respectively; the
unconditional kurtosis of the interest rate process is 5. For each approximation, the relative error is
shown, i. e., the approximation error divided by the exact yield. Since each possible value of the interest
rate corresponds to two different values of the state variable, we show results for four different initial
values of the state variable, two corresponding to an interest rate of 6% and two corresponding to an
interest rate of 600%. As shown, the approximations in τ are highly accurate with only a small number
of terms, across an extremely wide range of initial interest rates and maturities. By contrast, the
approximations in ∆, even with a larger number of terms, are accurate only for very short maturities,
and when the initial interest rate is not very large. For these parameter values, the series in ∆ diverges
for maturities greater than approximately 5.24 years. Entries of “N/A” indicate that the bond price
approximation is zero or negative, so that the corresponding yield approximation is not real-valued.
Table 3: Bond Yield Approximations in CIR Model
Maturity
(years)
Exact
Zero-coupon Bond Yields
Order 2 Approx. in τ
Order 4 Approx. in τ
Yield
Rel. Error
Yield
Rel. Error
Order 8 approx. in ∆
Yield
Rel. Error
Typical initial interest rate—r = 6%
1
5
10
20
50
100
1000
10000
6.409818%
7.124507%
7.379918%
7.523945%
7.611073%
7.640116%
7.666256%
7.668869%
6.416892%
7.138838%
7.388510%
7.528294%
7.612813%
7.640986%
7.666343%
7.668878%
1.10 × 10−3
2.01 × 10−3
1.16 × 10−3
5.78 × 10−4
2.29 × 10−4
1.14 × 10−4
1.13 × 10−5
1.13 × 10−6
6.409823%
7.124555%
7.379951%
7.523962%
7.611080%
7.640120%
7.666256%
7.668870%
7.82 × 10−7
6.82 × 10−6
4.47 × 10−6
2.24 × 10−6
8.85 × 10−7
4.41 × 10−7
4.39 × 10−8
4.39 × 10−9
6.409817%
6.751295%
−15.58451%
−35.51708%
−28.93838%
−20.01544%
−3.842947%
−0.568490%
−1.88 × 10−7
−5.24 × 10−2
−3.11 × 100
−5.72 × 100
−4.80 × 100
−3.62 × 100
−1.50 × 100
−1.07 × 100
1.749034%
5.416320%
N/A
N/A
−24.30220%
−18.86522%
−3.783540%
−0.563010%
1.07 × 10−6
8.26 × 10−2
N/A
N/A
−4.29 × 100
−3.51 × 100
−1.49 × 100
−1.07 × 100
−480.3992%
−363.4862%
−238.0046%
−146.9416%
−73.49005%
−42.29912%
−6.072789%
−0.791494%
−2.02 × 100
−2.66 × 100
−2.97 × 100
−3.28 × 100
−3.42 × 100
−3.22 × 100
−1.69 × 100
−1.10 × 100
Low initial interest rate—r = 0.06%
1
5
10
20
50
100
1000
10000
1.749033%
5.003263%
6.246237%
6.954521%
7.383299%
7.526229%
7.654867%
7.667731%
1.755732%
5.017505%
6.254826%
6.958871%
7.385039%
7.527099%
7.654954%
7.667739%
1
5
10
20
50
100
1000
10000
472.4884%
219.2489%
120.7479%
64.46632%
30.38848%
19.02882%
8.805126%
7.782757%
472.6014%
219.2735%
120.7569%
64.47067%
30.39022%
19.02969%
8.805213%
7.782765%
3.83 × 10−3
2.85 × 10−3
1.38 × 10−3
6.25 × 10−4
2.36 × 10−4
1.16 × 10−4
1.14 × 10−5
1.13 × 10−6
1.749037%
5.003311%
6.246270%
6.954538%
7.383306%
7.526233%
7.654867%
7.667731%
2.62 × 10−6
9.61 × 10−6
5.27 × 10−6
2.42 × 10−6
9.12 × 10−7
4.47 × 10−7
4.40 × 10−8
4.39 × 10−9
High initial interest rate—r = 600%
2.39 × 10−4
1.12 × 10−4
7.41 × 10−5
6.75 × 10−5
5.73 × 10−5
4.57 × 10−5
9.88 × 10−6
1.12 × 10−6
472.4887%
219.2490%
120.7480%
64.46634%
30.38848%
19.02882%
8.805126%
7.782757%
8.09 × 10−7
5.34 × 10−7
2.92 × 10−7
2.61 × 10−7
2.22 × 10−7
1.77 × 10−7
3.82 × 10−8
4.33 × 10−9
This table shows bond yields for the model of Cox, Ingersoll, and Ross (1985), calculated exactly, by
series approximation in ∆, and by series approximation in τ . For series in τ , approximations including
terms up to order τ 2 and τ 4 are included; for series in ∆, terms up to order ∆8 are included. The
parameters used are κ = 0.5000, θ = 0.0800, and σ = 0.1500. For each approximation, the relative error is
shown, i. e., the approximation error divided by the exact yield. As shown, the approximations in τ are
highly accurate with only a small number of terms, across an extremely wide range of initial interest
rates and maturities. By contrast, the approximations in ∆, even with a larger number of terms, are
accurate only for very short maturities, and when the initial interest rate is not very large. For these
parameter values, the series in ∆ diverges for maturities greater than approximately 8.24 years. Entries
of “N/A” indicate that the bond price approximation is zero or negative, so that the corresponding
yield approximation is not real-valued.
54