Python代写-INOMIYA1 AND
时间:2022-08-01
WEAK APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
AND APPLICATION TO DERIVATIVE PRICING
SYOITI NINOMIYA1 AND NICOLAS VICTOIR2
Abstract. The authors present a new simple algorithm to approximate weakly
stochastic dierential equations in the spirit of [9][13]. They apply it to the problem
of pricingAsian options under theHeston stochastic volatilitymodel, and compare
it with other known methods. It is shown that the combination of the suggested
algorithm and quasi-Monte Carlo methods makes computations extremely fast.
1. Introduction
1.1. The Problem and its Motivation. We consider a stochastic dierential equa-
tion written in the Stratonovich form
Y(t; x) = x +
Z t
0
V0 (Y(s; x)) ds +
dX
i=1
Z t
0
Vi (Y(s; x)) dBis;
V j 2 C1b

RN;RN

;
(1)
where B =

B1; ;Bd

is a standard Brownian motion, and C1b

RN;RN

denotes
the set of RN-valued smooth functions dened over RN whose derivatives of
any order are bounded. In particular, we will use the classical notation V f (x) =PN
i=1 V
i (x)

@ f=@xi

(x) for V 2 C1b (RN;RN) and f a dierentiable function from Rn
into R: This stochastic dierential equation can be written in It ˆo form:
Y(t; x) = x +
Z t
0
˜V0 (Y(s; x)) ds +
dX
i=1
Z t
0
Vi (Y(s; x)) dBis;
where
˜Vi0

y

= Vi0

y

+
1
2
dX
j=1
V jVij

y

:
Now, given a function f with some regularity, how can one approximate e-
ciently E

f (Y(1; x))

? It is equivalent to the following deterministic problem: if L is
the dierential operatorV0+ (1=2)
Pd
i=1 V
2
i and u is the solution of the heat equation
@u
@t
(t; x) = Lu; u (0; x) = f (x);
2000Mathematics Subject Classication. 65C30, 65C05.
Key words and phrases. Hestonmodel, numericalmethods for stochastic dierential equations, math-
ematical nance, quasi-Monte Carlo method.
This research was partially supported by the Japanese Ministry of Education, Science, Sports and
Culture, Grant-in-Aid for Scientic Research (C), 15540110, 2003.
1
2 S. NINOMIYA AND N. VICTOIR
howdoes one approximate u (1; x) (which is equal toE

f (Y(1; x))

by Feynman-Kac
theorem [7]).
This problem has had a lot of attention because of its practical importance: it
gives the evolution of the temperature in some media, and also represents price of
nancial derivatives under stochastic nancial models such as Black-Scholes [1].
Non-probabilistic methods to solve the PDE (such as nite dierence methods)
seem to only work well when L is elliptic and in low dimension. We refer to [11]
for a more detailed discussion on the subject. We will focus in this paper on
probabilistic methods.
1.2. Notation. If V is a smooth vector eld, i.e. an element of C1b

RN;RN

,
exp (V) (x) denotes the solution at time 1 of the ordinary dierential equation
dzt
dt
= V (zt) ; z0 = x:
For x 2 R, bxc denotes the integer part of x.
1.3. Probabilistic Methods.
1.3.1. Order 1. The most popular probabilistic method to approximate E
h
f

Yx1
i
is
called the Euler-Maruyama method [8]. We rst x n independent d-dimensional
random variables Z1; ;Zn such that, if X denotes a standard normal random
variables, E

p (Zk)

= E

p (X)

for all polynomial of degree less than or equal to 3.
1 Then one denes recursively the following random variables:
X(EM);n0 = x;
X(EM);n(k+1)=n = X
(EM);n
k=n +
1
n
˜V

X(EM);nk=n

+
1p
n
dX
i=1
Vi

X(EM);nk=n

Zik+1:
Then, one can show [8][17] that for all nice enough function f
(2)
E h f X(EM);n1 E f (Y(1; x)) C f 1n :
2 Of course, one needs an algorithm to compute E
h
f

X(EM);n1
i
: If the Zk are con-
structed from Bernoulli random variables, E
h
f

X(EM);n1
i
is a discrete sum, but one
would need to do 2nd additions, which can be rather lengthy when nd is large (one
is then forced to do someMonte-Carlo on a discrete measure). If theNi are normal
random variables, one then is forced to do use some Monte Carlo or quasi-Monte
Carlo techniques. When nd is big, quasi-Monte Carlomethod become less eective
thanMonte-Carlo, but if nd is not too high, quasi-Monte Carlo method can be very
ecient.
1Such random variable are easy to nd. One can take, for a xed i; Z ji to be d independent Bernoulli
or Gaussian random variables. A more elaborate choice of such random variables appeared in [13][16].
2Here, we have used the subdivision (k=n)k2f0; ;ng of [0; 1] : In no way taking equal time steps is
optimal. We do not want to address this problem in this paper, and we will always take subdivisions
with equal time steps.
WEAK APPROXIMATION OF SDE 3
Another method with the same rate of convergence appeared in [13], and is
called cubature on Wiener space of degree 3. It is dened with the following
recursive formula:
X(cub3);n0 = x;
X(cub3);n(k+1)=n = exp
0BBBBB@1nV0 + 1pn
dX
i=1
Zik+1Vi
1CCCCCA X(cub3);nk=n
Such algorithm can be seen as a practical application of the Wong-Zakai theo-
rem [7][18], when the Zk are normal random variables.
If Bnt = (B
n;1
t ; : : :B
n;d
t ) (n 2N) is the piecewise linear approximation of the Brown-
ian motion dened by
Bnt = (bntc + 1 nt)Bnbntc=n + (nt bntc)Bn(bntc+1)=n;
and Yn denotes the solution of the ordinary dierential equation
Ynt = x +
Z t
0
V0

Yns

ds +
dX
i=1
Z t
0
Vi

Yns

dBn;is ;
then the Wong-Zakai theorem states that Yn converges almost surely to Yx. It is
easy to see that X(cub3);n1 and Y
n
1 are equal in law, proving the convergence of the
weak algorithm cubature onWiener space of degree 3 (but this argument does not
provide the rate of convergence).
Remark 1.1. In the algorithm cubature on Wiener space of degree 3, one has to solve
numerically ODEs (unless one is lucky and one has a close form solution!). One possibility
is to take its Taylor approximation of order 1 for the approximation of exp (V) (x) and we
fall back on an Euler scheme. Taking a better approximation (Taylor approximation of order
2) will give a scheme sometimes described as the Milstein scheme. Not spending enough
care on the approximating method of the ODEs to be solved can result in some catastrophic
situations. A general case where that happens is when the diusion is almost surely on a
subset ofRn; that is, does not ll the whole space. If one has an approximation schemewhich
at some time provides an answer outside this set (which is what happen if one approximates
badly the ODEs), the algorithm may go very wrong or even bug. Increasing n (which is
costly) or articial techniques can be implemented to solve this problem [], while this can
be overcome by taking an appropriately good approximation of the ODEs which have to
be solved (we usually recommend a high order Runge-Kutta scheme, or an adaptive step
size scheme, but this may depend on the particular SDE to approximate). We will give an
example of this problem in Section 3.
1.3.2. Higher order. Away to obtain approximations of higher order is based on the
understanding of more terms in the stochastic Taylor formula (see [3] and [8] for
example).When the vector elds Vi commute, it is relatively easy to nd a scheme
of high order, see [8] and the references within. In the general case, one needs to
understand how to approximate weakly the increments of the Brownian motion
together with its rst few iterated integrals. This was rst successfully done, to
our knowledge, in [9][12], see also [10], and then generalized with the method
cubature on Wiener space [13].
4 S. NINOMIYA AND N. VICTOIR
1.4. Romberg Extrapolation. Consider a nice scheme of order p, that is, a scheme
X(ord p);nk=n such that for smooth f , there exists a constant K f such thatE h f X(ord p);n1 i E f (Y(1; x)) K f 1np
C f 1np+1 :
Then,
(3)
2p
2p 1E
h
f

X(ord p);2n1
i
1
2p 1E
h
f

X(ord p);n1
i
provides a scheme of order p+1. We refer once again to [17] formore details and the
proof that the Euler-Maruyama scheme and its successive Romberg extrapolations
are “nice” schemes.
It is strongly conjectured that the cubature on Wiener space algorithms and the
algorithm presented below are “nice” schemes, but this has not been proved yet.
1.5. An informal remark on the Monte Carlo method. Let X(ord p);n1 denotes a
scheme of order p of the type above. To calculate X(ord p);n1 numerically, one need to
approximate an integral over a nC(d) dimensional space (C(d) denoting a function
depending on d; for Euler or Cub3, C(d) = d. As we will see later, C(d) = d + 1
for our new algorithm). If one uses the Monte-Carlo method to approximate this
integrals, and usesM samples, we denote this new approximation by X(ord p;mcM);n1 .
Then, roughly speaking,E h f X(ord p;mcM);n1 i E f (Y(1; x)) = O np + pM
where is a random variable with bounded variance. Therefore, one should,
asymptotically at least, eectuate M = Kn2p simulations, to keep the precision
of the scheme of order p. To compute E
h
f

X(ord p;mcM);n1
i
, one actually generates
L = MnC(d) points. Therefore, choosing M = Kn2p; we obtain L = KC(d)n2p+1
to obtain a precision of C0dn
p = C00d L
p=(2p+1). Therefore, an algorithm of order p
provides, using Monte Carlo, an approximation whose error is of order Lp=(2p+1)
where L is the number of generated random variables. In our discussion, we have
assumed that the variance of does not depend on the dimension of the integral
domain andM, which is a fact which seems to be conrmed by experiments.
1.6. Aremarkon thequasi-MonteCarlomethod. Although there are some results
which justify the quasi-Monte Carlomethod and give theoretical errorwith respect
to the numberM of sample points and the dimension of the integral domain, those
results help little for error estimation in practice when we apply the quasi-Monte
Carlo method to weak approximation of SDEs (see [14] or [15]).
The quasi-Monte Carlo method is usually more ecient than the Monte Carlo
method in dimension not too high. The integral that we have to approximate to
obtain X(ord p);n1 is on a space of dimension nC(d). If the numerical method is of
high order and C (d) is not too big, one can then use quasi-Monte Carlo with this
numerical method to obtain a very fast algorithm.
Therefore, it seems optimal to look for a (simple) scheme of order greater than
the one of the Euler scheme (one), with C (d) remaining comparable to d (i.e. the
C (d) of the Euler scheme). This is the object of this paper, where we suggest a new
WEAK APPROXIMATION OF SDE 5
numerical scheme of order 2, with C (d) = d + 1: We will prove its eciency by
numerically pricing an Asian option under the Heston model.
2. Presentation of the new Algorithm
We present our new algorithm, of order 2.
Theorem 2.1. Let (i;Zi)i2f1; ;ng be n independent random variables, where each i is a
Bernoulli random variable independent of Zi, which is a standard d-dimensional normal
random variable. Dene fX(New);nk=n gk=0;:::;n to be a family of random variables as follows:
X(New);n0 = x;
X(New);n(k+1)=n =8>>>>>><>>>>>>:
exp
V0
2n

exp
0BBBB@Z1kV1pn
1CCCCA exp 0BBBB@ZdkVdpn
1CCCCA exp V02n X(New);nk=n if k = +1;
exp
V0
2n

exp
0BBBB@ZdkVdpn
1CCCCA exp 0BBBB@Z1kV1pn
1CCCCA exp V02n X(New);nk=n if k = 1:
(4)
Then, for all f 2 C1b (RN), E h f X(New);n1 i E h f Yx1i C fn2 ;
that is, our new algorithm is of order 2.
A few remarks before all: To compute
exp
V0
2n

exp
0BBBB@Z1kV1pn
1CCCCA exp 0BBBB@ZdkVdpn
1CCCCA exp V02n X(New);nk=n ;
one needs to solve d+ 2 ordinary dierential equations. First along the vector eld
V0 from t = 0 to t = 1=(2n) with starting point X
(New);n
k=n , then along Vd from t = 0
to t = Zdk=
p
nwith starting point the solution of the ODE we have just solved, and
we repeat similar operations d + 2 times. One would need an algorithm to solve
this ODE numerically (unless one has a close form solution), and we, once again,
strongly suggest that one pays a lot of attention to the quality of such algorithm.
One of course will have to use an algorithm to approximate E
h
f

X(New);n1
i
,
but this is just a (dicult but classical, common to Euler algorithm for example)
problem of integrating a function on a nite dimensional space. The simplest
but quite eective method is to do some basic Monte-Carlo simulation of the
random variables (i;Zi)i2f1; ;ng. One could also simulate the random variables
(i;Zi)i2f1; ;ng with some quasi-Monte Carlo techniques, or replace the random
variables Zi with some discrete random variables with the right moment up to
order 5. As this is a very classical problemand common to all the other probabilistic
solutions to our numerical problem, we do not provide anymore precisions here.
Proof. The proof is quite classical, so we will not go into details. The reader should
be convinced that the algorithm is of order 2 once we show that for f smooth
enough, E h f X(New);n1=n i E f (Y(1=n; x)) C fn3 :
6 S. NINOMIYA AND N. VICTOIR
The error over n steps, from the Markov property of Y, would then be n times n3.
We consider a smooth function f . First observe that, from of the Feynman-Kac
theorem, E f (Y(1=n; x)) x + 1nL f (x) + 1n2 L2 f (x)
C fn3:
Developing L2; that means
x +
1
n
L f (x) +
1
n2
L2 f (x) = x +
1
n
0BBBBB@V0 + 12
dX
i=1
V2i
1CCCCCA f (x)
+
1
2n2
0BBBBBB@V20 + 12V0
dX
i=1
V2i +
1
2
dX
i=1
V2i V0 +
1
4
dX
i; j=1
V2i V
2
j
1CCCCCCA f (x):
Now we need to approximate E
h
f

X(New);n1=n
i
: Using Taylor approximation of
the ODEs involved, we quickly see that the absolute value of
E
"
f

exp
1
2n
V0

exp

1p
n
Z1kV1
!
exp

1p
n
ZdkVd
!
exp
1
2n
V0

(x)
!#
minus
x +
1
n
0BBBBB@V0 + 12
dX
i=1
V2i
1CCCCCA f (x)
+
1
2n2
0BBBBBB@V20 + 12V0
dX
i=1
V2i +
1
2
dX
i=1
V2i V0 +
1
4
dX
i=1
V4i +
1
2
dX
i< j
V2i V
2
j
1CCCCCCA f (x)
is bounded by ˜C fn3: Inverting the order in which the vector elds are integrated,
we obtain that the absolute value of
E
"
f

exp
1
2n
V0

exp

1p
n
ZdkVd
!
exp

1p
n
Z1kV1
!
exp
1
2n
V0

(x)
!#
minus
x +
1
n
0BBBBB@V0 + 12
dX
i=1
V2i
1CCCCCA f (x)
+
1
2n2
0BBBBBB@V20 + 12V0
dX
i=1
V2i +
1
2
dX
i=1
V2i V0 +
1
4
dX
i=1
V4i +
1
2
X
i> j
V2i V
2
j
1CCCCCCA f (x)
is bounded by ˜C fn3: Adding up and dividing by 2, we obtain thatE f (Y(1=n; x)) x + 1nL f (x) + 1n2 L2 f (x)
˜C fn3:

Remark 2.1. Following [9], one could show the convergence of the algorithm with f just
continuous, under a condition on the vector elds weaker than Ho¨rmander condition.
This algorithm could be seen in a non-trivial way as a particular case of the
algorithm cubature on Wiener space of degree 5. One should also notice some
common features with splitting methods.
WEAK APPROXIMATION OF SDE 7
3. Numerical Example: Application to Finance
In this section,wenumerically compareournewalgorithmto theEuler-Maruyama
scheme and their Romberg extrapolation. We calculate the price of an Asian call
option with maturity T and strike K written on an asset whose price process Y1
satises the following two factor stochastic volatility model (Heston model [6]):
Y1(t; x) = x1 +
Z t
0
Y1(s; x) ds +
Z t
0
Y1(s; x)
p
Y2(s; x) dB1(s);
Y2(t; x) = x2 +
Z t
0
( Y2(s; x)) ds +
Z t
0

p
Y2(s; x) dB2(s);
(5)
where x = (x1; x2) 2 (R>0)2, (B1(t);B2(t)) is a 2-dimensional standard Brownian
motion, and , , are some positive coecients such that 2 2 > 0 to ensure
the existence and uniqueness of a solution to our SDE [5]. The payo of this option
is max (Y3(T; x)=T K; 0), where
(6) Y3(t; x) =
Z t
0
Y1(s; x) ds:
The price of this option becomes D E [max (Y3(T; x)=T K; 0)] where D is the
appropriate discount factor. We set T = 1, K = 1:05, = 0:05, = 2:0, = 0:1,
= 0:09, and (x1; x2) = (1:0; 0:09). We ignore D in this experiment. Let Y(t; x) =
t(Y1(t; x);Y2(t; x);Y3(t; x)). We transform the SDEs (5) and (6) into a Stratonovich
form SDE:
(7) Y(t; x) =
2X
i=0
Z t
0
Vi(Y(s; x)) dBi(s);
where
V0

ty1; y2; y3 = t y1 y22 ; ( y2) 24 ; y1
!
V1

ty1; y2; y3 = ty1py2; 0; 0
V2

ty1; y2; y3 = t0; py2; 0 :
(8)
3.1. Implementation of the algorithm. We apply the algorithm which we intro-
duced in Section 2 to this problem.
3.1.1. Solutions of the ODEs. We can easily get exp (sV1) and exp (sV2) (s 2 R) as
follows:
exp (sV1)

ty1; y2; y3 = ty1espy2 ; y2; y3 ;
exp (sV2)

ty1; y2; y3 = t0BBBB@y1; s2 + py2
!2
; y3
1CCCCA :(9)
As there exists no closed form solution to exp (sV0), we are forced to use an ap-
proximation and we choose:
(10) exp (sV0)

ty1; y2; y3 = ty1(t); y2(t); y3(t) ;
8 S. NINOMIYA AND N. VICTOIR
where
y1(t) = y1 exp

J
2

s +
y2 J
2

es 1! ;
y2(t) = J +

y2 J es;
y3(t) = y3 +
y1

eAs 1

A
+O

s3

;
J =
2
4
; A = y2
2
; and B =
(y2 J)
4
:
(11)
The error compared to the true solution isO

t3

in small time, creating anadditional
error of O

n3

at every step of the algorithm, but as the error of our scheme at
every step was also O

n3

; taking the above approximation of exp (sV0) does not
alter the convergence rate of the algorithm.
Here, we see that one of the advantages of this algorithm over the Euler-
Maruyama scheme is the one we mentioned in Remark 1.1. When we apply
the Euler-Maruyama scheme to this process (5), it may happens that the square
volatility process (Y2)
(EM);n
k becomes negative, and the algorithm then fails at the
next step (as we will have to take its square root). On the other hand, equations (9)
and (11) show that our new algorithm does not share this problem. 3
3.1.2. A remark on general implementation. In general, it is not always possible to
obtain the closed form solution to exp(sVi). Even in such cases, it is not dicult to
implement our new algorithm. All we have to do is to nd an approximation of
exp(sV0) whose error isO(s3) and approximations of exp(sVi); (i , 0) whose errors
are O(s6). This can be achieved by Runge-Kutta like methods [2].
3.1.3. Application of the quasi-Monte Carlo method. Our new algorithm has the virtue
that the application of the quasi-Monte Carlo method to this algorithm is possible
in a straight forward way, once we embed (i;Zi)i2f1;:::;ng into [0; 1)n(d+1).
3.2. Comparison to Euler-Maruyama scheme. We compare numerically our new
algorithm to the Euler-Maruyama scheme with and without Romberg extrapola-
tion. Such methods involve, as we saw, approximation of an integral over a nite
dimensional space; wewill do these approximation using theMonte Carlomethod
and theQuasi-MonteCarlomethod. Here,we considerE [max (Y3(T; x)=T K; 0)] =
6:04720626353478 102 which is obtained by our new algorithm with extrapola-
tion, quasi-Monte Carlo, n = 256 + 128, andM = 1:1 109.
3There exists a way of avoiding this problem with the Euler-Maruyama scheme [4].
WEAK APPROXIMATION OF SDE 9
1e-07
1e-06
1e-05
1e-04
0.001
0.01
1 10 100 1000 10000 100000
Er
ro
r
Num. of Partitions
Euler-Maruyama
Euler-Maruyama + Extrpltn
New
New + Extrpltn
5e-05
5e-06
O(1/N)
O(1/N2)
Figure 1. Error coming from the discretization
1e-06
1e-05
1e-04
0.001
0.01
0.1
1
10 100 1000 10000 100000 1e+06 1e+07

(fo
r M
C)
an
d A
bs
olu
te
dif
fer
en
ce
(fo
r Q
MC
)
Num. of Sample points
MC : Euler-Maruyama, 1/∆t=64
MC : New, 1/∆t=16
MC : Euler-Maruyama + Extrpltn, 1/∆t=16
MC : New + Extrpltn, 1/∆t=8
QMC : Euler-Maruyama 1/∆t=256
QMC : New 1/∆t=16
QMC : New + Extrpltn 1/∆t=4
QMC :Euler-Maruyama + Extrpltn, 1/∆t=16
5e-05
5e-06
Figure 2. Convergence Error from quasi-Monte Carlo and Monte Carlo
10 S. NINOMIYA AND N. VICTOIR
Method #Partition #Sample CPU time (sec)
E-M +MC 2000 108 1:09 105
E-M + Extrpltn +MC 16 + 8 108 2:20 103
New +MC 16 108 3:2 103
New + Extrpltn +MC 4 + 2 108 1:4 103
E-M + Extrpltn + QMC 16 + 8 5 106 1:10 102
New + QMC 16 5 104 1:6
New + Extrpltn + QMC 4 + 2 104 1:4 101
Figure 3. #Partition, #Sample, and CPU time required for 4 digits accuracy.
3.2.1. Discretization Error. Figure 1 shows the relation between the number of parti-
tions in our discretization of the interval [0; 1] (n in the description of the algorithm)
and the error of the algorithms. We observe that to achieve four digits accuracy, our
new method with Romberg extrapolation requires n = 6, our new method needs
n = 16, while the Euler-Maruyama scheme with Romberg extrapolation needs
n = 24, and the simple Euler-Maruyama scheme needs n 2000. In all algorithms,
consumed time is proportional to nM, whereM is the number of sample points.
3.2.2. Convergence Error from Monte Carlo. We have already mentioned in 1.5 that
the convergence performance of the Monte Carlo method is independent of the
number of partitions. We can see in Figure 2 that in this experiment this statement
holds. This gure also shows that to achieve four digits accuracy with 95% con-
dence level (2) by using Monte Carlo method, we need over 108 sample points.
We can also see in this gure that the Monte Carlo errors which come from al-
gorithms boosted by the Romberg extrapolation become greater than those of the
original algorithms.
3.2.3. Convergence Error fromquasi-MonteCarlo andMonteCarlo. Figure 2 also shows
that the performance of the convergence of the quasi-MonteCarlomethoddepends
on the number n of partitions and on the algorithms. Figure 2 seems to show that
the quasi-Monte Carlo method outperforms the Monte Carlo method specially
when used with our new algorithm and that the algorithm needs 5 104 sample
points for four digits accuracy, the algorithmwith extrapolation 104 sample points,
and Euler-Maruyama with extrapolation 5 106 sample points when we use the
quasi-Monte Carlo method.
3.2.4. Performance comparison with respect to consumed time. The elapsed time of
all methods required for four digits accuracy is shown in Figure 3. We nd in
this gure that our new algorithm with Romberg extrapolation and the quasi-
Monte Carlo method provides the fastest calculation. Our new algorithm with
Romberg extrapolation and quasi-Monte Carlo is about 800 times faster than Euler-
Maruyama scheme with Romberg extrapolation and quasi-Monte Carlo. We also
see that evenwithout Romberg extrapolation, our new algorithm is still faster than
any boosted Euler-Maruyama method.
References
[1] Black, F., and Scholes, M. The Pricing of Options and Corporate Liabilities. Journal of Political
Economy 81 (1973), 637–59.
WEAK APPROXIMATION OF SDE 11
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1Center for Research inAdvanced Financial Technology, Tokyo Institute of Technology, 2-12-1
Ookayama, Meguro-ku, Tokyo 152-8552 Japan
E-mail address: ninomiya@craft.titech.ac.jp
2Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB, UK
E-mail address: victoir@maths.ox.ac.uk
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