DEWET
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MIT
Sloan School of Management
Working Paper 4340-01
December 2001
PRICING AMERICAN OPTIONS:
A DUALITY APPROACH
Leonid Kogan and Martin Haugh
© 2001
by Leonid Kogan and Martin Haugh. All
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MASSACHucFlTS INSTITUTE
OF TECHNQLOGY_
J
UN
3 2002
LIBRARIES
Pricing American Options:
A
Duality Approach^
Martin B. Haugh^ and Leonid Kogan*
Abstract
We develop a new method
for pricing
American options. The main
practical contribution
a general algorithm for constructing upper and lower bounds on the true
price of the option using any approximation to the option price. We show that our
bounds are tight, so that if the initial approximation is close to the true price of the
of this paper
option, the
is
bounds are
also guaranteed to
be
close.
worst-case performance of the pricing bounds.
is
We
also explicitly characterize the
The computation
of the lower
bound
straightforward and relies on simulating the suboptimal exercise strategy implied by
the approximate option price.
The upper bound
is
also
computed using Monte Carlo
by the representation of the American option price as a
solution of a properly defined dual minimization problem, which is the main theoretical
result of this paper. Our algorithm proves to be accurate on a set of sample problems
where we price call options on the maximum and the geometric mean of a collection
of stocks. These numerical results suggest that our pricing method can be successfully
simulation. This
is
made
feasible
applied to problems of practical interest.
*An
earlier draft of this
Approach
.
We thank Mark
paper was
titled
A
Duality
Wang and
seminar
Pricing High- Dimensional American Options:
Broadie, Domenico Cuoco,
Andrew
Lo, Jacob Sagi, Jiang
Columbia University, INSEAD, London Business School, MIT, Princeton, Stanford, and
Wharton for valuable comments. We thank an anonymous referee for pointing out the alternative proof
of Theorem 1 in Section 3.1. Financial support from the MIT Laboratory for Financial Engineering, the
Rodney L. White Center for Financial Research at the Wharton School, and TIAA-CREF is gratefully
participants at
acknowledged.
We
are grateful to Anargyros Papageorgiou for providing the
generate the low discrepancy sequences used
in this
^Operations Research Center, MIT, Cambridge,
FINDER
software to
paper.
MA
02139; email; mhaugh@alum.mit.edu.
'Corresponding author: Sloan School of Management, MIT, 50 Memorial Drive, E52-455, Cambridge,
MA
02142; email: lkogan@mit.edu.
Introduction
1
Valuation and optimal exercise of American options remains one of the most challenging practical problems in option pricing theory.
The computational
cost of traditional
valuation methods, such as lattice and tree-based techniques, increases rapidly with the
number
known
of underlying securities
and other payoff-related
Due
variables.
to this well-
curse of dimensionality, practical applications of such methods are limited to
problems of low dimension.
In recent years, several
sionality.
methods have been proposed
to address this curse of dimen-
Instead of using traditional deterministic approaches, these methods use Monte
Carlo simulation to estimate option prices. Tilley (1993) was the
first
Other important
that American options could be priced using simulation techniques.
work
in this literature includes
mar and Zwecher
to demonstrate
Barraquand and Martineau (1995), Carriere (1996), Ray-
(1997), Ibanez
and Zapatero (1999) and Garcia (1999). Longstaff and
Schwartz (2001) and Tsitsiklis and Van Roy (1999, 2000) have proposed an approximate
dynamic programming approach that can compute good
in practice. Tsitsiklis
and Van Roy
success of approximate
price estimates
and
very fast
is
also provide theoretical results that help explain the
dynamic programming methods. The
price estimates these tech-
niques construct, however, typically only give rise to lower bounds on the true option
price.
As a
result, there
is
usually no formal
method
for evaluating the
accuracy of the
price estimates.
In an important contribution to the literature, Broadie
develop two stochastic mesh methods
for
American option
and Glasserman
pricing.
tages of their procedure over the previously proposed methods
is
One
that
it
(
1997a, b)
of the advan-
allows
them
to
generate both lower and upper bounds on the option price that converge asymptotically
to the true option price. Their
bounds are based on an application of Jensen's inequal-
and can be evaluated by Monte Carlo simulations.
ity
necessarily generalize to other pricing methods.
drawback but nonetheless appears
of Broadie
method
of their first
suffer
demanding. In an
to be computationally
is
from
effort
and Tan (2001) generalize the stochastic
to address this drawback, Boyle, Kolkiewicz
mesh method
The complexity
The second approach does not
exponential in the number of exercise periods.
this
However, such bounds do not
and Glasserman (1997b) using low discrepancy sequences to
improve the efficiency of the approach.
The main
practical contribution of this paper
is
a general algorithm for constructing
upper and lower bounds on the true price of the option using any approximation to the
option price.
is
We
show that our bounds are
close to the true price of the option, the
addition,
tight, so that
bounds are
the
if
of the lower
bound
is
obtained by simulating a different stochastic process that
appropriate supermartingale.
We
In
of the pricing bounds.
straightforward and relies on simulating the
suboptimal exercise strategy implied by the approximate option
is
approximation
also guaranteed to be close.
we exphcitly characterize the worst-case performance
The computation
initial
justify this
is
price.
The upper bound
determined by choosing an
procedure by representing the American
option price as a solution of a dual minimization problem, which
is
the
main
theoretical
result of this paper.
In order to determine the option price approximation underlying the estimation of
bounds, we also implement a
dynamic programming
fast
and accurate valuation method based on approximate
(see Bertsekas
and
Tsitsiklis 1996)
where we use non-linear
gression techniques to approximate the value function. Unlike
Monte Carlo simulation
most procedures that use
to estimate the continuation value of the option, our
deterministic and relies on low discrepancy sequences as an alternative to
simulation.
For the examples considered
in this paper,
we
re-
method
is
Monte Carlo
find that low discrepancy
sequences provide a significant computational improvement over simulation.
While the duality-based approach to portfolio optimization problems has proved
successful
and
is
now widely used
1998), the dual approach to the
in finance,
(see, for
example, Karatzas and Shreve
American option pricing problem does not seem
been previously developed other than
in recent
to have
independent work by Rogers (2001).
Rogers establishes a dual representation of American option prices similar to ours and
applies the
new
options using
approach
representation to compute upper bounds on several types of American
Monte Carlo
simulation. However, he does not provide a formal systematic
for generating tight
upper bounds and
his
computational approach
is
problem
specific.
Andersen and Broadie (2001) use the methodology developed
in this
paper to
for-
A
dis-
malulate another computational algorithm based on Monte Carlo simulation.
tinguishing feature of their approach
They
the use of an approximate exercise policy, as
price, to estimate the
opposed to an approximate option
option.
is
bounds on the true
price of the
also observe that a straightforward modification (taking the martingale
part of the supermartingale) of the stochastic process used to estimate the upper
bound
also applies to the algorithm
we use
leads to
in this
more accurate estimates. This observation
paper where we begin with an
led to a significant
improvement
initial
in the
approximation to the option
price.
computational results that were presented
This
in
an
earlier draft of this paper.
The algorithm
riods, given
of
Andersen and Broadie
is
quadratic in the number of exercise pe-
knowledge of the approximate exercise
policy, while
given knowledge of the approximate option price. While
results are not currently available,
it
it is
our approach
is
linear
true that formal complexity
does seem to be the case that accurate approxi-
mations to the option price can be computed very quickly.
(See, for
example, Longstaff
and Schwartz 2001,
gramming methods such
that pricing
and Van Roy 2000, and other approximate dynamic pro-
Tsitsiklis
as the
method used
methods based on an
initial
in this paper.)
As a
approximation to the option
we beheve
result,
price, rather
than
the exercise policy, will be more efficient in practice. This should hold in particular for
problems with many exercise problems. For a particular
exercise dates,
it is
The
compare the
difficult to
paper
rest of the
In Section 3,
we
is
derive the
We
problems with more than 10 exercise periods and
for
practical tradeoffs
between our alternative approaches.
new
duality result for
price. In Section 4
American options and use
it
to derive
we describe the implementation
report numerical results in Section 5 and
we conclude
of the
in Section 6.
Problem Formulation
2
In this section
we formulate the American option
Information Set.
pricing problem.
We consider an economy with a set of dynamically
markets, described by the underlying probability space,
risk-neutral valuation
if
ours.
organized as follows. In Section 2 we formulate the problem.
an upper bound on the option
algorithm.
sample problems with 10
Andersen and Broadie compute pricing bounds that are similar to
However, they do not report results
so
set of
financial
measure Q.
It is
well
known
il,
complete financial
the sigma algebra T, and the
(see Harrison
and Kreps 1979) that
markets are dynamically complete, then under mild regularity assumptions
there exists a unique risk-neutral measure, allowing one to
compute
prices of
contingent claims as the expected value of their discounted cash flows.
economy
More
we assume that Tt
specifically,
is
is
^
-.t
€ [0,r]}.
'
t
€
[0.7"]}.
generated by Z(, a d-dimensional standard Brow-
nian motion, and the state of the economy
process {Xt e
state-
The information
represented by the augmented filtration {Tt
structure in this
all
is
represented by an .F(-adapted Markovian
Option Payoff.
Let
/i,
=
/j(A',)
be a nonnegative adapted process representing the
payoff of the option, so that the holder of the option receives
at time
t.
We
also define a riskless account process Bt
We
denotes the instantaneously risk-free rate of return.
/i,
=
if
the option
exp
(
j^
/-^c/s-
is
j,
exercised
where
r^
assume that the discounted
payoff processes satisfies the following integrability condition
max
(1)
(=0,1,. ..,T
where E,
ditional
[]
denotes the expected value under the risk-neutral probability measure, con-
on the time
t
information,
JF,.
Exercise Dates. The American feature
at
any of the pre-specified exercise dates
of the option allows its holder to exercise
T = {0,
in
and T. Equal spacing of the exercise dates
is
1,
.
,
.
.
assumed
it
T}, equally spaced between
to simplify the notation
and
mapped
not restrictive. Moreover, a unit time increment within the model can be
is
into
any period of calendar time.
Option Price. The
value process of the American option,
the option conditional on
it
not having been exercised before
Vt
=
Vt, is
t.
the price process of
It satisfies
Bthr
sup Et
(2)
Br
where r
is
any stopping time with values
in the set
Tn[t,T].
This
is
a well-known
characterization of American options (see, for example, Bensoussan 1984, Karatzas 1988
and Pliska 1997).
3
Theory
Our approach
Step
1.
to the
American option pricing problem consists of the following
Compute an approximation
to the
steps.
market price of the option as a function of the
time and state. Specifically, we use an approximate dynamic programming algorithm to
determine the continuation value of the option,
on not exercising
Step
2.
it
i.e.,
the value of the option conditional
at the current time period.
Estimate the lower bound on the option price by simulating the approximate
exercise strategy based on the option price approximation from Step
Step
it
3.
Based on the option price approximation, define a martingale process and use
to estimate the
new dual
the
We
of the
upper bound by Monte Carlo simulation. This
last step is
based on
representation of the option price presented below.
begin this section with our main theoretical result on the dual representation
American option
price.
compute bounds on the option
We
then show
how
to use this price characterization to
and study the properties
price,
of these bounds.
The Dual Problem
3.1
The problem
an American option, the primal problem,
of pricing
Vo
=
Fit.^)._^^
Then the dual problem
gales,
TTj.
is
iTt,
max
se{\t,T]nT}
Bt
that of computing
we
I
define the dual function F{t,
—
+t7r
^TT
it)
as
(3)
\Bs
to minimize the dual function at time
over
all
supermartin-
Let Uq denote the optimal value of the dual problem, so that
Uo
=
inf F(0,
tt)
=
inf
Eq
max
—
—
TT,
+^
following theorem shows that the optimal values of the dual
coincide.
is
sup Eo
rer
For an arbitrary adapted supermartingale,
The
1.
(D)
and primal problems
Theorem
(Duality Relation) The optimal values of
1
the dual
problem (D) are
problem
is
given by
=
tTj*
equal,
Vq
i.e.,
=
Vt
is
the value process for the
Vq
=
sup
Br
—
teT
where the
TTf,
Vt/Bt
is
<
sup Eo
4-(yr
+(yr
(TT
(4)
\Bt
inequahty follows from the optional sampling theorem
first
gales (see Billingsley 1995)
gales,
tt;,
hr
<En max —
on the
right
hand
option,
Et
sup Eo
r6T
sup Eo
T&T
American
'hrB,'
=
re{[t,T]nT}
Proof For any supermartingale
and
Moreover, an optimal solution of the dual
f/o-
Vt/Bt, where Vt
the primal problem (P)
and condition
(1).
for
supermartin-
all
supermartin-
Taking the infimum over
side of (4) implies Vq
<
f/o.
On
the other hand, the process
a supermartingale, which implies
Uo<Eo max
{ht/Bt
teT
Since Vt
>
attained
when
ht for all
Theorem
1
=
tt^*
t,
-Vt/Bt)
we conclude that Uq <
Vq.
+Vo.
Therefore, Vq
=
Uo,
and equality
is
V/Bt.
shows that an upper bound on the price of the American option can be
constructed simply by evaluating the dual function over an arbitrary supermartingale
TTt-
In particular,
bounded above by
if
such a supermartingale
tto.
Theorem
1
satisfies nt
ht/Bt, the option price Vq
is
therefore implies the following well-known character-
ization of the
American option
Proposition
1
cess Vt/Bt
the smallest supermartingale that
is
>
price (see, for example,
Phska 1997).
(Option Price Characterization) The discounted option
price pro-
dominates the discounted payoff of the
all exercise periods.
option at
The
reverse
is
also true,
i.e.,
one can use Proposition
1
to prove
Theorem
1.
that since both processes on the right-hand side of (3) are supermartingales, so
discounted dual function F{t,n)/Bt. Clearly F(i,
-^(0,
tt)
>
Vo by Proposition
1.
When
tt^
is
>
tt)
ht/Bt for any
Theorem
1
The
pricing problem
American option, which
we use
is
is
to evaluate the
closely related to the
method
in
in a
new
price.
While various methods
in the literature,
for
computing
remained a challenging problem. The result of Theorem
for super-replicating the
American option.
Since the discounted dual function F{t, 7r)/Bt
markets
1
problem of dynamic replication of the
equally important in practice.
reliable replication strategies has
suggests a
and the dual problems
upper bound on the option
approximating American option prices have been suggested
1
implying that
therefore expresses the well-known result of Proposition
constructive form, which
the
is
chosen to be the discounted option price, the
reverse inequality holds, implying that the values of the primal
coincide.
t,
Note
is
a supermartingale and financial
our model are dynamically complete, there exists a self-financing trading
strategy with an initial cost F(0, n) which almost surely dominates F{t,
1996, Section
2.1).
tt)
(see Duffie
Such a trading strategy super-replicates the payoff of the American
option, since F{t, n) dominates the price of the option
Using an approximation to the option
price,
we can
and hence
define
tt,
its
payoff at exercise.
and F(^7r)
explicitly, so
that super-replicating the option can be a relatively straightforward task. In particular,
Boyle et
al.
(1997) and Garcia et
al.
(2000) describe
Monte Carlo methods
for estimating
the portfolio strategies replicating the present value process of a state contingent claim.
This claim could correspond to a derivative security or some consumption process. Their
results are directly applicable to (3).
The Upper Bound on the Option Price
3.2
When
the supermartingale ni in (3) coincides with the discounted option value process,
Vt/Bt, the upper
bound F(0,
TTf
in
the exact price,
equals the true price of the American option.
such a way that when the approximate option price,
V(,
tt^
equals the discounted option price, Vt/ Bt-
to use either of the following
Tfo
This
bound can be obtained from an accurate approximation,
suggests that a tight upper
by defining
tt)
=
two definitions of
It
V,,
Vt,
coincides with
seems natural then
ttj:
Vo
(5)
(6)
(7)
where
TT,,
Vt
(x)"*"
implying
=
Vf,
Tit
:= max(x,0).
tt^
=
is
By
construction, E^
an adapted supermartingale
[ttj+i
for
Vt/Bt, because the latter process
is
— (Tt
bound corresponding
the properties of the
to (5,6)
bound under the
is
first
<
for either definition of
any approximation,
Also,
Vj.
when
a supermartingale and the positive
part of expectations in (5,6) and (5,7) equals zero. While
the upper
]
we cannot claim a
superior to the
priori that
bound determined by
(5,7),
definition are considerably easier to analyze.
Note also that the upper bound can be tightened further by omitting the positive
part in the definition of
of
TTf)
The
resulting process
and therefore a supermartingale, so that
bound.
It
tt;.
It
coincides with nt at
t
=
and
is
we used the formulation
a martingale (the martingale part
too can be used to construct an upper
always greater than or equal to
therefore leads to a lower value of the upper
of this paper,
it
is
bound defined by
(5,6) to
(3).
(In
an
nt for
t
>
0.
earlier draft
compute upper bounds on the option
price.
Andersen and Broadie (2001) point out, however, that
in general tighter
upper
bounds can be obtained by using the martingale component of the supermartingale,
component
In our framework, the martingale
tt^.
obtained by simply omitting the positive
is
in (5,6)).
For the remainder of the paper we
TTo
T^t
+l
=
the upper
bound
Fn =
Vn
+
=
TT,
+
max
Vt
Bt
(9)
to (8)
and
(9).
Then
it is
easy to see that
-^
-;:;
\Bt
+
y
(10)
tjj.
Bj
Bt
Bj_i
bound
deter-
Upper Bound) Consider any approximation
to the
following theorem relates the worst-case performance of the upper
mined by
Theorem
(8)
and
(9) to
the accuracy of the original approximation,
2 (Tightness of the
American option
price, Vt, satisfying Vt
Ko <
Proof
Vt+i
Bt+i
E(
-Df
by
(
t€T
The
-
-5-
bound corresponding
En
to be defined by
(8)
V„
explicitly given
is
ttj
Vo
•D(+l
Let V'o denote the upper
take
will therefore
K)
+
>
hf.
Vf.
Then
2 Eo
;ii)
Bt
See Appendix A.l.
Theorem
2 suggests
in practice. First,
it
two possible reasons
suggests that the
exercise periods. However, this
Eo
max
\
Tif'
\
is
for
why
the upper
bound may deteriorate
bound may be
linearly in the
limited
number
of
a worst case bound. Indeed, the quantity of interest
——
TIT
Bt
Bt
+
2_]
10
Ej-
V
VJ-1
5,
Bj-i
(12)
and while we would expect
clear that
options
it
should increase
when we
typically
of exercise times in
[0,
The second reason
Van Roy
to increase with the
it
(2000) have
This
linearly.
is
number
of exercise periods,
particularly true for pricing
want to keep the horizon, T,
fixed, while
we
let
it
not
is
American
the
number
T] increase.
is
due
to the
approximation
shown that under
dynamic programming algorithms,
certain conditions,
it
assumes that
—
\Vt
and
and
Tsitsiklis
Vt\/Bt.
for certain
approximate
can be bounded above by a constant,
this error
dependent of the number of exercise periods. This
to perpetual options since
error,
T
result,
however,
is
in-
only applicable
-^ oo while the interval between exercise
times remains constant. As mentioned in the previous paragraph, however, we are typically interested in
decreases.
problems where
T
is
In this case, Tsitsiklis and
fixed
and the
interval
Van Roy show
bounded above by a constant times \/N, where
A'' is
between exercise periods
that the approximation error
the
number
is
of exercise periods.
These two observations suggest that the quality of the upper bound should deteriorate
with
A^,
when we
3.3
but not in a linear fashion. In Section 5 we shall see evidence to support this
successfully price options with as
many
as 100 exercise periods.
The Lower Bound on the Option Price
To construct a lower bound on the
true option price,
we
define the Q-value function to
be
"
Qt(Xt) := Et
The Q-value
at time
t
is
being exercised at time
available, iov
t
=
1, ...
Bt
R
Dt+l
Vt+AXt+i)
(13)
simply the expected value of the option, conditional on
t.
,T —
it
not
Suppose that an approximation to the Q-value, Qt{Xt),
1.
Then, to compute the lower bound on the option
we simulate a number of sample paths originating
11
at A'q. For each
is
price,
sample path, we
find
the
first
exercise period
exists, in
if it
t,
which h(Xt)
>
exercised at this time and the discounted payoff' of the path
this
is
a feasible Tt
-
adapted exercise poHcy,
is
it
The option
Qt{Xt).
is
is
then
given by h{Xt)/ Bt- Since
clear that the expected discounted
payoff from following this policy defines a lower bound, Vq, on the true option price, Vq.
The
T
f = min{f E
Formally,
:
Qt
<
ht}
and
following theorem characterizes the worst-case performance of the lower bound.
Theorem
3 (Tightness of the
Lower Bound) The
lower bound on the option price
satisfies
Vo
>
Vo
-
Eo
y-
\Qt-
'
(14)
Proof See Appendix A. 2.
While
theorem suggests that the performance of the lower bound may deteriorate
this
linearly in the
Theorem
not the case.
in
of exercise periods, numerical experiments indicate that this
exercised,
it is
i.e.,
is
3 describes the worst case performance of the bound. However,
order for the exercise strategy that defines the lower
performance,
it
number
bound
to achieve the worst case
necessary that at each exercise period the option
the condition Qt{Xt)
must be the case that
<
h{Xt)
< Qt{Xt)
is
is
satisfied.
mistakenly
left
un-
For this to happen,
at each exercise period, the underlying state variables are close
to the optimal exercise boundary. In addition, Qt
true value Qt so that the option
is
must systematically overestimate the
not exercised while
it is
optimal to do
so. In practice,
the variability of the underlying state variables, Xt, might suggest that Xt spends
time near the optimal exercise boundary.
This suggests that as long as Qt
approximation to Qt near the optimal exercise
12
frontier, the lower
is
little
a good
bound should be a
good estimate
of the true price, regardless of the
of exercise periods.
Implementation
4
we describe
In this section
that
we use
algorithm we describe
it
initial
now standard
example, Bertsekas and
algorithms,
some
in
obtaining the
for
of this kind are
for
number
is
is
detail
approximate Q-value
approximation to the value function,
in the
Vj.
Algorithms
approximate dynamic programming literature
Tsitsiklis,
An
1996).
that, in contrast to
deterministic.
iteration, the algorithm
(see
interesting feature of the particular
most approximate dynamic programming
This deterministic property
is
achieved through the
use of low discrepancy sequences. While such sequences are used in the
same
spirit as
independent and identically distributed sequences of random numbers, we found that
their application significantly
They
4.1
As
improved the computational efficiency of the algorithm.
are described in further detail in
Appendix B.
Q- Value Iteration
before, the
problem
is
to
compute
Vo
In theory this
problem
is
=
sup Eo
T6T
easily solved using value iteration
where we solve
for the value
functions, Vt, recursively so that
Vt
Vt
The
price of the option
economy.
= HXt)
= max
is
(15)
h{X,),E
(16)
then given by V'o(Xo) where
In practice, however,
if
d
is
large so that
13
Xt
Xq
is
is
the initial state of the
high dimensional, then the
'curse of dimensionality' implies that value iteration
As an
is
not practical.
alternative to value iteration consider again the Q-value function, which was
defined earlier to be the continuation value of the option
=
QtiXt)
E,
Vt+i{Xt+,]
(17)
Bt+i
The
value of the option at time
t
+
Vt+i{Xt+,)
so that
we can
1 is
=
then
miix{h(Xt+i).Qt+AXt+,))
(18)
also write
Bt
msLx{h{Xt+i),Qt+i{Xt+i))
QtiXt
(19)
Bt+i
Equation (19) clearly gives a natural extension of value iteration to Q-value
The algorithm we
iteration.
use in this section consists of performing an approximate Q-value
iteration.
why
it is
preferable to concentrate on approximat-
ing Qt rather than approximating Vt directly.
Letting Qt and Vt denote our estimates
There are a number of reasons
of Qt
and
and value
Vt respectively,
for
we can write the
defining equations for approximate Q-value
iteration as follows:
Bt
QtiXt)
=
Vt{Xt)
= nmx{h(X,),E
max{hiXt+,),Qt+i{Xt+:,
E,
(20)
Bt
(21)
Vt+dXt^i]
Bt+i
The
functional forms of (20) and (21) suggest that Qt
more
easily
is
smoother than
14,
and therefore
approximated.
More importantly, however, Qt
is
the
unknown quantity
of interest,
and the decision
to exercise or continue at a particular point will require a comparison of Qt{Xt)
14
and
h{Xt).
we only have
If
difficult to
However,
if
make. For example,
Vt{Xt)
us then such a comparison will often be very
Vt available to
if
Vt{Xi)
>
h{Xt) then we do not exercise the option.
only marginally greater than h{Xt), then
is
> Qt{Xt) and
ation,
we misinterpret Vi{Xt) and assume that
is
is
the case that
it is
optimal to continue when
in fact
it
optimal to exercise. This problem can be quite severe when there are relatively few
exercise periods because in this case, there
value of exercising
now and
often a significant difference between the
When we
have a direct estimate of
arise.
Approximate Q- Value Iteration
4.2
The
is
the continuation value.
Qt{Xt) available, this problem does not
first
\ Qt{-',
is
may be
actually attempting to approximate h{Xt). In this situ-
h{Xt)
Vt{Xi)
it
step in approximate Q-value iteration
Pt)
:
/3t
e 5R^
[,
which
is
is
to select an approximation architecture,
a class of functions from which
we
select Qt. This class
parametrized by the vector Pt £ 5?^ so that the problem of determining Qt
to the problem of selecting Pt where pt
The
choice of architecture does not
sufficiently 'rich' to accurately
is
is
reduced
chosen to minimize some approximation error.
seem to be particularly important as long
as
it
is
approximate the true value value.
Possible architectures are the linearly parametrized architectures of Longstaff and
Schwartz (2000) and Tsitsiklis and Van Roy (2000), or non-linearly parametrized architectures such as neural networks or support vector machines (see
paper we use a multi-layered perceptron with a single hidden
of neural networks.
Vapnik 1999). In
layer,
a particular class
Multi-layered perceptrons with a single hidden layer are
to possess the universal approximation property so that they are able to
any continuous function over a compact
number
set arbitrarily well,
of neurons are used (see Hornick, Stinchcombe
15
this
known
approximate
provided that a sufficient
and White 1989).
The second
step in the procedure
is
to select for each
t
=
I,
.
.
.
,
K—
I,
a set
Sr.= {Pl.--.Ph,}
St in
do
=
makes sense to choose the
sets
such a way that they are representative of the probabihty distribution of Xt-
We
of training points
this using
points, then
where
G
P,[
n
for
JR*^
1,
.
.
.
low discrepancy sequences so that
we simply take Nt points from a
the technique described in Appendix B,
it is
,
A^j.
if
Nt
It
is
of training
then straightforward to convert these points
Of
course,
also possible to
it is
by simply simulating from the distribution of the state vector,
select the training points
Our Umited experience shows
work very
number
particular low discrepancy sequence. Using
into training points that represent the distribution of Xt-
A'(.
the desired
that both simulation and low discrepancy sequences
The performace
well in practice.
of the low discrepancy sequences, however,
appeared to be marginally superior when applied to the problems we consider in
this
paper.
The
with
t
third step
=
K—
1
is
and
to perform a training point evaluation. Defining
for
n =
Qt{Pl)
1,
.
= Et
.
.
,Nt,
_Bt
-
we use Qt{Pn)
max
1
The operator
This
is
E[
.
]
in (22)
necessary as
it
is
is
Qt =
0,
we begin
to estimate Qt{Pi) where
(h{Xt+i), Qt+i{Xt+i)
(22)
^
intended to approximate the expectation operator, E[
usually not possible to
.
].
compute the expectation exactly on
account of the high dimensionality of the state space. For example, E[
.
],
could corre-
spond to Monte Carlo simulation where we simulate from the distribution oi Xt+i given
that Xt
= PI Then
E[
.
]
is
defined by
M
Bt
m&x(^h{Xt+i),Qt+i{Xt+i))
:=^^^|^5]max(M:i--(),4+i(-i^())
Bt+\
16
(23)
where
M xis are drawn randomly from the conditional distribution of Xt+\-
lem with
this
method
which can be too slow
generate the
5,
x;'s,
with
M was set equal to
that the rate of convergence to the true expectation
is
for
M chosen
in
0(-^)
advance. For the 5-dimensional problems of Section
we
1000. For the 10-dimensional problems,
set
sinuilation, then in order to achieve a
accuracy, a substantially larger value of
we estimate Qt with
Q,(x;
is
prob-
our purposes. Instead, we use a low discrepancy sequence to
Had we used Monte Carlo
Finally,
The
Qt{,
M equal to 2000.
comparable
level of
M would have been required.
where
\t3t)
A) = Y.r,{j)e U(j) + J2 (^'O'-O^IO)
(24)
)
and
K
^=
n=
The parameters
9{.)
fe((j),
rt{j)
_
and
(25)
l
constitute the parameter vector Pt, while
r,(j, /) in (24)
denotes the logistic function so that
e{x)
K
2
&vgmmY,[Qt{K)-Q>{Pn-,Pt))
refers to the
number
of 'neurons',
—^
=
and d
is
.
the dimensionality of the input vector,
While the input vector x may simply correspond to a sample state vector,
to
augment x with certain
state vector.
features, that
From an informational
is,
easily
computed functions
point of view, features add no
the neural network. However, they often
make
it
it is
common
of the current
new information
easier for the neural
x.
network
to
(or other
approximation architecture) to approximate the true value function.
In practice,
we
usually minimize a variant of the quantity in (25) in order to avoid the
difficulties associated
with overfitting. This
is
17
done using cross validation, an approach
that requires the training points to be divided into three sets, namely training, vahdation
and
test sets. Initially, only the training
and validation
sets are used in the minimization
so that the quantity
Y,{Qt{P!)-Qt{P!;Pt)y'
is
minimized where the sum
tion
is
in (26)
is
taken over points in the training
performed using the Levenberg Marquardt method
(see Bertsekas
and
the validation set
Tsitsiklis 1996).
is
also
(26)
for least
squares optimization
At each iteration of the minimization, the error
computed and
as long as overfitting
is
validation error starts increasing at any point then
The algorithm then terminates
number
of iterations,
and
pt
is
if
it is
in
not taking place, then
However,
the validation error should decrease along with the training set error.
place.
The minimiza-
set.
likely that overfitting
is
if
the
taking
the validation error increases for a prespecified
then set equal to the value of pt in the
last iteration of
the minimization before the validation error began to increase.
There
addressed.
is
one further
difficulty
The neural network
with the neural network architecture that needs to be
have
will typically
case that the algorithm will terminate at a local
minimum.
In this case,
it is
a different starting value for
many
minima and
local
minimum
that
is
far
pt.
This
may be
repeated until a satisfactory local
minimum
set.
use this training algorithm for finding Qt-\- For the remaining Q- value functions,
however, the problem
Qt-\-
from the global
necessary to repeat the minimization again, this time using
has been found, where 'satisfactory' refers to performance on the test
We
often the
it is
We
network.
is
now somewhat
can therefore use
In practice, this
quickly and that
simplified since
it is
usually the case that Qt
pt as the initial solution for training the
time
t
—
1
~
neural
means that the other neural networks can be trained very
we only need
to perform the minimization once.
18
It also
means that we
can dispense with the need
for
having a test set
Once Qt has been found, we then
We
have found Qq.
for all
iterate in the
but the terminal neural network.
manner
we
of value iteration until
could then use
= max{h{Xo),Qo{X,
Vo{Xo)
While such an estimate may be quite accurate,
to estimate the value of the option.
of limited value since
(27)
we can say very
little
about the estimation
it is
error. In addition,
we
do not have at hand an exercise strategy that has expected value equal to V'o(Xo) nor
do we know
if
such a strategy even
little
information with
for the price of the
American option
Finally,
exists.
it
provides
regards to hedging the option.
Computing the Upper and Lower Bounds
4.3
In Section 3.1
is
we showed that an upper bound
given by (10).
However,
estimated lower bound.
and so Theorem
in (10)
(Since Vq
we can
>
alternatively set Vq
ho, this
new
Vn
+
max
En
ter
7^ - 1^ + >
\Bt
Bt
^
I
upper bound
K-
Kj-i
B,
Bj-i
cjj'
This then gives a natural decomposition of the upper bound into a
nents.
in
The
first
component
is
is
where V^
%
>
is
the
ht for all
t,
given by
(28)
sum
of
the estimated lower bound, while the second
some sense measures the extent
fails
Vq,
definition satisfies
2 continues to hold.) In this case, the
Vn =
=
two compo-
component
to which the discounted approximate value function
to be a supermartingale.
We
estimate
Vq by simulating sample paths
max
ter
Vt
I
-;:::
T^
I
Bt
Bt
+ V2_^ ^J"
j=i
19
of the state variables, evaluating
(29)
Bj
Bj-i
along each path and taking the average over
all
Evaluating (29) numerically
paths.
is
time consuming since at each point, {t,Xt), on the path, we need to compute
Any unbiased
V,
V,j-i
Bj
Bj-i
(30)
-j-i
noisy estimate of the expectation in (30), however, will result in an upwards
biased estimate of the upper bound, due to the application of the
It
maximum
operator.
therefore important that accurate estimates of the expectation in (30) can be
is
computed. As before, we again use low discrepancy sequences to estimate these highdimensional integrals.
bound,
it
is
we wish
Since
to
compute an accurate estimate
of the
upper
important to use a good stopping criterion to determine the number of
points that will be used to estimate the expectation in (30).
Let E(A^) denote the estimate of (30)
some
fixed value of
M, we
when
A^ low discrepancy points are used. For
then examine E{Mi) for
i
=
1,
.
.
,
.
L and
terminate either
when
|E(Af(z
or
when
i
=
L,
if
+ l))-E(Mz)| <e
the condition in (31)
In the numerical results of Section
and 10-dimensional problems,
L
—
is
5,
(31)
not satsified for any
we
respectively.
set
For
M
all
i
problems, we set
48000. These values typically result in estimates of the upper
experiments where
M
and
e
is
One
of,
e
=
.2
cents and
bound that appear
to
based upon further numerical
were increased and decreased respectively. The results of
these experiments invariably resulted in estimates of the upper
or 2 cents
L.
equal to 2000 and 4000 for the 5
be very accurate for practical purposes. This observation
1
<
and often considerably
bound that were within
closer to, the original estimate.
possible concern about the use of low discrepancy sequences for estimating
20
the upper
bound
might result
many
that even a small bias in the estimate of the expectation in (30)
is
This
exercise periods.
In particular, as the
is
number
not a problem, however, for reasons mentioned before.
of exercise periods increases, the time interval between
exercise periods decreases which
means that the
As a
next period also decreases.
result,
An Automated
variability of the value function in the
the conditional expectation in (30) can be
estimated more accurately for fixed values of
4.4
upper bound when there are
in a considerable bias in the estimate of the
M and
e.
Pricing Algorithm
Perhaps the obvious way to compute the lower and upper bounds
fashion so that after estimating the Q-value functions,
is
a sequential
in
we simulate a number
of sample
paths to compute the lower bound, and simulate another set to compute the upper
bound. One
Vq might be
When
difficulty
with this strategy, however,
significant so that there
this occurs,
ing points or a
more
we
is
is
that the difference between V^ and
a large duality gap.
are forced to re-estimate the Q/s, possibly using
flexible
approximation architecture.
This process
more
train-
may need
be repeated a number of times before we obtain a sufficiently small duality gap.
American options that need
to be priced regularly, this
is
to
For
unlikely to be a problem since
experience should allow us to determine in advance the appropriate parameter settings.
However,
for pricing
more exotic American options, we might need
ad hoc approach. This might be very
method
We
inefficient in practice, so
we now
to use such
an
briefly outline a
to address this problem.
have already mentioned that while the upper bound should work very well in
practice, the lower
Section 5
when we
bound should
still
be superior.
price call options on the geometric
these problems, where
it
is
actually possible to
21
This observation
mean
of a
compute option
is
number
borne out
in
of stocks. For
prices exactly,
we
see
that the lower
bound
is
closer than the
This then suggests that when there
upper bound
is
a large duality gap
is
not sufficiently close to Vq.
given in (28), suggests that
upper bound
upper bound to the true
if
E(_i
The expression
V(/5(— 14-i/5(-i
it
price.
is
usually because the
upper bound, as
for the
tends to be large, then the
not be very tight. Based on this observation, we propose the following
will
modification to the approximate Q-value iteration.
After Qt has been computed, we do not proceed directly to computing Qt-\- Instead,
we simulate a number
compute
E(
threshold,
Vt+\/ Bt+\
e^,
of points from the distribution of
—
V't/Bt
.
If
Xt and
the average value of these quantities
then believing that we have a good estimate of
Qt-i- Otherwise
we re-estimate
each point, we
for
l^,
we proceed
The
We
to estimate
by increasing the number of time
Qt, either
points or by refining the approximation architecture, depending on the
seems more appropriate.
below some
is
then repeat the process until the average
training
t
remedy that
than
is less
e(.
and upper
resulting estimates of the Q-value functions should lead to tight lower
bounds.
A
further advantage of this proposal
computational
effort
termine online how
is
is
many
how much
allows us to determine
training points are needed or
how complex
we can now
de-
the approximation
good estimates of the option
price.
Numerical Results
In this section
of 5
it
required to obtain a good solution. In particular,
architecture needs to be in order to obtain
5
that
we
and 10 stocks
illustrate our
respectively,
results for call options
easier to solve
methodology by pricing
call
and the geometric mean of
options on the
5 stocks.
on the geometric mean of 10 stocks since
than the 5 stock
case.
this
We
do not present
problem
While somewhat counterintuitive,
22
maximum
this
is
is
in fact
explained
by noting that the
volatility of the
geometric
mean
decreases as the
number
of stocks
increases.
In the problems that follow,
we assume that the market has
A'^
traded securities with
price processes given by
dSl
where Z[
Z/
is
is
Each security pays dividends
pij.
interval [0,T].
The
first
T
-
5,)dt
+,a dZl\
(32)
at a continuous rate of
We
(5j.
assume that
and that there are n equally spaced exercise dates
date occurs at
the n"' exercise date occurs at
let r
Sl[{r
a standard Brownian motion and the instantaneous correlation of Z] and
the option expires at time
and
=
=
i
T.
i
=
We
which we
call
is
the P' exercise period and
use k to denote the strike price of the option
be the annual continuously compounded interest
constant though this assumption
in the
rate. It is
assumed that
r
is
easily relaxed.
In order to generate the initial approximation to the option price, certain problem
specific information
extraction,
network.
was
also used.
For example, we have already mentioned feature
where functions of the current state vector are used as inputs
In
all
the problems of this section, we found
stock prices before using
them
it
advantageous to order the
as inputs to the neural network.
In addition,
the current intrinsic value of the option as a feature for options on the
for
we used
maximum,
while
options on the geometric mean, the corresponding European option value was used.
Though
will
for the neural
it is
true in general that the exact
not be available, this
is
European value of a high-dimensional option
not important since
it is
known
(see
Hutchinson
et al (1994)
that European options prices can be quickly and accurately approximated using learning
networks
.
(This observation also suggests another
dynamic programming algorithm. Instead
way
of
implementing the approximate
of using the approximation architecture to
23
estimate the Q- value, we could use
to estimate the early exercise
it
premium, conditional
on not exercising the option at the current exercise period. This conditional early exercise
premium
plus the estimated
European option
Q- value function.)
of the
Another method by which problem
Broadie and Glasserman 1997).
is
more accurate estimate
price might give a
We
specific information
know,
for
was used
European option. Such
easily be incorporated to the
We
by simply redefining the estimated Q-value to be the
maximum
timated value function one time period ahead, and a European option that
bound on the Q-value.
infor-
approximate dynamic programming algorithm.
mation can
this
policy fixing (see
example, that the American option price
greater than or equal to the price of the corresponding
do
is
of the es-
a lower
is
For options on the geometric mean, we again use the corre-
sponding European option value to bound the Q-value from below. For options on the
maximum, we
The
use the European option on the stock that
most
'in
the money'.
approximation to the option price was obtained using 2000 and 2500
initial
training points per period for the 5
assigned 65%,
is
25% and 10%
and 10-dimensional problems,
of the training points for the time
respectively.
T-
I
cent.
and stopped as soon
The remainder
of the time
t
+
these networks,
1
as the
network as the starting point
70%
lower
test set
maximum
was
less
than
of
1
of the neural networks were trained only once, using the solution
for training the
time
t
network.
For
of the training points were assigned to the training set with the
remainder assigned to the validation
The
mean-squared error on the
also
neural network to
the training, validation and test sets in turn. This network was trained a
5 times
We
set.
bound was obtained by simulating the approximate
8 million sample paths.
exercise strategy along
The upper bound was computed using 1,000 sample paths
estimate the expectation in (28).
24
to
We
(7,
on the
Call
5.1
Maximum
assume that there are
=
The
0.2
and
p,j
=
for
5 assets, r
^
i
of 5 Assets
=
T=
0.05,
assumed
All stocks are
j.
results are given in Table
to have the
problems where there are
for
1
=
and k
3 years
We
100.
same
=
let 5,
0.1,
initial price Sq.
50 and 100 time
10, 25,
periods.
It
can be seen that the estimated lower and upper bounds are typically very
thereby providing very accurate estimates of the true price.
duality gap widens with the
number
of exercise periods
it
even for the problems with 100 exercise periods, we can
As we argued
before, this widening of the duality
deterioration of the upper
bound
as the
see further evidence to support this
In Table
allow us to
we
1
is
typically
less.
due to the gradual
We
of exercise periods increases.
will
when we examine options on the geometric mean.
early exercise premia of the
approximately
exercise periods or
duality gap
is
obtain very good estimates.
also report the prices of the corresponding
compute the
the duality gap
number
true that the
is
it
does so quite gradually so that
still
gap
While
close,
1%
American options.
of the early exercise
For problems with as
many
European options which
premium
for
We
see that
options with 25
as 50 or 100 exercise periods, the
between 1% and 3%. Using the midpoint of the lower and upper bounds
is
should therefore enable us to price the early exercise premium of these options to within
1%
or 2%.
lower
a,
bound
is
Call
5.2
We
Even more accurate
usually closer to the true price than the upper bound.
on the Geometric Mean of
assume that there are
=
0.4
price So-
and
The
price estimates could be obtained by noting that the
pij
=
for
5 assets, r
i
^
j.
=
0.03,
T=
1
5
years and k
All stocks are again
results are given in Table
1
for
25
Assets
=
100.
We
let J,
=
assumed to have the same
problems where there are
10, 25,
0.05,
initial
50 and
100 time periods.
For
tlie
American
call
price of the option can be
option on the geometric
mean
computed using a standard binomial
mean
process that describes the evolution of the geometric
We
motion.
mean
therefore report the true price of the
together with our niunerical results in Table
The
results are similar to those in Table
percentage of the early exercise premium,
for the options that start
number
5.3
1,
tree, since
is itself
the stochastic
a geometric Brownian
American options on the geometric
2.
though the duality gap, measured as a
now tends
to be
somewhat wider than before
out of the money. Again, this duality gap increases with the
of exercise periods though at a very gradual pace.
on the Mciximuni of
Call
We make
the same assumptions for the
did for the 5 asset case except
3.
of a collection of stocks the true
Measured
now
10 Assets
call
T =
option on the
year.
I
in absolute terms, the duality
gap
is
The
maximum
of 10 assets as
we
results are displayed in Table
again very small for these problems.
However, measured as a percentage of the early exercise premium, the duality gap,
though
still
quite small, can be as large as 20%.
This could be due to a number of
reasons.
First, the early exercise
overall option vahie
represents a
much
smaller component of the
than before, so that using the duality gap (measured as a percentage
of the early exercise
errors.
premium now
premium)
to
measure performance
is
likely to
accentuate pricing
This problem could possibly be overcome by approximating the early exercise
premium
rather than the continuation value of the option, as mentioned earlier.
The more
likely
reason for the wider duality gap
training points or neurons were used.
the state space,
Indeed, while
we made only a moderate
26
number
of
we doubled the dimensionality
of
is
that an insufficient
increase in the
number
of training points
and did not increase
at all the
number
As a
of neurons.
estimated lower bound for this problem, $15,181,
bound, $15,178,
for the
corresponding 50 period problem,
100 period problem
is
is
it is
inferior.
is
greater than the true option price for the
clear that the initial option price
Since the upper bound
decreased to $15,228.
is
quite sensitive to the
for the
100 period
The
gap
resulting duality
is
upper bound
to $15,196, while the
only 3 cents which, for
all
practical
very tight.
Conclusions
In this paper
options.
we have developed a new method
Our approach
is
tion of the
for pricing
and exercising American
based on approximate dynamic programming using nonlinear
regression to approximate the option price.
on
is
approximation
resolved the 100 period problem using 4,000 training points per period
and 25 neurons, the estimated lower bound increased
6
The
considerably wider than the duality gap for the 50 period problem.
When we
purposes,
$90.
greater than the estimated lower
is
approximation, we are not surprised to see that the duality gap
problem
=
corresponding 100 period problem. Since we know that the true
option price for the 100 period problem
initial
we expect performance
Consider, for example, the 50 exercise period problem with So
to suffer.
for the
result,
American option
Our main
theoretical result
is
a representa-
price as a solution of a dual minimization problem.
this dual characterization of the price function,
Based
we use Monte Carlo simulation
to
construct tight upper and lower bounds on the option price. These bounds do not rely on
a specific approximation algorithm and can be used in conjunction with other methods
for pricing
American options.
We characterize
the theoretical worst-case performance of
the pricing bounds and show that they are very accurate on a set of sample problems
where we price
call
options on the
maximum and
27
the geometric
mean
of a collection
of stocks.
These numerical
results suggest that our pricing
applied to problems of practical interest.
28
method can be
successfully
A
Proofs
Proof of Theorem 2
A.l
Simplifying (8) and
Vo
as
=
V'o
an upper bound on
Vo
=
K)
+
and using Theorem
(9),
+
^;
I
—
price of the
max EEj-.
j=i
<
V^o
<
+
Eo
American option.
At time
Vf,
then have
5,_i
S,_,
B,_i
-j-i
5j
5j-l
-^J
-^J-l
K)+|Vb-K)|+EoJ](E,_i
option price process,
A. 2
B,
B,
We
V,
E
where the second inequaUty
Theorem
(Al)
Ej-
j=i
is
5,
5.
j=i
of
>
Bt
B,
ter
we obtain
+
-^
\Bt
teT
tlie
Eo
max
Eo
1,
Bj-i
Bt-.,
due to the supermartingale property of the discounted
and the
last step follows
from the triangle
inequality.
The
result
2 then follows.
Proof of Theorem 3
t,
the following six mutually exclusive events are possible:
(ii)
Qt<Qt<
We
define
f,
=
ht,
(iii)
min{s €
Qt <
[f,
T]
ht
n
<
T
(iv)
Qt.
:Q~,
Vt
<
=
Qt
<
ht
<
Qt,
(v) ht
<
(i)
Qt
Qt
<
<Qt <
Qt,
ht,
(vi) Ih
<
/ij and
BtEt
For each of the six scenarios above, we establish a relation between the lower bound and
the true option price.
(i),(ii)
The algorithm
for
estimating the lower bound correctly prescribes immediate
29
Qt
<
Qt-
.
=
exercise of the option so that Vt— y^
(ill)
In this case the option
Vt-V<
is
0.
exercised incorrectly.
=
]4
ht
and
=
Qt implying
\Qt-Qt\.
(iv) In this case the option
is
not exercised though
it is
optimal to do
Bt
Vt
= -^Et \VtJ
while
Vt
= ht<Qt+ {Qt-
Bt
Qt)
= -^Et
+[Qt-Q
[Vt+,]
This implies
Vt-V<\Qt-Qt\
(v),(vi) In this case the option
is
+^E,
correctly
Vt-Vt =
left
-^Et
\Vt+x
-
Vt+i]
unexercised so that
\Vt+i
-
Vt+i]
.
Therefore by considering the four possible scenarios, we find that
Vt-Vt<
Iterating
Vt
and using the
\Qt
fact that
-
Qt\
Vr= Vt
+^Ef
\Vt+i
-
Vt+x]
implies the result.
30
.
so.
Therefore
B
Low Discrepancy Sequences
A
low discrepancy sequence
in
some
fixed
unit cube
is
a deterministic sequence of points that
domain. Often, and without
Because the points
[0, 1]''.
in
loss of generality,
we take
this
where
{y,
quence.
=
:i
An
1,
.
,
.
.
A^},
is
evenly dispersed
domain
to be the
a low discrepancy sequence are evenly dispersed,
they are often used to numerically integrate some function /(.) over
/
is
[0, Ij'^so
that
f{x)dx^^^^l^
(Bl)
a set of A^ consecutive terms from the low discrepancy se-
important property that low discrepancy sequences possess
terms are added, the sequence remains evenly dispersed.
is
that as
This property implies that,
in contrast to
other numerical integration schemes, the term
determined
advance and can therefore be chosen according to some termination
terion.
in
Because these sequences are evenly dispersed, their use
often results in a rate of convergence that
where the convergence rate
is
is
much
faster
0(-^). For the technical
new
A'^
in
in
(Bl) need not be
cri-
numerical integration
than Monte Carlo simulation
definition of a low discrepancy
sequence and a more detailed introduction to their properties and financial applications,
see Boyle, Broadie
and Glasserman (1997). See Birge (1994),
and Paskov and Traub (1995)
In this paper
for
some
and Tan (1996)
of these applications.
we use low discrepancy sequences
ing point evaluation.
Joy, Boyle
for training point selection
The low discrepancy sequences
and
train-
are of particular value for training
point evaluation since a good estimate of
_Bt
E,
^max(M.Yt+i),Q,4-i(A-,+i))
B
can usually be computed much faster by using a low discrepancy sequence
Monte Carlo
simulation.
31
in place of
Even though a low discrepancy point y G
interpret
it is
it
as being
[0, 1]"^ is
sampled from a uniform distribution
then straightforward to convert y into a point,
variable
Z
deterministic
in
[0, 1]''.
it
can be useful to
With
this in
x, that is representative of
with cumulative distribution function F(.).
mind,
a random
For example, suppose
Z
is
a
d-dimensional standard normal random variable with correlation matrix equal to the
identity.
We
can then construct a point
2
a multivariate normal
is
5R'^
that
is
representative of
by setting
(B2)
taken componentwise in (B2).
random
Z
= F-\y)
z
where the operation F~^
6
variable,
X
If
instead
we wish
to 'simulate'
with covariance matrix E, then we simply
premultiply z by the Cholesky decomposition of S.
In financial applications
tributed.
able
is
random
variables are often
Since transforming a normal
easy, however,
we can do
random
likewise with
assumed
to be lognormally dis-
variable into a lognormal
z.
We
we use
in this
paper are Sobol sequences.
32
vari-
can therefore easily convert a
d-dimensional low discrepancy sequence into a sequence of points that
of a d-dimensional multivariate log-normal distribution.
random
is
representative
The low discrepancy sequences
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35
Table
Table
We
1
1:
Call on the
mciximum
contains estimates of the price of an American
use the following set of parameter values: r
=
call
0.05,
of 5 assets
option on the
T —
3,
k
—
maximum
100,
6^
—
of 5 assets.
0.1, a^
—
0.2
and pij =
for i,j = 1, ...,5. All stocks are assumed to have the same initial price Sq. The
columns "Lower Bound" and "Upper Bound" contain estimates of the lower and upper bounds,
respectively.
for
and
The standard
problems with
10, 25,
errors of these estimates are given in brackets. Results are displayed
50 and 100 time periods.
We
report estimated options prices in $'s
their standard errors in cents.
So
Lower Bound
Upper Bound
10 exercise periods
90
100
110
90
100
no
90
100
no
90
100
no
16.640
European Price
Table
2:
Call on the geometric
Table 2 contains estimates of the price of an American
5 assets.
We
mean
call
use the following set of parameter values: r
—
of 5 assets
option on the geometric
0.03,
T=
1,
k
=
100,
6i
mean
—
of
0.05,
= 0.4 and pij — for i,j — 1, ..., 5. All stocks are assumed to have the same initial price Sq.
The columns "Lower Bound" and "Upper Bound" contain estimates of the lower and upper
bounds, respectively. The standard errors of these estimates are given in brackets. Results
CT(
are displayed for problems with 10, 25, 50
prices in $'s
and
Sq
and 100 time periods.
Lower Boundd
Upper Bound
True Price
10 exercise periods
90
We
report estimated options
their standard errors in cents.
European Price
Table
3:
Call on the
maximum
Table 3 contains estimates of the price of an American
assets.
We
use the following set of parameter values: r
of 10 assets
call
=
option on the
0.05,
T =
I,
k
maximum
—
100,
Sj,
of 10
=
0.1,
and pij = for i, j = 1, ..., 5. All stocks are assumed to have the same initial price SoThe columns "Lower Bound" and "Upper Bound" contain estimates of the lower and upper
ai
=
0.2
bounds, respectively.
ai'e
The standard
errors of these estimates are given in brackets.
displayed for problems with 10, 25, 50 and 100 time periods.
prices in $'s
and
We
their standard errors in cents.
5o
Luwer Bound
Upper Bound
European Price
10 exercise periods
90
100
110
15.082
15.092
(0.40)
(0.37)
26.860
26.863
(0.46)
0.60)
39.076
39.076
(0.51)
(0.48)
14.747
26.403
38.522
25 exercise periods
90
100
110
15.158
15.172
(0.39)
(0.51)
26.960
26.970
(0.45)
(0.74)
39.182
39.193
(0.50)
(0.64)
14.747
26.403
38.522
50 exercise periods
90
100
110
15.181
15.216
(0.38)
(0.79)
26.978
27.092
(0.45)
(3.46)
39.225
39.261
(0.49)
(0.96)
14.747
26.403
38.522
100 exercise periods
90
100
110
15.178
15.283
(0.39)
(1.36)
26.996
27.070
(0.45)
(1.17)
39.223
39.306
Results
report estimated options
14.747
26.403
38.522
3
Date Due
3 9080 02246 0726