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DEWEY
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
The Tractable "Quadratic" Class of
Growth and Interest Rate Processes
Xavier Gabaix
Working Paper 06-1
June 12,2006
Room
E52-251
50 Mennorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=907867
MASSACHUSETTS INSTITUTE
OF TECHNOLOGv
JUN
2 u
I
LIBRARIES
The Tractable "Quadratic"
Class of
Growth and
Interest
Rate
Processes
Xavier Gabaix
MIT
and
NBER
March 23, 2006
This version: June 12, 2006*
Very Preliminary and Incomplete - Comments Welcome.
First version:
Abstract
We
propose a new class of growth and interest rate processes with appealing properties
the paper-and-pencil theorist.
tuities,
It yields
for
simple closed-form solutions for stocks, bonds, and perpe-
and can accommodate an arbitrary number of
factors. Expressions are linear in the state
premium and the
stock's growth rate. Surprisingly,
one can change the volatility of the process, or force the interest rate to be positive, without
changing bond prices. The process generaUzes to discrete-time settings, and has a number of
economic modeling applications. These include (i) macroeconomic situations with changing trend
growth rates, (ii) asset pricing with time-varying risk premia or time-varying dividend growth
rates, and (iii) yield curve analysis that allows flexibihty and transparency. (JEL: G12, G13)
variables, such as the interest rate, the equity
Keywords: Long term
risk.
Growth
rate risk. Perpetuity, Modified
Gordon growth model. Yield
curve. Interest rate processes, Stochastic Discount Factor, Quadratic processes.
*xgabaix@mit.edu. Oleg Rytchkov provided very good research assistance. For helpful conversations,
thank John Cox, Barrel Duffie, Alex Edmans, Leonid Kogan, Jun Pan and seminar participants at MIT.
thank the NSF for support.
I
I
Introduction
1
new growth and interest rate process that has a number of attractive properties.
Most notably, it is simple and tractable, allowing closed-form solutions for the prices of stocks, bonds
and perpetuities. It is also flexible, can be applied to an arbitrary number of factors with a rich
correlation structure, and generalizes well to discrete time. All of these features make it particularly
This paper proposes a
accessible to the paper-and-pencil theorist.
Quadratic processes have the
drift of
an Ornstein-Ulenbenck processes, with a squared term
added. This squared term does not change the drift much, so that quadratic processes are in effect
an Ornstein-Uhlenbeck process. However, adding the quadratic terms greatly simplifies
the expressions of expected values of stocks and bonds. The expressions are linear in the factors.
We obtain closed forms for the price of stocks with a time-varying growth rate, and the price of
fairly close to
perpetuities.
Such closed forms
for perpetuities
is
not not available with the available processes, such
Ornstein-Uhlenbeck / Vasicek (1977), Cox, IngersoU, Ross (1985), the Heath, Jarrow,
Morton (1992), or models in the affine class (Duihe and Kan 1996).
as those of
Quadratic processes are meant to be a practical tool
process
is
varying;
likely to
(ii)
be useful
in:
macroeconomics,
(i)
asset pricing, for models
the trend growth rate of dividends
is
for several topics in
for
where the equity premium
varying; and
(iii)
is
The quadratic
fixed-income analysis. Also, the process
be useful for thinking about other situations where the uncertainty
e.g., in
economics.
GDP
growth rate is timetime-varying, and models where
models in which
in
may
the discount factor matters,
environmental economics.
Several literatures motivate the need for a tool such as the quadratic process.
Many
recent
and macroeconomics, e.g.,
Bansal and Yaron (2004), Barro (2006), Croce, Lettau and Ludivgson (2006), Gabaix and Laibson
(2002), Hansen Heaton and Li (2005), Hansen and Scheinkman (2005), Julliard and Parker (2004),
Parker (2001). The quadratic process offers a way to model long-term risk, while keeping a closed
form for stock prices.
In addition, there is debate about the existence and mechanism of the time-varying expected
stock market returns, e.g., Campbell and Shiller (1988), Cochrane (2006), and many others. Because
of the lack of closed forms, the literature relies on simulations and approximations. The quadratic
process offers closed forms for stocks with time-varying equity premium, which is useful for thinking
studies investigates the importance of long-term risk for asset pricing
about those
Finally,
issues.
we contribute
flexible process.
to the vast literature
It is as flexible as
on
interest rate processes,
the afline class of Duffle and
Kan
by presenting a new,
which includes the
(1996),
Vasicek (1978) and the Cox, IngersoU, Ross (1985) process as special cases. Section 4.3 develops the
between the quadratic class and the affine class.
This paper stipulates a process for finding desirable properties for the pricing kernel, and in this
follows a productive literature represented by, e.g., Campbell and Cochrane (1999), Cox, IngersoU,
Ross (1985)., and particularly Menzly, Santos and Veronesi (2004) and Pastor and Veronesi (2005).^
The last two papers are the closest to this paper, as they contain several useful closed forms.
link
The core insight, that the quadratic
new to the literature.'^
process (72) yields particularly tractable formulas, appears
to be
Those papers do provide an economic foundation (via external habit formation) for the postulated processes.
The Fisher- Wright process (e.g., Karlin and Taylor 1982) does contain a quadratic term, but it has not been applied
Section 2 presents the basic one-factor process. Section 3 applies the one-factor process to stocks.
Section 4 presents the multifactor version of the quadratic process, which
is
useful for talking about
processes with fast and slowly mean-reverting components, or interest rate processes with short and
long rates.
Section
5,
which
is
more
technical, studies the range of admissible initial conditions.
Section 6 presents the discrete-time version of the process. Section 7 concludes.
The
2
one-factor quadratic process
and basic properties
Definition
2.1
We fix a probability space
(fi,
T
^
P) and an information filtration !Ft satisfying the usual technical conand Shreve (1991)). We start from a simple Ornstein-Uhlenbeck
ditions (see, for example, Karatzas
process, but
add a quadratic term:
drt
= -{P -a)
{rt
- a)dt+ {n - af
dt
+ dMt,
with a </?, for rt < f £ (a,/?) where rt is the short-term interest rate.
— (/3 — a) {rt — a) dt is the standard term which leads to mean reversion to the long run value
a,
with a speed
(3
—
a.
The second
When
Ornstein-Uhlenbeck process.
standard
—
(/?
—
a)
(r
—
a) dt^
.
term,
rt
{rt
—
a)
,
represents a small departure from a classical
not too far from a, the drift
is
However,
it
is
turns out that the correction
little
{rt
changed from the
— a)
dt substantially
improves the tractability of the process.
We assume that the martingale {Mt,t > 0} has properties that ensure the interest rate remains
below some upper bound r e {a,P), and thus does not explode. One example is dMt = o' {h) dzt
where zt is a Brownian process and a {r) ^ k{r — r)'^, for r in a left neighborhood of f, k > 1/2
and A; > 0. Given the drift is negative around f, that will ensure that f is a natural boundary, and
{Vf
,
rt
<
f} almost
We
surely.''
have the option to
also
set a lower
bound
r to the interest rate,
by
>
adding an additional restriction on dMt. For instance, if one wishes to ensure rt
r for some r < a,
— o" {rt, t) dzt, with at ^ k' {r — r)'^ k' > 1/2 for r in a right neighborhood of 0,
we could have dMt
and k' > 0.
The above
,
process can be rewritten:
drt
Appendix
=
{rt
-
a)
{rt
-
+ dMt.
P) dt
(1)
A contains a formal statement
and justification of these assumptions, and further purely
technical assimiptions. Given these assumptions, and an initial value tq, the stochastic differential
equation
The
(1)
has a unique strong solution for
following
Theorem
rt,
defined for
gives a basic result. Its proof
is
all
in
times
t
>
0.
Appendix B.
bonds or stocks. Also, it is more special than the quadratic class, because it imposes a specific functional
form on the variance. Driessen, Maenhout and Vilkov (2005) apply the Fisher-Wright process to options.
is less than 0.25%
''in many applications, it is plausible that \rt — q| < 5%, and thus the quadratic term (rj — a)
to the pricing
per year.
"See Karlin and Taylor (1981,
treatment.
Lemma
15.6.3,
Table 15.6.2), and the appendix of Ait-Sahalia (1996)
for a concise
Theorem
The price
1
at time
t
=
of a zero-coupon bond of maturity T, Zt (T)
Et exp
(
—
rsds
J^
is:
Z.(T)
§^e-^+^e-'^^.
—a
— a
=
p
(2)
p
(2) is particularly simple, as it is linear in the current rate rt- Remarkably, the bond
depend on the noise dMt and thus the volatility of the process. ^ The intuition is as
for all t. Theorem 1 still
follows. Consider the case where the process is deterministic, i.e., dMt =
=
A + Br, where A and B are constants that depend on T
applies, and we rewrite Eq. 2 as: Z {T,r)
but not on r. Suppose that at time t the interest rate is rt, and apply a shock to the interest rate at
time t + dt so that rt+dt = ''"t+ut, where ut is a mean-zero random variable. After t + dt, the process
remains deterministic, and evolves according to (1) with dMt = 0. Given an initial value rt + ut of
Expression
price does not
the interest rate, the bond price at
t
+
dt
Z
is
+ ut) = Z {T,rt) +
(T, rt
But, because
Z
is
affine in
r. Hence the value at i is £ [Z {T, rt + Ut)] = Z {T, rt). The expected value of the impact of uj is 0,
because the bond price is linear in uj. A zero mean addition of volatility does not affect the bond
price.
The cnrx
is
=
that, in the deterministic case {\ft,Mt
of the interest rate:
= A{T) + B
Z{T,rt)
{T)rt.
By
0),
the
bond
price
is
an afhne function
contrast, intuition based on the expression
t+T
would suggest a convex function.
exp (-/;
„ds
'The independence of bond prices from volatility greatly simplifies the analysis.
E,
dMt
could have jumps, which model a decision by the central bank.
the volatility process to get the prices of bonds: only the drift part
margin of
flexibility to
cahbrate
volatility, for instance
on
is
In particular,
One does not need
to specify
necessary. This leaves a high
interest rate derivatives, a topic
we do not
pursue here.
Equally surprisingly, we can impose a lower bound without affecting bond prices, provided the
lower bound
less
is
than
r<
a.
The
intuition
is
process simply by setting the variance to be
the
new lower bound does not
Existing models, such as
We
the following.
below
r.
bond prices.
Vasicek and Cox-Ingersoll-Ross,
lutions for zero-coupon bonds.
can make
r a lower
bound
for the
Since variance does not affect bond prices,
affect
are able to generate closed-form so-
However, the quadratic process introduced by
this
paper can also
generate closed-form solutions for perpetuities.
Proposition
1
The price of a perpetuity
is
roo
a+
.J,
f3
-
rt
aP
Proof. This follows directly from Eq.
2
roo
V
Zt
/
Jo
Zt{T) dT
_
1
a
a -
rt
(3)
a/3
and
-rtl
P —a a
13
,
rt- al
P—aP
a+Pap
rt
We now analyze the dynamics of the process in some further detail. If we start
(T) = e~°''^ which is the value the bond would have if rt remained pinned at
,
'Hence the process exhibits "unspanned
volatility" in the sense of
from rt = a, then
a. However, rt is
ColUn-Dufresne and Goldstein (2002).
With a < (3, Eq.2 shows that Zt (T) ~ j^e~°''^ for T — oo. Hence a is the long run
interest rate, a = — limr-.cxj Z' (T) /Z (T). Given the drift process (1), E [drt] >
if r is below a,
if r is above a. Hence, r is mean-reverting towards a.
and E [drt] <
P has two roles. First, it represents an upper bound to the interest rate: rt < f < p. Second,
—
P a is the speed of mean-reversion of the process. When /3 is high, the process mean-reverts faster.
The process is defined only for rt <r < j3. If r^ > f, the process may explode in finite time.
Indeed, when the process is deterministic (M = 0) and rj > f, then the process explodes in finite
time, i.e., there is ii > t such that lims^fj r^ = oo. The assumption r^ < ^ is needed in the proof
stochastic.
>
to ensure that the process
is
defined at times in
Appendix A.
we require the leading terms
\t,t
+ T].
This
guaranteed
is
if rj
<
/3,
under the
conditions of
Finally,
kr^ for a
/c
to
be r^ in the definition
the quadratic process, not
(1) of
7^ 1.
The empirical
relevance of the quadratic term
unclear.
is
It
could be, for instance, that the
quadratic term exists under the risk-neutral probability, though not in the physical probability. Also,
and in many cases, for instance for options,
assume Gaussian innovations rather than fat-tailed ones
2003, 2006). Quadratic processes allow for jmnps and fat-tailed innovations, as
of course all processes are an approximation of reahty,
the major counterfactual assumption
(see
Gabaix
et al.
the shape of the innovations (the
The
2.2
of
to
dMt term) does not
rest of this section considers
Sum
is
more
affect the prices.
specialized topics in the one-factor quadratic process.
two quadratic processes
sum of two quadratic processes. This apphcation is
premium and the dividend growth rate are time-varying.
This section studies the
when both
the equity
useful, for instance,
Proposition 2 (Sum oj two independent quadratic processes) Consider two one-factor quadratic
processes:
drt
dRt
where n and
N
= -cp{rt - rt)dt + {rt -r^)'^ dt + dnt
= -^ {Rt - R») dt + {Rt ~ R^f dt + dNt
are martingales with uncorrelated innovations, d (n,
t+T
Et exp
calling S„
+ Rs)ds
=
+
r^,
Et
exp
and such
0,
that (n.
A'')
is
interest rate rt
rt-r,
S.
first
exp
t+T
r^ds
+
Rt, Pt
=
Et
/q°°
exp
(
part
S,
is
+
4>
Rt-R.
5* + $
immediate using
Ito's
{n
-
(5.
r,) {Rt
- R,)
+ 0) (5. +
(r^
+
(4)
Rs)ds\ dT
(f)
(5)
lemma. The second part comes from the calculation
-4>T
e-(-+^-)^(l + (r,-r.
1^
^^
+ + ^)
+ <^ + $) J
{2S,
*) (5.
of:
P,=
—
R.ds
R^,
Pt
Proof. The
=
t+T
{rs
and the price of a perpetuity with
is,
N)^
Then,
a martingale.
_
„-*T
I
l
+ {Rt-
R.)
$
-1
dT.
Application to the pricing of stocks with a stochastic growth rate
3
This section applies the quadratic process to model the dividend growth rate of stocks. We start
with a stochastic trend growth rate of dividends, continue with a stochastic equity premium, and
finally analyze
Time- varying dividends
3.1
The
more complex examples.
stock has a dividend Dt, which grows at a stochastic rate
^
=
gtdt
ct>{g,-gt)dt-{gt-g,fdt
(dMt,dNt)/dt
=
=
9t
>
g>P
dgt
where P <
6
=c=
0,
a,
g*
Nt
is
is
a.
gt:
Mt
martingale, and
cgt
is
+ dNt
(6)
+ dMt
(7)
+b
(8)
(9)
a martingale that ensures
gt>g
for
some g >
p.
When
the long-run growth rate of the stock.
Innovations to dividends, dNt, can be correlated with innovations to the growth rate, dMtstrength of that correlation can vary with the level of the growth rate
The
discount factor
is
constant at R.
The
stock price
=E
Vt
The
gt-
is:
e-^'^'-'^Dsds
t
We
start
with the case
b
=c—
0, i.e.,
of no correlation between dividend level and growth rate.
Proposition 3 (Modified Gordon growth model, with time-varying dividend growth
there is no correlation between innovation the level of dividend and innovations to
rate
({dMt,dNt)
=
0), the
price/dividend ratio of the stock
V^./A
where g^
{gt)
from
The
is
the long
=
^
R- g*
(l
\
term growth of dividends, and
+
gt
When
is:
^^^^)
R- +
g^
— g*
rate)
their growth
is
(10)
4>J
the deviation of the current growth rate
the trend (g*).
first
term,
R — g^,
is
the
P/D
ratio
from the Gordon growth model, since 5,
—
is
the long term
growth rate of the dividend. The second term, gt
g*, is the deviation of the current growth rate
(gt) from the trend (5*). It increases the P/D ratio, by an duration factor 1/ (/? — g, + (p) that is
decreasing in the discount rate R, and the speed of mean reversion (p.
We now consider the case where {dMt, dNt) /dt = cgt + b.
Proposition 4 (Modified Gordon growth model, with correlations between innovations
and growth rate of dividends) The price/dividend ratio of the stock is:
R - g* +
+ c + {gt - g*)
{R-g,){R-g, + (l>)-cR-b'
Vt/Dt
The stock
price
is
to the level
(p
(11)
increasing in the correlation between the level of dividend and
its
trend growth
rate.
3.2
Time- varying equity premium
Consider the stochastic discount factor:
=
Qt
The
We
price at time
t
exp
(
-ri -
-^ds -
/
Dsds
of any asset yielding a dividend
X^dB,
/
at
time
(12)
s
=
Vt
is:
Et [J^ QsDsds] /Qt-
seek to price a stock with dividend:
Dt =^exp(gt-
Assume
that the risk premium, nt
dnt
where Nt
is
=
=
-(p
XtO't,
-
{-Kt
^ds +
/
TV^
Dq.
)
(13)
follows a one-factor quadratic process:
TT,)
+
(ttj
-
a martingale that ensures that the process
that processes
a^dBs
/
TT,)^ dt
+ dNt
well-defined
is
and Bt are independent. Then, the price of a stock
(tt^
<
tF
<
tt*
+
0)-
Assume
calling J^/^ the filtration
is,
associated with Nt,
=
Vo
Eo
Uo
QtDtdt /Qo
\2
*
=
Eo
=
Eo
/
Hg -
exp
r) i
-
/
oo
2
,
'^
/
rt
expUg-r)t-
'
\2
Xi
ft
ds+
+ a',,
ds
^
=
^o
/
expl{g — r)t—
so that, applying Proposition
1
to rt
/
Xsasdsjdt
1
r
This
TTf is
is
a
equal to
---
= —g + r + nt, we
Vt/Dt
+ 7T»
/.
-
=
rrt
+
,
/_
^
^2
{as-Xs)
^^U,|^N
Jo
Eq
exp{{g
/
—
r
—
Trt)t)dt
get:
- TT*
+ 7rt - g +
TTt
1
g
ias-Xs)dBs]dt
r
(14)
'
Gordon growth formula, with time- varying risk-premium ttj. When
its central value, tt*, we get the traditional formula, Vt/Dt — 1/ (r -|-
the risk
tt,
—
g).
premium
When
ttj
is
different
from
tt,,
we
get an adjustment, equal to
the effective rate of discounting, r
+ tt*
—
5,
ttj
— tt,
times the duration of that term, which
plus the speed
is
of mean-reversion of the risk-premium.
Time- varying equity premium and dividend growth rate
3.3
We now
merge the two previous examples, to incorporate both a time- varying equity premium and a
time-varying dividend growth rate. The stochastic discount factor and dividend are given as follows:
where
gt follows
=
exp
{-rt-
Dt
=
exp
(/' gsds - -^ds + asdBs
risk
-^ds -
/
XsdBs
/
)
(15)
Dq,
the quadratic process
dgt
The
rt
ft )2
Qt
premium,
ttj
=
= -$ (gt - g*) dt -
- g*f
{gt
+ dNt.
dt
XtCti follows a one-factor quadratic process:
d-Kf
-(j)
{iXt
-
TT*)
dt
+
(TTt
-
TT,)
dt
+ dnt
nt, Nt are martingales satisfying the familiar quadratic conditions. Assume that the processes
nt,Nt and Bt are independent. Then, the price of a stock, Vt = Eq [J^ QtDtdt] /Qq, is, by the
where
reasoning of the previous section.
Vt
=
exp
Et
{r
Define 5,
=
r
-|-
tt,
V.ID.
—
=
Formula
5*.
^
1
_
+ Tru-
9u)
du
1
ds
u=t
L"'5=f
(5) gives:
^i ""'^*
3t
,
5*
*
+$
_
(TTt
-
{S.
ttQ
+
(j>)
{gt
(5*
-
+ + ^)
+ ^){S, + 4> + $)
g,) (25.
cj)
(16)
= 1/ (1 + tt, — g^). It is modified by the
premium, and the growth rate of the stock. A stock with a currently high
growth rate gt exhibits a higher price-dividend ratio, and this is amplified when the equity premium
is low, as shown by the term (ttj — T^*){gt — 9*)In^the above process, ^("ffleaTFreverts, so the^rice/dividend ratio mean-reverts in (11). There
is much recent literature on the predictabiUty of returns, in particular the predictabihty of the P/D
ratio (Campbell and Shiller 1988; Cochrane 2006). The above setup may serve as a useful benchmark
The
central value
is
again the Gordon formula, Vt/Dt
current level of the equity
to evaluate the debate.
^Menzly, Santos and Veronesi (2004, Eq. 20) have a similar expression, even though the processes are quite different
in their details.
.
Correlation between the dividend and the discount factor
3.4
We
next study an example where the innovations to the dividend are correlated with the discount
factor.
We
suppose:
-—^ = gdt-V dNt
We assume a quadratic process:
with 7 is a fixed number, and Nt a martingale.
dMt, and
'
dDt
\
-—-,drt) /dt
,
for
two constants a and
Proposition 5 The price
E
at time
=
J\
\e- f' -'^"D J
L
where
A'
and
f
>t
+
=
=
^'
X
+
e-^'(^-')
"-^)
- \
ji
/J.
—
g
—
/x)
D,
(17)
J
a
(18)
{X-g)iix-g) +
b
(19)
are the solution of
e-lX + ii-g-a]( +
Proposition 6 The
price at time
t
iX-g){i.L-g)
of a dividend claim yield
,-i:ruduj^
E
Proof. By direct integration of
A
When gt
the P/D
0.
at every s
>
t
is:
+ ii-g-a-rt
Dt.
{X-g)ifi-g) + b
(17).
quantitative illustration
is
at its long
run value,
g^,
ratio also varies with the
Eq. 10 gives the simple price dividend ratio 1/ {R — gt). However
rate. To assess the quantitative importance of this effect,
growth
consider the following calibration using aimualized values:
2% and
an equity premium of 6%; g^
typical values of long-run
also take
Ds
+b=
X
,
LJs=t
of
—
is:
X
^
AV
3.5
A) (r^
^' satisfy
X'
i.e.,
s
+
4^
—
e-^'(-')
—
correlated with the interest rate.
of a claim at date
t
[rt
= an + b
,
may be
Dividend growth
b.
,
=
drt
</>
=
GDP
5%, which gives a
Vt/Dt
=
=
R=
growth. This yields a central
half-life of
1
R-g.
+
mean
g.)
P/D
reversion of \ii2/4>
9t
{R -
8%, which
reflects a risk-free rate
3%, the long-run growth rate, which
{R -
=
g,
+ 4>)
20-1-
ratio of:
=
1/
is
proportional to
{R —
g,)
=
14 years. This gives
200
{gt
-
a)
20.
We
dt+
With a
—
volatility ag^
deviation of
4,
and the
2%, we get a{V/D)
volatility of returns
-.„(v./oo
is,
200(7g
=
=
g*
at gt
^3^ =
=
4.
The P/D
10-.
=
ratio has
an annual standard
20%.
Application: Long term gro^vth risk
3.6
=
Consider dCt/Ct
such a model
with
gt,
=
Thus Ct+r/Ct
=
exp
gsds).
(J^
dT=j exp(-p5-(7-l)y
^^'^i-^]
J
we
gt stochastic.
The
price of a stock in
is:
Vt/Ct
If
—
gsdsUT.
define:
n = P+(7we obtain Vt/Ct
=
/q°°
e^Jt
'^"'^^ds,
l)5i,
the perpetuity price.
We study the process in two representative cases.
(Ttdzt, with A =
+ 75* < we have:
For a quadratic process, drt
=
[rt
—
A) (rj
—
/z)
(it+
/i
/O
A \
Therefore, the volatility of the
P/E
ratio
is,
/U
at the central value rt
=
A:
0"r|r=A
'^dlVi/Wt
/^
We
calibrate using
which
ratio of 1/A,
dr
= —
i.e.,
(/J
—
A) (r
—
is
3,
14.3.
A) dt
a time scale of 1/
=
7
{jj,
+
—
=
5
cr\dzt.
X)
2%,/)
Around
=
We
A,
=
1%, which yields A
=
7%.
We
choose a central
P/E
the process behaves like a Ornstein-Uhlenbeck process,
select
jj.
= 15%
and a speed of mean-reversion
jj,
— X = 8%,
12.5 years.
We pick (Jr = 0.5%. This yields a standard deviation of growth rate of approximately: ar {2{fj, — X))
= (Tj./0.4. Taking 1.5% as the standard deviation of the growth rate, we have:
ar (2 8%)~
ar = 0.4 1.5% = 0.6%. With a growth rate volatility of ag = 0.5%, we obtain interest rate volatihty
= 15%, this yields a stock price volatility aw = y^ — 10%.
of ar = "fCTg = 1.5%. Choosing
'
fj.
4
The
multifactor quadratic process
Several factors are needed to capture the dynamics of bonds (Litterman and Scheinkman 1991) and
stocks
(Fama and French
1996).
Accordingly,
we study
process.
10
the multifactor version of the quadratic
'
.
Definition and basic properties
4.1
A
.
M" has
quadratic process X^ S
=
n =
the form
- AXt)
+ {n /3'(Xt-X.) + r,
dXt
{b
dt
r^) (Xt
- X,)
dt
+ dMt
(20)
(21)
n X n matrix with positive eigenvalues {5i)-_-^ „. b,P, G ]R",X, = A~^b e lR",r* £ R.
E[dMt] = 0, but its component dMu, dMjt can be correlated. As in
the one-factor process, the volatility of dMt must go to zero in some limit regions for the process to
be well-defined. We defer this more technical issue until later.
with:
yl is
Mt 6 M"
We
a
is
a martingale, so
calculate the price of a zero coupon, Zt (T)
Theorem
2 (Price of a bond for a
maturity T, given the state Xt,
n— factor
=
Et exp
(
—
process) The price at time
=
e-^*^
+ e-'-'
V
p-AT _
1
{Xt
J
We use a function / (A) for a matrix. The notation is standard.
= Yl fn^^, then / (A) = J2 InA'^. If A is in diagonal
QDiag
(/ (Ji)
...,
,
t
j
.
of a zero coupon bond of
is:
Zt (T)
pansion f (x)
then / (A) =
rgds
J^
/ (5„))
- X,)
If
/
is
form,
(22)
analytic and admits the ex-
A = QDiag {5i,
Some
4.2
perpetuity,
-- -/3' {A + rJn)-' {Xt - X,)
Pt
=
(23)
exEimples
Example
1.
Example
2.
and a
=
5n) ^~^,
fi-^.
Proposition 7 (Price of a perpetuity for a n— factor process) The price of a
L Zt (T) dT, is, with In, the identity matrix of dimension n:
Pt
...,
For n
=
1,
we obtain the
expressions of the one- factor quadratic process.
Dividend growth rate as a sum of mean-reverting processes
(e.g.,
a slow
fast process)
The growth
rate gt
is
gt
=
ff*
+ X] -^^^
dXit
=
-(piXudt
-
{gt
-
g*)Xitdt
+ dMu
a steady state value g^, plus the sum of mean-reverting processes Xu- Each
Ai, and also has the quadratic perturbation {gt — g^) X^dt.
Xit mean- reverts with speed
We
apply the Theorem
dividend
T
periods ahead
2,
with P'
=
(1,...,1),
A = Dia^
((;/>i, ...,
0„).
is:
"
Et [Dt^T] /Dt
=
e^-^
+ e^'^
^
1=1
11
1-e-*--^
^
X
The expected
value of a
'
and, for the price-dividend ratio of a stock:
=
^'/^^
Each component Xit perturbs the
^
R-9*
baseline
A
3.
dLt
cf>{Lt-rt)
=
.
R — g^,
(24)
]
which
+ {rt-r,f
[c^{r^-Lt)
the short term rate. Lt
The second equation
—
E
~{R- f\.
9*+9i)
is
1/
{R —
g*).
the term 1/
The perturbation
(i?
—
g*
is
Xit,
+ (^J.
short rate and a long rate
dr,
Tt is
+
\
Gordon expression
times the duration of Xi, discounted at rate
Example
f1
is
dt
+ dM}
+ {rt-r^){Lt-r^)]dt + dM\2
the long term rate. Because of the
expresses that Lt goes to a steady-state value
—
—
first
r,.
equation,
drawn to
rj is
The process
is
Lt-
enriched by
which make the process quadratic, hence making bond
prices tractable. In practice, it is reasonable to expect \rt — r^\ and \Lt — r,| both less than 5%, so
that the magnitude of the quadratic terms is less than 5%^ = 0.25% per year.
the terms
We
and
r,)
(rj
apply the Theorem
with
get,
{rt
r't
= rt-
r*, L't
=
2,
Lt
r*) {Lt
with
-
Xi'=(^'j,/3=(M,A=(^
Example
~
r,
^
ip
4.
rt
ip
J
'^'^
L
(
I'
(j)
—
ip
\(j)
(p
— ip
"^^'^
'''
+
r. (r,
r^ ^
"J'
r,,
(j)
Pt
r»),
(j>)
r, (r,
+
4>)
(r,
having a time-varying trend
-|-
ip)
In the post-Volcker era, interest rates tended
to have predictable trends of increase or increase, which may be captvued by the following trend
growth rate Sj of the interest rate:
with
first
A,
/U
>
0,
if
=
dst
=
and A
expression.
negative
drt
St
-|-
^ >
- r^f dt + dMu
[-XfJ.{rt- r^,) - {X + iJ.)st]dt+ {rt-r^)stdt + dM2t
stdt
0.
+
in
Economically,
Sj is
mean-reverts for two reasons:
the predicted trend in interest rates, as per the
first,
because of the
interest rates are too high); second, because of the
12
—
{X
—X/j, {rt
+ ^) st
—
r,)
term.
term
{st
becomes
We
apply Theorem
Z{T)
=
with Xi
2,
=
{n,
1
=
+
St
(r,
r,
(r*
Those examples show
Comparison
4.3
(1,
e-^'^
V
Pt
=
/?'
st),
it is
+
+
A
—
A
(/i
-
r»)
/
+ ^) (rt
+ ;u)
0),A={^
^
^
A)
We
.
)
'
fj.[fj.
get:
—
\)
A) (r.
quite easy to get closed forms with easy to interpret processes.
\vith existing processes
Comparison with the
The
affine class
afhne class (Duffie and
Kan
1996; Duffie,
Pan and
Singleton 2000; Dufhe 2002) comprises processes of the type:
dXt
wtw[
with
Tt
before,
=
u+ 0' {Xt - X,).
X, = A~^b, assumed
Under mild technical
with
r, where
situation
is
e R"^", {Ho,Hi) e M" x
]R">^",cr
e E.
We
(T)
=
bond
prices have the expression:
exp {-r.T
+ T {T)'
{Xt
- X,) + a^a (T))
if
Hi =
0.
In that case,
{T)
=
T
exp {-r.T
(T)
=
7
+ 7 (T)'
(T), with
{Xt
7
(T)'
=
P' [e'^'^
- X,) + a^a {T))
-
l)
M-
The
Then:
(25)
.
express can be contrasted with the expression for the quadratic process (20):
Z^^{T) = e-^'^{l+'r{TyiXt-X.)).
If
note, as
a (T) and T (T) satisfy a typically compUcated ordinary differential equation.
simpler
Z^^
The
A
b,Xt e M",
to exist.
conditions,
Z^^
= {b- AXt) dt + wtdzt
= a^{H[Xt + Ho)
7 (T) Xt
is
small, the two expressions are the same,
second order in
in fact affine,
a.
and
Hence, a quadratic process
vice-versa.
is
up
(26)
7 (T) Xt, and
to terms of second order in
a good approximation
if
the underlying process
is
In most cases, the two values are likely to be close, so that existing
estimates of parameters in the affine class can be used to calibrate quadratic processes.^
Comparison with Heath, Jarrow and Morton
forward rates are factors.
In the Heath, Jarrow and
In the quadratic process, bond prices are factors.
The
Morton model,
HJM
does not
'That equivalence gives a useful way to calculate easily functionals of quadratic processes, that can be expressed
One first works with the affine process, setting volatility to 0, doing a first order
= a + b{Xt — X,) + o{Xt — X,) -'r o\u ),
Taylor expansion of terms in {Xt — X.). One gets an expression; P
for some constant a, 6.
Then, one knows that for the corresponding quadratic process, the value of the asset is;
as a linear combination of bonds.
P^^
=a + h{Xt-X.),
exactly.
13
.
yield essentially
any closed forms, but has proven very useful to price options. Applications of the
quadratic process to options are not investigated in this paper.
4.4
Additional remarks
The
following relation highlights
Lemma
why
1 Suppose that process yt G
VT >
C. Then,
the quadratic process yields linear expressions.
'
=
-ffr^ds
Et
dLs
As Lt =
As a
yt,
We
linear
Ls ^
get
=
^''^'^
-
/dt
=
Cytdt
+ rtytdt,
for
some
q ^ q
matrix
e^^
T ds
yt+T
e'^^yt.
(27)
ys
^-ffr,ds
Et
corollary, consider
Lemma,
price of
we
[dyt]
0,
Et
Proof. Call Ls
Et
satisfies
M.''
+ CysdS + rsysdS)
^_rsysdS
m
^^'^yt.
which would have a constant
Dg
that an asset yielding a dividend
=
r* (l
price.
We
CLs
can easily
— P'A~^Xs)
using the above
see,
at each date
has a constant
s,
1.
close
bond
with the following Conjecture.
If true,
the quadratic term
is
the only
way
to obtain
prices.
Conjecture
1 Suppose that
Xt
is
a
Markov process
E" such
with values in
that the interest rate is
—
p {Xt), and bond prices are affine in Xt, i.e., there are deterministic functions a (T)
and r (T) with values in M and R" respectively, such that, for all X in the domain of definition,
a function r
\/T
> 0,E exp —
(
Jj
p{Xs)ds]
\
Xt
—
X =
a(T) +r{T)'X.
Then, Xt follows a quadratic
process.
5
5.1
Conditions to keep the process well-defined
The
With one
practical
bottom
factor, the process
is
line
well-defined
stays within r
if it
<
with f
f,
<
fx.
For an
n— factor
process, the following algorithm gives the conditions for the initial state vector to yield a well-defined
process.
First,
one diagonalizes the matrix A,
root not necessarily ordered.
Then, one
The
sets:^
process
is
The
Q = Diag
well-defined,
i.e.,
({n'l3)j /6j]
if
sets
A = QAQt^,
A=
with
Diag{5\,
eigenvector corresponding to eigenvalue 5j
This rescales the eigenvectors, so that
in
...,5„),
(Oij)^_j
with the
^.
fi-^
the initial value of
Vi= L..n,^
is
mm{5i,5j]
Xt
satisfies the
Qjk {Xk - X^k) <
the diagonal representation, r
14
n
=
5'
X
inequalities:
1-
(28)
which we
being in the "Root" region.
call define
=
In matrix form, one defines Tij
'"'"^^'' ''
(
and the
,
j
n
the sense that the inequality holds for each of the
If
criterion
FQ {Xt —
is:
X«) <
(1,
...,
1), in
coordinates.
the initial value of Xt satisfies (28), and increments are continuous, then
future
all
Xgyt
also
satisfy (28).^
The above
conditions are simply sufficient, but one conjectures that they are actually optimal,
domain
in the
of conditions requiring only
Some examples
For n
=
n
linear inequalities.
the condition
1,
just
is
condition for the stock with time- varying growth rate
With n
factors,
if
the process
Q =
the identity matrix,
is
wV^2^j=l
'
^jj
\~^^
(^Pj/5j),
A=
i.e.,
fj,,
gt
Diag
<
r,
The
(f>.
equivalent
with r — Y2Pi^i^
^"_j """^" ^ -^Xj <
{6i, ...,5n),
and the condition
is
:
Vi,
''hen
1,
^
i.e.,
^
'^ ^
1
max((5.,(5j)
When n —
Diag
diagonal,
is
i.e.
<
rt —
— g* > —(p-
rt
is:
1,
with
A = Diag (61,62)
X
I
and
r
=
J2i=i ^i^ii the root region
Zi + X2 <
and -iXi
1
+ X2 <
is:
1
62
In
Example
i?
—
with Pi
gt,
—
—1. Therefore, the condition
Vz
=
Example
rate, so to speak),
Diag
T=
(
{\/(j),(f>/'ip)Q,~^
(
-,
I
,
.
^1^=1
-^it,
,
one applies the
When
>
cp
and the long
ip
we
rate,
(the short rate
get
moves
5'
=
faster
{(f>,ifj),
fl
=
than the long
and the conditions become:
—— <
1
and
—
—<
;
;
\
and L cannot deviate from
1.
ip
(p
Intuitively, r
+
max(0i,0j)
3 of section 4.2, with the short rate
Q =
)i
J,
g*
is:
^^>-l.
l...n,V
^-r'
In
=
2 of section 4.2 with a dividend growth rate gt
=
criterion to rt
their central value, r,
by a value order of magnitude the
,
speed of their mean-reversion.
__
.
5.2
We
The
rest of this section justifies these claims. It
skipped in a
first
reading.
_
_
__
Notation and motivations
consider a diagonal process,
open subset of
We
may be
rescale
X
cases,
if
we can
rely
i.e.,
on
needed, so that P
=
A=
Diag{5i,
...,(5„),
with
6\
<
6,
and so that
r
=
^ 6iXi.
...
We
this case after diagonalizing A.
We
<
5n,
define
reason
and n > 1. In a dense
= and 5n+i = 6n(5o
first in
the deterministic
version of the process.
One can show
that another, strictly more stringent, sufficient condition,
1.
15
is
Vi
=
l...n,
X^"=i ^Si<Sj Qjk {^k
— X.k) <
.
We
use
n
tests vectors, F^^^
wr
...,
r("\ with Vi
= {x
G M"
=
1...,
=
l...n, F^')
Vz
e M".
n, F'*^
X
I
We
study regions of the type:
< l|
(29)
We
want to make sure that, if the process hits the boundary of to, noted duj, then it goes back inside
To express this, call m{X) = E [dXt/dt Xt = X] = {r - A) X the drift of Xt at X. If X £ du,
then call j the coordinate such that F'-^^ X = 1; then, we want to ensure F'^^ m[X) < 0. We
w.
\
calculate:
f(^)
to
•
=
(X)
f(^)'
{r-A)X =
= r-1we
So,
require, for all
i
>
1:
^
{Vj
i, F^^')
•
X
(aT^'A'
<
1
and
(1,
...,
impose this condition only for i > 1, because
1)', which requires a special treatment. Call St =
{X
region
The
iOr to
and
if St^^
X
l}
F'^^
|
following
Lemma
<
<
is
1,
at
an
initial
-
T^'^'AX
(S-AT^'A-X.
X=
l}
=^
Xt
<
date St^
=
1,
J2j
AF^'))
boundary
in practice, a
F'^^
-
{S
-^jt^
then for
X
then dSt/dt
all i
>
<
0.
of choice
=
to, S'f
is
F'^'
=
—
1).
rj {St
<
1.
So, the
automatically non-repelling.
completes the reasoning, by providing a sufficient condition for the region
be non-repelling.
Lemma
each
e R"
1,
X=
F^^)
We
So, St cannot cross
rV^'^'X
=
i
2 (Sufficient condition to get a non-repelling region) Suppose
2...n, there is a ai
>
and a non-negative vector
ip^^'
T'^^'
=
G M", with
(1,
^
...,
1)',
ipj
=
and that for
1;
Vi
<
1;
such that:
ai5+{l-
aiA)
F^')
=
^
i>fr^^\
(30)
j
Then, the process does not escape from region w.
Proof. Indeed, then
a, [5
5.3
- AF«)
X =
[
-F(^)
+ J2 ^f r(^)
•
X
The Root region
-.(i)
wr defined by F^.J = 5i/s,j/5i, where i Aj = min(i,j). Because it involves
the eigenvalues of A, we call it the "Root region". Heuristic arguments lead to the conjecture that
the Root region is optimal, in the sense that it may be the largest non-repelling region that can be
defined by the intersection of no more than n inequalities expressed as linear combinations of the
We
consider the region
factors.
16
,
Theorem
0Jr=
is
=
3 The Root region wp, defined by the boundary vectors Tj
{x eW\yi=
<
l...n,r(^'-X
5i/^j/5i
and
l|
non-repelling.
Conditions on volatility
5.4
In dimension
some
for
£:
0, or
(r)
o"
~
A;
(f
—
In dimension n, the analog
should be
v^^'
near the boundary.
is
wfH [X
i^(')of
=^ var
\
that
dMt) =
(r'^''
X=
Mi,T^^^
0.
It
a > 1/2 or a = 1/2 and
when r('''X( = 1, (while
r)° with
near the boundary. The simplest condition
neighborhood
Xt G
=
we require that var [dMt)
1,
>
is
can be
in a region
—
[//
e, /x],
k large enough.
X
€
aJr),
that, for each
i
=
then var
l...n,
(r''^'
there
•
dMt)
an open
is
l], such that
In other terms, the noise along the projection on
becomes
T^^'
0.
A
a simple
way
to ensure that the process
we
practice very small,
Ke (X) =
where, as always,
1
{A)
We start from an
well-defined
is
is
the following.
>
For an e
0,
in
define the killing function
equal to
is
A
1 if
1
I
is
max
T^^
•
X
<
1
- ej
(31)
otherwise.
true, or
arbitrary continuous martingale process, dMt, and then define a
new continuous
martingale:
dMt = K,{Xt)dMt.
become
In other terms, aU variances
region wp-
Otherwise, the process
if
is
X
is
(32)
within a sraaU neighborhood of the limit of the root
not modified.
This "killing" procedure might be the most
convenient to use in practice.
Condition with diagonalizable matrices
5.5
we diagonalize
and define the new
In the general case, given a diagonalizable matrix A,
Diag
{5i, ...,(5n)-
We set Q = Diag ((J7'/3)j /5i)-Q.''^,
it,
A = nAn~\
with
state vector to be: Yt
A =
= QXt-
Then:
=
=
5'Yt
As
r(')
rt
•
= 5'Yt,
y < 1.
5'
Diag{{n'(3)J5i)n-^Xt
l3'n-n-^Xt
=
l3'Xt
=
=
Diag{{a'f3)^)n-'^Xt
rt.
Yt satisfied the conditions of this section for the tests,
As
r(')
•
Yt
=
r'-'^'QXt
=
Ylj
^QjkXk,
that does not depend on the prior ordering of the roots: Vi,
condition announced in (28).
17
and the Root region
is:Vi
=
l...,n,
the condition can be re-expressed in a
^
'^
^
QjkXk <
1,
way
which gives the
The process
6
time
in discrete
This section studies the discrete-time version of the quadratic process, which
many
reasons. In
applications, discrete-time
technical issues
by discrete-time examples.
are clarified
The key
object
the stochastic discoimt factor
is
Ds
a stochastic dividend
of maturity
useful for several
is
The
simpler than continuous time.
is
T
S >t
at date
is:
mt 6
The
K!^.
price at time
Pt
= E [mt+i...msDs]- The
=
Et [mt+i...mt+T]
t
of a claim yielding
price of a zero
coupon bond
is:
Zt (T)
For instance, the gross short term interest from
=
time process, mt
exp
f
—
j^_.^
Vudu
1
to
t
-|-
i
1 is
(33)
l/Et [mt+i\- In terms of the continuous
.
Discrete time, one factor
6.1
The most obvious generalization
we propose is:
Emt+\ quadratic
of the process,
in
does not work. Instead, the
tti^,
generalization
Etmt+i
=
a
< a < 6. We assume that
The expected value of m,t+i is
with
the other hand,
1
^
mib (mt
—
if
there
is
no
-), so that, for
the process
-^
such that for
is
in a
then
m,t goes to the attractive root, 1/a.
1
p \mt
a
=
p
—
a/h.
Indeed, Et [mt+i]
On
— l/a
However,
its
j
+oimt
a/feG(0, 1).
(34) behaves like a autoregressive process
lation coefficient p
mt > 1/b}^
times,
all
neighborhood of 1/a,
Et mt+i
Hence
(34)
abrrit
increasing in mt, so the process has a positively autocorrelation.
noise,
mt
+ \-
m, = 1/a, and autocorrebond and perpetuity prices,
AR(1), with central value
functional form yields tractable
shown by the following Proposition.^^
as
Proposition 8 (Price of
There are many ways
series of
a zero-coupon bond in the 1-factor, discrete time case) The price of a zero
to ensure that
mt remains above
independent, positive random variables
rj^,
with
E
a lower
[tjJ
=
1,
bound
and
m
set;
S [1/6, 1/a).
mt+i
=
[^
+
The
j
simplest one
— ^^
m)
rj ^
is
to get
+ rn. As
m > and rrtt+i > m. The reason to impose mt > 1/6 is that, if we had mt < 1/6
mt > me [1/6, 1/a), ^ + ^ — ^i^
some t, in the deterministic version of the process, there would be a time s > t such that m., < 0, which is not
for
allowed for a discount factor.
"The
1/ (1
-
correspondence with the one-factor process
T/3)
,mt
=
rEt
i.e.,
E [n+i -
Tt]
1
-
[rt+i
=
{n
rrt.
is
the
following;
for
a
small t,
a
-
/3)
= 1/(1—
Then, (34) reads;
-rt]
= - {E [mt+i] - mi) =
-a){n-l3)T +
(r),
i.e.,
I
\a
E
[dn]
mt)
{
- - mt]
J \o
=
18
(n
-
—=
J m.1
a) (n
-
/3) dt.
(ri
q) (n
-
1
—
TTi
ra),
6
=
coupon bond of maturity
T
is:
=
Zt (T)
Summing
over the maturities,
Proposition 9 (Price of
we
—- — a'— +
_j^u
a-'
iilf
~
,-T''^t
b-'
"^
-ib — a
get the price of a perpetuity.
(35)
.
12
The price of a perpetuity
a perpetuity in the 1-factor, discrete time case)
is:
a + b — I — m7
„
^-= (a-l)(6-l)
The above discount
factor can be enriched
by a stochastic component,
This section studies the case with a number of factors n
a stochastic discount factor
with eigenvalues in
i?*
(2«)
as in Eq. 15.
Discrete time, n factors
6.2
(i)
.„.
•
-
=
+ r, >
1
0,
(
— 1,1);
(iv)
mj S
R*^_;
a vector
/3
>
€ R"; and,
A
1.
quadratic process
Xt £ M";
a state vector
(ii)
(iii)
a matrix
is
given by:
F £ E"^",
a central value of the gross interest rate
(v)
with:
Rtirit
mt
We
assume that the process
is
= ^{l-p'Xt).
well-defined for
all ^'s.
The
conditions that ensure this are analo-
gous to the continuous case, and will be specified in the next iteration of the paper.
To
get
some
intuition for the process,
steady state value of
roots, then
Lemma
vr >
Xt
is
X
is 0,
we
yt
£
R'^satisfies
Et
[yt+i]
is
=
Proof.__Call Lt- the left-hand _side._I/i
=
=
= _E\yt+\\
=
Pt
—
is
In general,
contractive, the
if
T has positive
mt term.
Et [mt+i...mt+Tyt+T+i]
=
Et {mt+i...mt+T-iCyt+T]
Et [mt+i
(1
+
= C^
qxq
matrix
—
mt
C
.
Then,
(37)
=.Cytlmt, and
Et {mt+i...mt+T-iEt+T [mt+TVt+T+i]]
= CEt
Alternatively, one can try a price process of the type Pi
equation:
1//J,.
Cyt/mt, for some
0,
Et [mt+i...mt+T-i2/t+r]
'
mt
positively autocorrelated, with a small modulation introduced by the
3 Suppose that process
Lt+i
As T
consider the steady-state.
first
so the steady-state value of
=
Pt+i)]-
19
A-\-
= CLt-
[mt+i...mt+T-iyt+T]
B /mi,
and solve
for A,
B
that satisfy the functional
Theorem 4
(Price of a zero-coupon bond in the multifactor, discrete time case) The price of a
zero-coupon bond of maturity
Zt (T)
Proposition 10 (Price
price of a perpetuity, Pt
Proof.
a
+
= r:^ - R-J-'p'T (/„ -
=
'}Zt'=i
(38).
b'Xt/mt, and solve for a 6
+
Pt+i)].
with /„ the identity matrix of dimension n:
is,
r)-i (/„
- r^)
^.
^t (^)'
^'''
''^'''l-^
(38)
mt
of perpetuity in the multifactor, discrete time case)
One simply sums
Et[mt+iil
7
T
Assume
i?,
>
1.
^n the identity matrix of dimension n:
Alternatively, one can try a price process of the type:
K
and
6
The
e R" that
satisfy the functional equation:
=
Pt
Pj
=
—
m
Conclusion
Quadratic processes are very tractable, as they yield closed forms
are linear in factors.
They
for perpetuities,
are likely to be useful in several parts of economics,
rates, or risk premia, are time-varying.
and
when
prices that
trend growth
Hence quadratic processes might be a useful addition to the
economists' toolbox.
20
.
Appendix A. Regularity conditions
for the one-factor process
This appendix details conditions for the existence and uniqueness of the solutions.
We recommend
Katatzas and Shreve (1991 Chapter 5.5) and Revuz and Yor (1999, Chapter IX) for systematic treatments, and Ait-Sahalia (1996, Appendix) and Dufhe (2001, Appendix E) for pedagogical overviews.
of
V=
(r, r) the domain of existence of r, and c an arbitrary point in V. We call fi (r) the
and assume dMt = a (rj) dzf. We make the following assumptions.
(i) The drift and diffusion functions are continuously differentiable in r in V, and a^ (r) >
We
call
drift
r,
in
D.
The integral of m (r) = exp ( JJ^ 2/i {u) ja"^ (u) duj /a'^ (r) converges at both boundaries
(iii) The integral of s (r) = exp (/J 2^ (u) /a^ (u) du) diverges at both boundaries of V.
of V.
(ii)
(iv)
for
is
ij,
any e >
If
Lipschitz continuous, and there
0, /,q
conditions
.
p {x)~ dx
=
+oo, and
a function p {x) R-|- —>
(x) - a (y)\ < p{\x - y\)
is
|cr
are satisfied, then there
(i)-(iv)
R-j.,
:
is
a unique Ito process {rt,t
solution of the stochastic differential equation (1) with initial condition ro
is
Markov.
The key substantive point
This condition
is
crucial, as
if
that the process
is
we
started with ro
>
with p
is
/?,
defined for
(0)
=
0,
such that
.
>
all i
=
>
is
a strong
Moreover, {rt,t
r.
0,
0} which
>
0}
and does not explode.
the process would explode in finite time with
would not be defined for all times.
guarantee the existence and uniqueness of the solution up to the
positive probability, so that the process
Conditions
variable
The
may
intuition
(i),
(ii)
and
(iv)
hit the boundaries.
is
Condition
(iii)
implies that the boundaries are actually not reached.
as follows. Consider the correct boundary. Condition
the process tends to return inside P, and also requires that
Sufficient conditions to ensure (i)-(iv)
differential equation (1)
Conditions
(iii)
imphes
tends to
a'^
(r)
(i)
and
(ii)
fast
p.
(f)
<
enough
0,
so that
as r 1
f
guarantee that the stochastic
admits a unique strong solution. Those conditions are verified in the following
(iii) guarantees that the end points V of are natural boundaries.
assume p{f) < and limr^r p (r) > 0, so that close to the end points of V, the process tends
to go back inside V. In the case p{r) — [r — a)[r — /3), with a < P, this corresponds to f G (q,/3)
and r e [— oo, a).
Conditions (ii) and (iii) are verified if the following conditions [C-V) hold. For r in a leftneighborhood of r, cr^ (r) ~ /c (f - ry, with (k > 1 and fc > 0) or (k = 1 and
< fc < -2m (f)). If
cases.
Condition
We
> — oo, for r in a right neighborhood of r, a'^ (r) ^ k' {r — r)'^ with (k' > 1 and k' > 0) or (k' = 1
and
< fc' < 2m (r)). If r = — oo, then r is a natural boundary if, for r in a neighborhood of — oo
_(j^(r) ~
\r\_, w[th k>
and P < 3, Those last conditions imply assumptions (ii), (iiij.
For r = — oo, the situation is complex for condition (iv), as the standard conditions found in
r
,
fc
textbooks do not apply,
p
(r) is
not Lipschitz continuous, as p'
that a simple weakening of condition (iv) will allow the case r
If r
>
— oo, the
for a large
above conditions (C-V) also imply
enough constant K.
21
(iv), as
=
[r) is
unbounded.
We
conjecture
— oo.
one can take p{x)
=
K max (x"'^,
x'^
'
,x
Appendix B. Proofs
Proof of Theorem
7.1
It
is
sufficient to
Vt
=
Et exp
(
—
Jj
1
prove the statement for an
r^ds)
,
for
G
t
[0,T].
The
-nVt +
We
seek a solution Vt
= Wt
condition
Wt =
arbitrage equation of Vt
wirt,t)
=
A{t)rt
imphes A (T) =
and
which in the case dMt =
+
=
(r
—
a) (r
in the functional
fusion)
is 0.
The
—
form
PDE
is
/3),
we have
{r, t)
+ Aw (r, t) +
Ati; (r, t)
= A (t)
(41), the Ito second-order
Q=
B{t)
(41)
cr
1
.
We
call
A
the infinitessimal
[rt) dzt is:
^/"
(r)
=
=
—w {w,
(r
—
a)
{r
t)
=
—
0). Importantly,
term (equal to ^—^d'^w/dr'^
—w {w,
in the case of a dif-
t)
+ B{t)] + A{t){r-a){r-p)+[A'{t)r + B'{t)]
+ B' it)] + r {-B [t) - (a + /3) A (t) + A' (t)]
-r[A{t)r
[aPA
(i)
Importantly, the r^ terms cancel out. This explains
why adding
a quadratic term to the standard
Ornstein-Uhlenbeck process substantially improves tractability. The
and only
because d'^w/dr'^ =
with:
Q = —rw (r, t) + Aw (r, t) +
if
is;
(40) becomes:
—rw
Given Ar
and define
(40)
B (T) =
Af (r) = {r-a)ir- P) f (r) +
PDE
T >
fix
E \dVt] /dt =
1
diffusion operator of the process,
The
We
0.
with the functional form:
Wt =
The boiondary
time equal to
initial
PDF Q =
is
satisfied for all r
if:
al3A{t)
+
B'
{t)
-^Bii)^- (q +-,0)-A(t)-+ -A^i)Differentiating the last equation,
we
=
^
get:
-B'{t)-{a + p)A'{t) + A"{t) =
which imphes, via
a^A
(t)
+ B' [t) =
a/3
-0
Q
0:
A (t) -
(q
+ /3) A' (i) + A" (i) =
22
0,
-(^)-
,
which
satisfied
is
by a function
e^* if
=
a and
so that the roots are
C
equations for c and
it
is
that
- nWtdt] =
0,
(a
Wt
=
Eq
=
^'
-
(^
-
a) (^
/3)
some constants c and C, A
1 and (42). We get:
=
(t)
'-
^e"^'-^
P-a
+
—rtWt
satisfies
[dLt]
=
[Lt],
i.e.
+
'
E[dWt]/dt
To conclude the
we have Et
martingale, which implies, Lq
+ /3) e +
(i)
=
ce"'
+
Ce^'.
The
A (T) = 0,5 (T) =
= A (t) n + B
the equal to the bond price.
Et [dWt
-
if:
Therefore, for
^.
are given by
Wt
The above shows
a/3
and only
—
we
proof,
P-a
exp
(
—
Wt = 1, but not yet that
Lj = exp — /q Tsds\ Wt- As
and
0,
define
Et [exp (- /J r,ds) {dWt
Wq = Eq
j^
e
(
- nWtdt)
rgds
Wt =
j
0.
Hence Lt
is
a
En
I.e.
p — a
exp
p - a
r.ds
the statement of the Theorem.
which
is
7.2
Proof of Proposition 3
The Proposition is a direct re-expression of (3), applied to the process rt = R — gt which follows,
= R - 5«: drt = -4>{t* - r) dt + {rt — r^)^ dt — dMt.
also
present an alternative, arguably more direct derivation. We look for a formula: Vt of the
We
with r^
type
Vt
=
v {Dt,gt)
= Dt{a + b {gt - g.))
which implies:
dDt
= E
E\dVt]/dt
/dt
Dt
=
gt{a
Expressing the above in terms of
_
--Q -
-
is
{gt
—
5
-J-- ^(a-+
-(^j"^^))
= [1-Ra + g,a] +
which
satisfied if
+ hE \dgt]
/dt
Dt
{gt
and only
-
+
[(^^
5.)
+ b{gt- g,)) + b
g*) rather
than
_ ^^y+
^^] [(a
+
+ bg,-
[-Rb
a
gt,
+
-4>{gt
the
i (^-
60]
+
-
g*)
PDE
is
-gM+
{gt
- g^f
-
{gt
-
g*)
Dt — RVt
b [(^ (gt
[b
-
-
+ E [dVj]
g»)
-
{gt
/dt
=
0:
-'g*f
b]
if:
— Ra + 5,a =
-Rb-^a + bg^-bcf) =
1
The term
{gt
—
g*)
cancels out, which justifies
its
0.
inclusion in the definition of the quadratic process.
23
.
The
last
two equations
give:
= \/{R-g.)
= a/{R-g^ + 4>),
a
b
I.e.
''-'*
,^(i+
R- \ R-
=
v,
g^
g*
Proof of Proposition 5
7.3
This
is
Theorem
a simple corollary of
the initial time to
0,
g
=
0,
Dq =
and Girsanov's theorem. To simphfy notation, we normalize
1
Define:
1.
^
jj^
where IN]^
is
the quadratic variation of
A''
up
eiv.-lN]./2
to time
t.
dM^ = dMt-d (Mt, Nt) =
Define
dMt - [an +
h) dt
then
dn
(r
-
{r'^
(r-
drt
Let
Q
Girsanov's theorem,
7.4
^-J^'r^dujy
dM^
-
n') dt
+ dM^
is
(43)
dQ =
-S^Tudu Nt-\N]J2
a martingale under Q, so
=
e^'
'^"/•^dP:
-
S<3
Theorem
1,
/('
Tu du
applied to process (43) gives:
Proof of Theorem 2
It is sufficient
to consider
to consider r*
=
bond
A') {r
id)
denote the probability with Radon-Nikodym derivative
V=E
By
-
+ dMt
{X + IJ.-a)r + Xfi + b)dt + dM^
A) {r
-
0,
X« =
by redefining
of maturity T.
The
6
=
rj
by redefining Xt to be Xt — X^ if needed. Also, it is enough
to be Tt — r,. We consider the Z {Xt, T) the price of zero-coupon
0,
arbitrage equation for Z\s:
=
-TtZ
+ E [dZt]
/dt.
guess the functional form Z {Xt,T) = 1 + 7' (T) Xt, for a function 7 (T) with values in IR" to
be determined. With this functional form, E [dZt/dt] = 7' (T) {-AXt + nXt) - drZ {Xt, T), so that
We
24
PDE
the
becomes:
= -r,{\+^'{T)Xt)+i'{T){-AXt + rtXt)-^l{T)'Xt
= -n-j'mAXt-j^yiTyXt.
The rtXt terms
we obtain:
One
cancel out, which justifies their inclusion in the quadratic process. Using
=
-/3'-y(T)A-^7(r)' =
^
jt^iT)'
7.5
We
bond
finally the earlier expression for the
=
e-^^ -
1
/3'
A
price.
Proof of Proposition 7
integrate (22):
Pt
=
/
Zt{T)dT=
Jo
=
e-'-'^dT+
Jo
= ^+P'
7.6
boundary condition 7
gives:
7(T)'
(f
e-'--^/?'
Jo
e-('--+^)^
- e-^'^dx)
J
(Xt
A
(Xt-X,)dT
- X.)
+ P'(^r^--)U^t-X.) = --P'
r.
\r^+A rtJA
r.
r^ [r^\
+ A)
Proof of Theorem 3
=
We
use the convention Jq
We
check the conditions of
and Sn+i
Lemma
J
2.
=
ip-
5n-
>
We
0,
=
(3'Xt,
-'y'(T)A-p'.
integrates this matrix equation hke a scalar equation, using the
which
and
=
rj
define qj
and
=l
25
=
1/(5,4. i
^^'^
(Xt-X.)
(0)
=
0,
We
next calculate each side of (30), evaluated at the k—th. coordinate.
LHS,-1 = (a.5+(l-a,^)rW)
^i+i
and, using J2j'^j
=
-
1
/^ +
=
fl
-
We
obtain:
i^)
^
-
1
Si
J \
!>
RHS,-i = E^fr^^-E^;' =
J
E^f (ri'^-i
3
3
^J]<5,-(<5,+i-5,_i)l,<,(^-l
SiSi
If
k
> L RHSk -1 =
= LHSk -l.llk<i,
Sk
RHSk-1 = yj—y]Sj{Sj+i-6j_,)l k<j<i \j:-i
,
"^
k+l<i<i
1
5i5i+i
k+l<j<i
\k+l<j<i
1
[(5j+i
+ 5^-
5k+i
[((^i+i
+ 5i -
(5fc)
-
5k) 6k
-
6i+i5i
+
Sk+iSk
5i5i+i
1
-
5k
1
5i+i5i
{5i+i
5i5i+i
LHSk So,
7.7
We
we
get
V/c,
i^iJ^t
-
1
-
5k) {5k
-
5i5i+i
1
= LHSk -
1, i.e.,
(30) holds.
Proof of Proposition 8
have:
Zt (T
+
1)
=
Et [mt+i...mt+T+2]
=
Et [mt+i...mt+T+iEt+T+i ['mt+T+2]]
1
Ef.
mt+i...mt+r+i
I
a
a
ZtiT +
l)
=
{l
+ -]
+
1
1
b
abrnt-f-T+i
Et [mt+i...Tnt+T+i]
T-Et
ab
I
+ l)zdT)-^^Z,iT-l).
26
[mt+i...mt+T]
5i)
- ^, which has roots l/a and 1/6. Hence, the solution
is: ^^ =
(^ + ^) ^
=
Aa~^ + Bb~^ for some constants A and B. Using the values Zt (0) = 1,
is of the type:
Zt (T)
and Zt (1) = Et [mt+i] = l/a + 1/6 — 1/ {abmt), we get the announced formula.
The
characteristic equation
,
Proof of Theorem 2
7.8
= Etlmt+i...mt+T] = Et[mt+i...mt+T-iK^{l-p'Xt+T)]
= R~^Et \mt+i...mt+T-i] - K^P'Et lmt+i...mt+T-iXt+T]
ZtiT)
V Xt
r
\R*J
as the previous
Lemma,
applied to Yt
=
Xf and
mt
C =
F/i?,, gives Et [Tnt+i...mt+T-iXt+T]
=
(i^j'^^^-Thus,
R'^Z (T)
= R^-^Zt
(T
-
-
1)
R^^pT^—
mt
and
so,
summing from
1
to T,
RlZt
Appendix C.
A
(T)
=
1
- R-^P'
=
y
^ r^^
mt
1
- R-^/3'r^JL^^,
-T
In
mt
admitting a closed form for per-
class of processes
petuities
The
following Proposition shows a
way
to generate processes that have closed form for perpetuities.
However, typically that functional process does not yield a closed form
for finite-maturity claims,
unlike quadratic processes.
Proposition 11 (Class of processes admitting a
a process iptt <^''^d constants a and /3 such that
dipt
where
dMt
is
=
+
{rtipt
an adapted martingale, and
is
art
is
a solution of the perpetuity
that
make
PDE:
1
—
rtVt
the process well-defined, then Vt
Proof. Call
Vt
=
(V't
1
-
-
=
dt
perpetuities) Suppose that there
(44)
/dt
=
0.
If
dMt
satisfies regularity conditions
the price of a perpetuity, Vt
=
Et
J^oOe-/>udu^5
+ a) //3.
rtVt
+ E [dVt] /dt=\-
is
+ dMt,
^
+ E [dVt]
is
(3)
form for
essentially arbitrary except for technical conditions.
'Then:
V.
closed
rt'^^^j^
27
+^
(r^^t
+
o^n
-
/3)
=
0.
Vt
A first example is the, quadratic process: Tp{r) = r, a = —
= (X + — rt) / (A/i), the formula 3 for perpetuities.
A second example is
d{l/rt) = <P{rt-r*)dt + dMt
(A
+ /x),5 =
Eq.
—Xfi.
44 gives
fi
where dMt ensures
tuities
equal
rt
>
0, i.e., rt
mean-reverts to a central value
r*.
This yields a price for perpe-
to:
F.=
by applying Proposition 11 to
ip
interpretation. If rt remains at
long run central value r*
,
=
(r)
its
1/r,
^)/(l + ^)
(^ +
a = ^,b = l +
The formula
(f)r*
.
(45)
(45)
current value, the perpetuity price would be
admits the following
l/r-f
.
If rt
was at
its
the perpetuity price would be 1/r*. In general, the perpetuity price (45)
sum
of those two perpetuity values,
indicates the speed of mean reversion,
and gives the weight between the "current value" 1/r-j and the "long run value" 1/r*.
is
a simple weighted
(f)
References
Ait-Sahalia, Yacine, "Testing Continuous-Time Models of the Spot Interest Rate," Review of Fi-
nancial Studies, 1996,
Bansal, Ravi and
9,
Amir Yaron
385-426.
A potential resolution of asset pricing
(2004). "Risks for the long run:
puzzles," Journal of Finance 59, 1481-1509.
Barro, Robert, 2006, "Rare disasters and asset markets in the twentieth century". Quarterly Journal
of Economics, 121.
Brigo,
Damiano and Fabio Mercuric,
New
Interest rate models, Theory
and
practice.
Springer (Berlin,
York), 2001.
Brown, Stephen
J.,
William N. Goetzmann, Stephen A. Ross, "Survival", Journal of Finance,
50,
1995.
Campbell, John Y. and John Cochrane "By Force of Habit:
A
Consumption-Based Explanation
of
Aggregate Stock Market Behavior", Journal of Political Economy, 1999.
Campbell, John Y. and Robert J. Shiller, 1988, "The Dividend-Price Ratio and Expectations of
Future Dividends and Discount Factors," Review of Financial Studies 1, 195-228.
Cochrane, John, 2006, "The Dog That Did Not Bark:
A
Defense of Return Predictability,"
NBER
WP.
Colhn-Dufresne, Pierre and Robert
Theory and Evidence
for
S.
Goldstein, 2002,
Unspanned Stochastic
Cox, John C, IngersoU, Jonathan E,
Jr,
"Do Bonds Span the Fixed Income Markets?
Volatility,"
Journal of Finance 57.
and Ross, Stephen A, 1985.
"A Theory
Structure of Interest Rates," Econometrica, vol. 53(2), pages 385-407, March.
28
of the
Term
Croce, Massimiliano, Martin Lettau and Sidney Ludvigson, "Investor Information,
and the Duration of Risky Cash Flows,"
Driessen, Joost, Pascal
NYU WP,
Long-Run
Risk,
2006.
Maenhout and Grigory Vilkov, "Option
Working Paper, 2005.
-
Implied Correlations and the Price
of Correlation Risk," Insead
Dynamic Asset Pricing
Duffie, Darrell,
Duffie, Darrell
Theory, Princeton University Press, 2001.
and Rui Kan, "A Yield-Factor Model of
Interest Rates," Mathematical Finance, vol.
6 (1996), no. 4, 379-406.
Duffie, Darrell,
Jump
Jun Pan and Kenneth Singleton "Transform Analysis and Asset Pricing
for Affine
Diffusions," Econometrica, Vol. 68 (2000), pp. 1343-1376
Fama, Eugene and Kenneth R. French, "Multifactor Explanations
of Asset Pricing Anomalies,"
The
Journal of Finance,1996.
Gabaix, Xavier and David Laibson, "The
economics Annual,
MIT
Bias and the Equity Premium Puzzle," NBER MacroBen Bernanke and Ken Rogoff ed., vol. 16, pp. 257-312.
6D
Press, 2002,
Gabaix, Xavier; Gopikrishnan, Parameswaran; Plerou, Vasiliki and Stanley, H. Eugene. "A Theory
of
Power Law Distributions
in Financial
Market Fluctuations." Nature, 2003, .^25(6937), pp.
267-230.
Gabaix, Xavier; Gopikrishnan, Parameswaran; Plerou, Vasiliki and Stanley, H. Eugene.
tional Investors
and Stock Market
Volatility."
"Institu-
Quarterly Journal of Economics, 1999, 121
(2),
pp. 461-504.
Hansen, Lars, John C. Heaton and Nan Li "Consumption Strikes Back?; Measuring Long
Risk,"
WP,
Hansen, Lars, and Jose Scheinkman, "Long Term Risk: an Operator Approach,"
Heath, David, Robert Jarrow,
Rates:
Run
2005.
A New
(Jan., 1992)
,
Andrew Morton, "Bond
Methodology
for
Pricing and the
Term
WP,
2005.
Structure of Interest
Contingent Claims Valuation," Econometrica, Vol. 60, No.
1
pp. 77-105
Julliard, Christian,
and Jonathan Parker, "Consumption Risk and the Cross-Section of Expected
Returns," Journal of Political Economy, 113(1), February 2005.
Karlin,
Samuel and Howard M. Taylor, 1981, A second course
in stochastic processes,
Academic
Press.
Litterman, Robert and Jose Scheinkman,
"Common
factors affecting
bond
returns,"
Journal of
Fixed Income, June, 54-61, 1991.
Menzly, Lior, Tano Santos and Pietro Veronesi, "Understanding Predictability," Journal of Political
Economy, February 2004, pp. 1-47
Parker, Jonathan, "The Consumption Risk of the Stock Market," Brookings Papers on Economic
Activity, Vol
2,
(2001) 279-348.
29
Pastor,
Lubos and Pietro Veronesi, "Rational IPO Waves," Journal of Finance,
60,
August 2005,
1713 - 1757.
Vasicek, Oldrich
Economics
"An equilibrium
characterization of the term structure,"
5 (1977) 177-188
30
Journal of Financial
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