Fast strong approximation Monte Carlo schemes for stochastic volatility models

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Fast strong approximation Monte Carlo schemes forstochastic volatility modelsChristian Kahl a b & Peter Jäckel aa Quantitative Analytics Group, ABN AMRO, 250 Bishopsgate, London EC2M 4AA, UKb Department of Mathematics, University of Wuppertal, Gaußstraße 20, Wuppertal,D-42119, GermanyPublished online: 18 Feb 2007.

To cite this article: Christian Kahl & Peter Jäckel (2006): Fast strong approximation Monte Carlo schemes for stochasticvolatility models, Quantitative Finance, 6:6, 513-536

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Quantitative Finance, Vol. 6, No. 6, December 2006, 513–536

Fast strong approximation Monte Carlo schemes

for stochastic volatility models

CHRISTIAN KAHLyz and PETER JACKEL*y

yQuantitative Analytics Group, ABN AMRO, 250 Bishopsgate,London EC2M 4AA, UK

zDepartment of Mathematics, University of Wuppertal, Gaußstraße 20,Wuppertal, D-42119, Germany

(Received 13 December 2005; in final form 5 May 2006)

Numerical integration methods for stochastic volatility models in financial markets arediscussed. We concentrate on two classes of stochastic volatility models where the volatilityis either directly given by a mean-reverting CEV process or as a transformed Ornstein–Uhlen-beck process. For the latter, we introduce a new model based on a simple hyperbolictransformation. Various numerical methods for integrating mean-reverting CEV processesare analysed and compared with respect to positivity preservation and efficiency. Moreover,we develop a simple and robust integration scheme for the two-dimensional system using thestrong convergence behaviour as an indicator for the approximation quality. This method,which we refer to as the IJK (137) scheme, is applicable to all types of stochasticvolatility models and can be employed as a drop-in replacement for the standard log-Eulerprocedure.

Keywords: Stochastic volatility models; Stochastic numerical integration; Strong approxima-tion error; Hyperbolic Ornstein–Uhlenbeck process; Hyperbolic volatility

1. Introduction

Numerical integration schemes for differential equationshave been around nearly as long as the formalism ofcalculus itself. In 1768, Euler devised his famous steppingmethod (Euler 1768), and this scheme has remained thefallback procedure in many applications where allelse fails as well as the benchmark in terms of overallreliability and robustness any new algorithm must com-pete with. Many schemes have been invented since, andfor most engineering purposes involving the numericalintegration of ordinary or partial differential equationsthere are nowadays a variety of approaches available.

With the advent of formal stochastic calculus inthe 1920s and the subsequent application to real worldproblems came the need for numerical integrationof dynamical equations subject to an external force ofrandom nature. Again, Euler’s method came to therescue, first suggested in this context by Maruyama

Maruyama (1955) whence it is also sometimes referredto as the Euler–Maruyama scheme (Kloeden and Platen1992, 1995, 1999).

An area where the calculus of stochastic differentialequations became particularly popular is the mathematicsof financial markets, more specifically the modellingof financial movements for the purpose of pricing andrisk-managing derivative contracts.

Most of the early applications of stochastic calculus tofinance focused on approaches that permitted closed formsolutions, the most famous example probably being theNobel prize winning article by Black and Scholes (1973).With increasing computer power, researchers andpractitioners began to explore avenues that necessitatedsemi-analytical evaluations or even required fullynumerical treatment.

A particularly challenging modelling approach involves

the coupling of two stochastic differential equations

whereby the diffusion term of the first equation is explic-

itly perturbed by the dynamics of the second equation:

stochastic volatility models. These became of interest to

financial practitioners when it was realized that in some

markets deterministic volatility models do not represent*Corresponding author. Email: [email protected]

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online # 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/14697680600841108

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the dynamics sufficiently. Alas, the first publications onstochastic volatility models (Scott 1987, Wiggins 1987,Hull and White 1988) were ahead of their time:the required computer power to use these models ina simulation framework was simply not available,and analytical solutions could not be found. One of thefirst articles that provided semi-analytical solutionswas published by Stein and Stein (1991). An unfortunatefeature of that model was that it did not give enoughflexibility to represent observable market prices, i.e. itdid not provide enough degrees of freedom forcalibration. Heston (1993) published the first model thatallowed for a reasonable amount of calibration freedompermitting semi-analytical solutions. Various otherstochastic volatility models have been published since,and computer speed has increased significantly.However, despite the fact that at the time of this writingcomputer power makes fully numerical treatment ofstochastic volatility a real possibility, comparativelylittle research has been done on the subject of efficientmethods for the numerical integration of these models.In this article, we present and discuss some techniquesthat help to make the use of fully numerically integratedstochastic volatility models a viable alternative tosemi-analytic solutions, despite the fact that majoradvances on the efficient implementation of Heston’smodel have been made (Kahl and Jackel 2005). In sec-tion 2, we present the specific stochastic volatility modelsthat we subsequently use in our demonstrations ofnumerical integration methods, and discuss someof their features in the context of financial marketsmodelling. In section 3, we elaborate on specific methodssuitable for the volatility process in isolation. Next,in section 4, we discuss techniques that accelerate theconvergence of the numerical integration ofthe combined system of stochastic volatility and thedirectly observable financial market variable both withrespect to the discretization refinement required andwith respect to CPU time consumed. This isfollowed by the presentation of numerical results insection 5. Finally, we conclude.

2. Some stochastic volatility models

We consider stochastic volatility models of the form

dSt ¼ �Stdtþ Vpt StdWt, ð1Þ

where S describes the underlying financially observablevariable and V, depending on the coefficient p givenby the specific model, represents either instantaneousvariance ( p ¼ 1=2) or instantaneous volatility ( p¼ 1).

As for the specific processes for instantaneous varianceor volatility, we distinguish two different kinds. The firstkind is the supposition of a given stochastic differentialequation directly applied to the instantaneous varianceprocess. Since instantaneous variance must never benegative for the underlying financial variable to remainon the real axis, we specifically focus on a process for

variance of the form (Cox 1975, Cox and Ross 1976,Beckers 1980, Chan et al. 1992, Andersen andAndreasen 2000)

dVt ¼ �ð� � VtÞdtþ �VqtdZt, Vt0 ¼ V0: ð2Þ

with �, �, �, q > 0, and p ¼ 1=2 in equation (1). Weassume the driving processes Wt and Zt to be correlatedBrownian motions satisfying dWt � dZt ¼ �dt.

The second kind of stochastic volatility model we con-sider is given by a deterministic transformation

�t ¼ �0 � fð ytÞ, f : R ! Rþ, ð3Þ

with f(�) being strictly monotonic and differentiable, of astandard Ornstein–Uhlenbeck process

dyt ¼ ��ytdtþ �ffiffiffiffiffi2�

pdZt, yt0 ¼ y0, ð4Þ

setting Vt ¼ �t and p¼ 1 in equation (1). The transforma-tion f(�) is chosen to ensure that �� 0 for the followingreason. It is, in principle, possible to argue that instanta-neous volatility is undefined with respect to its sign.However, when volatility and the process it is drivingare correlated, a change of sign in the volatility processimplies a sudden change of sign in effective correlation,which in turn implies a reversal of the conditional for-ward Black-implied volatility skew, and the latter is arather undesirable feature to have for reasons of eco-nomic realism. As a consequence of this train of thought,we exclude the Stein–Stein/Schobel–Zhu model (Stein andStein 1991, Schobel and Zhu 1999) which is encompassedabove by setting fðyÞ ¼ y.

In order to obtain a better understanding of the differ-ent ways to simulate the respective stochastic volatilitymodel we first give some analytical properties of thedifferent approaches.

2.1. The mean-reverting CEV process

By mean-reverting CEV process we mean the familyof processes described by the stochastic differentialequation (2). Heston’s model, for instance, is given byq ¼ 1=2 with p ¼ 1=2 in the process for the underlying (1).The family of processes described by (2) has also beenused for the modelling of interest rates (Chan et al. 1992).

For the special case q ¼ 1=2, i.e. for the Hestonvariance process, the stochastic differential equation isalso known as the Cox–Ingersoll–Ross (1985) model. Inthat case, the transition density is known analytically as

pðt0, t,Vt0 ,VtÞ ¼ �2d �Vt, ð Þ ð5Þ

with

� ¼4�

�2 1� e��t� � , ð6Þ

¼4�e��t

�2 1� e��t� �Vt0 , ð7Þ

�t ¼ t� t0, ð8Þ

d ¼4��

�2, ð9Þ

514 C. Kahl and P. Jackel

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where �2dðx, Þ denotes the non-central chi-square densityof variable x with d degrees of freedom and non-centralityparameter . Broadie and Kaya (2004) used this transitiondensity for the Monte Carlo simulation of Europeanoptions.

With q¼ 1, equation (2) turns into a stochasticdifferential equation which is affine in the drift and linearin the diffusion also known as the Brennan–Schwartz(1980) model. To the best of our knowledge, thereare no closed form explicit solutions for this equationallowing for a fully analytical expression, despiteits apparent simplicity. A formal solution for equationsof the form

dXt ¼ a1ðtÞXt þ a2ðtÞð Þdt

þ b1ðtÞXt þ b2ðtÞð ÞdWt, Xt0 ¼ X0, ð10Þ

is described in Kloeden and Platen (1992, 1995, 1999,Chap. 4.2 equation (2.9))

Xt ¼ �t0, t � X0 þ

Z t

t0

a2ðsÞ � b1ðsÞb2ðsÞ

�t0, s

dsþ

Z t

t0

b2ðsÞ

�t0, s

dWs

� �ð11Þ

with �t0, t given by (Kloeden and Platen (1992, 1995,1999, Chap. 4.2 equation (2.7))

�t0, t ¼ exp

Z t

t0

a1ðsÞ �1

2b21ðsÞ

� �dsþ

Z t

t0

b1ðsÞdWs

� �: ð12Þ

Applying this to equation (2) with a1ðtÞ ¼ ��, a2ðtÞ ¼ ��,b1ðtÞ ¼ � and b2ðtÞ ¼ 0 leads to

�t0, t ¼ exp � �þ1

2�2

� �ðt� t0Þ þ �ðWt �Wt0Þ

� �ð13Þ

as well as

Xt ¼ exp � �þ�2

2

!tþ �Wt

" #

� X0 þ

Z t

t0

�� exp �þ�2

2

!s� �Ws

" #ds

" #:

ð14Þ

The functional form of solution (14) is somewhatreminiscent of the payoff function of a continuouslymonitored Asian option in a standard Black–Scholesframework, and thus it may be possible to derive theLaplace transform of the distribution of Xt analytically

following the lead given by Geman and Yor (1993).However, whilst this is noteworthy in its own right, it isunlikely to aid in the development of fast and efficient

numerical integration schemes for Monte Carlosimulations, especially if the ultimate aim is to use theprocess X to drive the diffusion coefficient in a secondstochastic differential equation.

Beyond the cases q¼ 0, q ¼ 1=2, and q¼ 1, as far as weknow, there are no analytical or semi-analytical solutions.Nevertheless, we are able to discuss the boundarybehaviour solely based on our knowledge of the driftand diffusion terms:

(1) 0 is an attainable boundary for 0 < q < 1=2 and forq ¼ 1=2 if �� < �2=2;

(2) 0 is unattainable for q > 1=2;(3) 1 is unattainable for all q>0.

These statements can be confirmed by the aid of Feller’sboundary classification which can be found in Karlin andTaylor (1981). The stationary distribution of this processcan be calculated as (see Andersen and Piterbarg (2004,Prop. 2.4))

ð yÞ ¼ CðqÞ�1y�2qeMðy, qÞ, CðqÞ ¼

Z 1

0

y�2qeMðy, qÞdy

ð15Þ

with the auxiliary function M( y, q) given by

(1) q ¼ 1=2

Mð y, qÞ ¼2�

�2� lnð yÞ � yð Þ; ð16Þ

(2) q¼ 1

Mð y, qÞ ¼2�

�2��=y� lnð yÞð Þ; ð17Þ

(3) 0 < q < 1=2 and 1=2 < q < 1

Mð y, qÞ ¼2�

�2�y1�2q

1� 2q�

y2�2q

2� 2q

!: ð18Þ

The above equations can be derived from theFokker–Planck equation which leads to an ordinarydifferential equation of Bernoulli type. The first momentof the process (2) is given by

EVt ¼ ðVt0 � �Þe��t

þ �: ð19Þ

We can also calculate the second moment for q ¼ 1=2 orq¼ 1:

This means that in the case q¼ 1, for �2 > � which istypically required in order to calibrate to the marketobservable strongly pronounced implied-Black-volatility

E½V2t � ¼

e�2�t e�t � 1� �

2V0 þ e�t � 1� �

��

�2 þ 2��

2�for q ¼

1

2

2e�2�t��ðe2�t� �� 2� �

þ e�t V0 � �ð Þ 2�� 2� �

þ e�2t V0 �

2� 2�

� �þ ��Þ

� ��4 � 3�2�þ 2�2

for q ¼ 1:

8>>>><>>>>:

ð20Þ

Fast strong approximation Monte Carlo schemes for stochastic volatility models 515

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skew, the variance of volatility grows unbounded, despitethe fact that the model appears to be mean-reverting.For long-dated options, this is a rather undesirablefeature to have. On the other hand, in the case q ¼ 1=2,for �2 > 2��, instantaneous variance can attain zero,which is also undesirable for economical reasons. In addi-tion to that, for the modelling of path-dependent deriva-tives, model (2) requires the use of numerical integrationschemes that preserve the analytical properties of thevariance process such as to remain on the real axis, orto simply stay positive. In the next section, we discussalternatives for the generation of the stochastic volatilityprocess that make the integration of volatility itselfpractically trivial.

2.2. Transformed Ornstein–Uhlenbeck

The origin of this process goes back to Uhlenbeck andOrnstein’s (1930) publication in which they describe thevelocity of a particle that moves in an environment withfriction. Doob (1942) first treated this process purelymathematically and expressed it in terms of a stochasticdifferential equation. In modern financial mathematics,the use of Ornstein–Uhlenbeck processes is almostcommonplace. The attractive features of an Ornstein–Uhlenbeck process are that, whilst it provides a certaindegree of flexibility over its auto-correlation structure, itstill allows for the full analytical treatment of a standardGaussian process.

In this article, we chose formulation (4) to describethe Ornstein–Uhlenbeck process since we prefer aparametrization that permits complete separationbetween the mean reversion speed and the variance ofthe limiting or stationary distribution of the process.The solution of (4) is

Yt ¼ e��t y0 þ

Z t

0

e�u�ffiffiffiffiffi2�

pdZu

� �ð21Þ

with initial time t0 ¼ 0. In other words, the stochasticprocess at time t is Gaussian with

Yt � N y0e��t,�2 1� e�2�t

� �� ð22Þ

and thus the stationary distribution is Gaussian with var-iance �2: a change in parameter � requires no rescaling of� if we wish to hold the long-term uncertainty in theprocess unchanged. It is straightforward to extend theabove results to the case when �(t) and �(t) are functionsof time (Jackel 2002). Since the variance of the drivingOrnstein–Uhlenbeck process is the main criterion thatdetermines the uncertainty in volatility for the financialunderlying process, all further considerations are primar-ily expressed in terms of

�ðtÞ :¼ � �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e�2�t

p: ð23Þ

There are fundamental differences between therequirements in the financial modelling of underlyingasset prices, and the modelling of instantaneous

stochastic volatility, or indeed any other not directlymarket-observable quantity. For reasons of financialconsistency, we frequently have to abide by no-arbitragerules that impose a specific functional form for theinstantaneous drift of the underlying. In contrast,the modelling of stochastic volatility is typically moregoverned by long-term realism and structural similarityto real-world dynamics, and no externally given driftconditions apply. No-arbitrage arguments and theirimplied instantaneous drift conditions are omnipresentin financial arguments, and as a consequence, mostpractitioners have become used to thinking of stochasticprocesses exclusively in terms of an explicit stochasticdifferential equation. However, when there are noexplicitly given conditions on the instantaneous drift, itis, in fact, preferable to model a stochastic process in themost analytically convenient form available. In otherwords, when preferences as to the attainable domainof the process are to be considered, it is in practicemuch more intuitive to start with a simple process offull analytical tractability, and to transform its domainto the target domain by virtue of a simpleanalytical function. For the modelling of stochasticvolatility, this means that we utilize the flexible yettractable nature of the Ornstein–Uhlenbeck process (4)in combination with a strictly monotonic and differenti-able mapping function f : R ! Rþ.

One simple analytical transformation we consider isthe exponential function, and the resulting stochasticvolatility model was first proposed in Scott (1987,equation (7)). The model is intuitively very appealing:for any future point in time, volatility has a lognormaldistribution which is a very comfortable distribution forpractitioners in financial mathematics. Alas, though,recent research (Andersen and Piterbarg 2004) hascast a shadow on this model’s analytical features.It appears that, in its full continuous formulation, thelognormal volatility model can give rise to unlimitedhigher moments of the underlying financial asset.However, as has been discussed and demonstrated atgreat length for the very similar phenomenon ofinfinite futures returns when short rates are driven by alognormal process (Hogan and Weintraub 1993,Sandmann and Sondermann 1994, 1997a, b), thisproblem vanishes as soon as the continuous processmodel is replaced by its discretized approximationwhich is why lognormal volatility models remain numeri-cally tractable in applications. Still, in order to avoid thisproblem altogether, we introduce an alternative tothe exponential transformation function which is givenby a simple hyperbolic form.

In the following, we refer to

fexpð yÞ :¼ ey �expð yÞ :¼ �0 � fexpð yÞ ð24Þ

as the exponential volatility transformation also known asScott’s (1987) model, and to

fhypð yÞ :¼ yþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiy2 þ 1

q�hypð yÞ :¼ �0 � fhypð yÞ ð25Þ


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as the hyperbolic volatility transformation. The densitiesof the exponential and hyperbolic volatilities are given by

expð�exp, �0, �Þ ¼’ f�1

expð�exp=�0Þ, ��

d�exp�dy

, ð26Þ

hypð�hyp, �0, �Þ ¼’ f�1

hypð�hyp=�0Þ, ��

d�hyp�dy

ð27Þ

with

f�1exp

�

�0

� �¼ ln

�

�0

� �, ð28Þ

f�1hyp

�

�0

� �¼

�

�0��0�

� �2, ð29Þ

d�expdy

¼ �exp, ð30Þ

d�hypdy

¼2�0�

2hyp

�20 þ �2hyp

ð31Þ

and

’ð y, �Þ :¼exp½�1=2 y=�ð Þ

2�

� �ffiffiffiffiffiffi2

p : ð32Þ

The hyperbolic transformation has been chosen to matchthe exponential form as closely as possible near the origin,and only to differ significantly in the regions of lowerprobability given for jy=�j > 1. The functional forms ofthe exponential and hyperbolic transformation are shownin comparison in figure 1.

2.2.1. Exponential versus hyperbolic transformation. Infigure 2, we compare the densities of theOrnstein–Uhlenbeck process transformed with (24)and (25) given by equations (26) and (27). At first glanceon a linear scale, we see a reasonable similarity betweenthe two distributions. However, on a logarithmic scale,the differences in the tails of the distributions becomeclear: the hyperbolic transformation has significantlylower probability for both very low values as well as forlarge values.

Returning to the analytical form of the density func-tions (26) and (27), it is interesting to note that, given thaty ¼ f�1

ð�=�0Þ is Gaussian, the volatility distributionimplied by the inverse hyperbolic transformation (29)is nearly Gaussian for large values of instantaneousvolatility � � 0 since we have

�hyp � 2�0 � y for �hyp � 0: ð33Þ

This feature is particularly desirable since it ensuresthat the tail of the volatility distribution at the higherend is as thin as the Gaussian process itself, and thusno moment explosions are to be feared for the underlying.Conversely, for small values of instantaneousvolatility � 1, the volatility distribution implied by

the hyperbolic volatility model is nearly inverseGaussian because of

�hyp � ��02

.y for �hyp�!0: ð34Þ

0

1

2

3

4

5

6

7

8

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

fexp(y)

fhyp(y)

Figure 1. The exponential and hyperbolic transformationfunctions.

0

0.5

1

1.5

2

2.5

3

3.5

4

0% 20% 40% 60% 80% 100%σ

ψexp(σ,σ0,η)

ψhyp(σ,σ0,η)

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0% 50% 100% 150% 200%σ

ψexp(σ,σ0,η)

ψhyp(σ,σ0,η)

(a)

(b)

Figure 2. Densities of instantaneous volatility using theexponential and the hyperbolic transformation of the drivingOrnstein–Uhlenbeck process for �0 ¼ 25% and � ¼ 1=2. Notethe distinctly different tails of the distributions.


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In a certain sense, the hyperbolic model can be seenas a blend of an inverse Gaussian model at the lowerend of the distribution and a Gaussian density at theupper end, with their respectively thin tails. In contrast,exponential volatility is simply lognormally distributed,which in turn gives rise to distinctly fatter tails than thenormal (at the high end) or inverse normal (at thelow end) density.

From the basis of our complete analytical understand-ing of both the exponential and the hyperbolic volatilityprocess, we can use Ito’s lemma to derive their respectivestochastic differential equations. For the exponentialtransformation (24) we obtain Scott’s original SDE(Scott 1987, equation (7))

d� ¼ �� 2 � lnð�=�0Þ� �

dtþ ��ffiffiffiffiffi2�

pdZ: ð35Þ

It is remarkable to see the difference in the stochasticdifferential equation we obtain for the hyperbolicvolatility process (25):

d� ¼ ��6 þ �4�20 � ð8�2 þ 1Þ�2�40 � �

60

�2 þ �20� �3 dt

þ ��ffiffiffiffiffi8�

p ��0�2 þ �20

dZ:

ð36Þ

The complexity of the explicit form (36) of the hyperbolicvolatility process may help to explain why it has, to thebest of our knowledge, not been considered before.As we know, though, the resulting process is of remark-able simplicity and very easy to simulate directly, whilstovercoming some of the weaknesses of the (calibrated)CIR/Heston process (namely that zero is attainable), aswell as the moment divergence when volatility or varianceis driven by lognormal volatility as incurred by theBrennan–Schwartz process for volatility and Scott’smodel.

The moments of the exponential transformationfunction are

E fexpð yÞ� �m

¼ e1=2m2�2 : ð37Þ

For the hyperbolic transformation we obtain the generalsolution

E fhypðyÞ� �m �

¼23=2nffiffiffiffiffiffi4

p �n�1þ n

2

� �1F1 �

n

2, 1� n,

1

2�2

� �

þ2�3=2nffiffiffiffiffiffi

4p ��n�

1� n

2

� �1F1

n

2, 1þ n,

1

2�2

� �ð38Þ

in terms of Kummer’s hypergeometric function 1F1.The first two moments, specifically, are given by

Efhypð yÞ� �1�

¼ffiffiffi2

p� � �U �

1

2, 0,

1

2�2

� �, ð39Þ

Efhypð yÞ� �2�

¼ 1þ 2�2, ð40Þ

where Uða, b, zÞ is the logarithmic confluent hypergeom-etric function. More revealing than the closed form forthe moments of the respective transformation functions isan analysis based on their Taylor expansions

fexpð yÞ ¼ 1þ yþy2

2þO y3

� �, ð41Þ

fhypð yÞ ¼ 1þ yþy2

2þO y4

� �: ð42Þ

Thus, for both of these functions,

f��ð yÞð Þn¼ 1þ n � yþ

n

2

� þn

2

h i� y2 þO y3

� �: ð43Þ

Since y is normally distributed with mean 0 and variance�2, and since all odd moments of the Gaussiandistribution vanish, this means that for both theexponential and the hyperbolic transformation we have

Ef��ðyÞð Þ

n�¼ 1þ

n

2

� þn

2

h i� �2 þO �4

� �: ð44Þ

The implication of (44) is that all moments of theexponential and the hyperbolic transformation functionagree up to order Oð�3Þ. We show an example for this infigure 3. As � increases, the moments of the exponentialfunction grow faster by a term of order Oð�4Þ.

3. Numerical integration of mean-reverting

CEV processes

The numerical integration of the coupled stochastic vola-tility system (1) and (2) is composed of two different parts.First, we have to find an appropriate method for theapproximation of the stochastic volatility process itself,and second we need to handle the dynamics of the finan-cial underlying (1) whose diffusion part is affected by thestochasticity of volatility.

Since the volatility process does not explicitly dependon the underlying, we can treat it separately. In order toretain numerical stability and to achieve good conver-gence properties, it is desirable for the numerical integra-tion scheme of the volatility or variance process to

01

2

3

4 0

0.1

0.2

0.3

0.4

0.5

1.2

1.4

1.6

01

2

3

E[ fexp(y)n]for y ~ (0, h2)

E[ fhyp(y)n]

h

n

Figure 3. Comparison of the moments of the exponential andthe hyperbolic transformation functions. Note that the floorlevel is exactly 1.


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preserve structural features such as positivity. For theexponentially or hyperbolically transformed Ornstein–Uhlenbeck process, this is trivially taken care of by thetransformation function itself. For the mean-revertingCEV process (2), however, the design of a positivity pre-serving scheme is a task in its own right. The simplestapproach, for instance, namely the explicit Euler scheme

Xnþ1 ¼ Xn þ �ð� � XnÞ�tn þ �Xqn�Wn, ð45Þ

fails to preserve positivity. The same deficiency is exhib-ited by the standard Milstein and the Milsteinþ schemewhose formulæ we give in the first two subsections of theappendix, respectively. The balanced implicit method(BIM) as introduced by Milstein et al. (1998), however,

Xnþ1 ¼ Xn þ �ð� � XnÞ�tn þ �Xqn�Wn

þ CðXnÞðXn � Xnþ1Þ, ð46Þ

CðXnÞ ¼ cBIM0 ðXnÞ�tn þ cBIM1 ðXnÞj�Wnj ð47Þ

with control functions

cBIM0 ðxÞ ¼ �, ð48Þ

cBIM1 ðxÞ ¼ �x1�q, ð49Þ

is able to preserve positivity as is shown in Schurz (1996).Alas, this scheme only achieves the same strong order ofconvergence as the Euler scheme, i.e. 1=2. This means,whilst the BIM helps to overcome the problem of spuri-ous negative values for variance, it does not increase theconvergence speed. In fact, when a step size is chosen suchthat for the specific set of parameters at hand the explicitEuler scheme is usable,y the BIM often has worse con-vergence properties than the explicit Euler method. Thisfeature of the BIM is typically caused by the fact that theuse of the weight function cBIM1 effectively increases theunknown coefficient dominating the leading error terms.

Another scheme that has been shown to preserve posi-tivity for certain parameter ranges is the adaptive Milsteinscheme (Kahl (2004, equation 4.10)) with suitable stepsize��n and ~z � N 0, 1ð Þ

Xnþ1 ¼ Xn þ �ð� � XnÞ��n þ �Xqn

ffiffiffiffiffiffiffiffi��n

p~z

þ1

2�2qX2q�1

n ��n ~z2�1� �

: ð50Þ

Unfortunately, this scheme requires adaptive resamplingand thus necessitates the use of a pseudo-random numberpipeline which in turn disables or hinders a whole host ofindependently available convergence enhancement tech-niques such as low-discrepancy numbers, importance sam-pling, stratification, Latin-hypercube methods, etc. Anadvanced method that obviates the use of pseudo-random

number pipelines is based on the combination of theMilstein scheme with the idea of balancing: the BalancedMilstein Method (BMM)

Xnþ1 ¼ Xn þ �ð� � XnÞ�tn þ �Xqn�Wn

þ1

2�2qX2q�1

n �W2n ��tn

� �þDðXnÞ Xn � Xnþ1

� �,

ð51Þ

DðXnÞ ¼ d BMM0 ðXnÞ�tn þ d BMM

1 ðXnÞ �W2n ��tn

� �: ð52Þ

As in the BIM we can control the integration steps byusing weighting functions d BMM

0 ð�Þ and d BMM1 ð�Þ. The

choice of these weighting functions strongly depends onthe structure of the SDE. It can be shown (see Kahl andSchurz (2005, Theorem 5.9)) that the BMM preservespositivity for the mean-reverting CEV model (2) withthe following choice

d BMM0 ðxÞ ¼ ��þ

1

2�2qjxj2q�2, ð53Þ

d BMM1 ðxÞ ¼ 0: ð54Þ

The parameter � 2 ½0, 1� provides some freedom forimproved convergence speed but it has to be chosensuch that

�tn <2q� 1

2q�ð1��Þ: ð55Þ

It is always safe to choose � ¼ 1, though, for improvedperformance, we used � ¼ 1=2 whenever this choice waspossible.z

The above-mentioned integration methods, namely thestandard explicit Euler scheme, the BIM, the BMM, andthe adaptive Milstein scheme, deal with the stochasticdifferential equation in its original form (2). Anotherapproach to integrate (2) whilst preserving positivity isto transform the stochastic differential equation to loga-rithmic coordinates using Ito’s lemma as suggested byAndersen and Brotherton-Ratcliffe (2001). Applying thisto the mean-reverting CEV process leads to

d lnVt ¼2�ð� � VtÞ � �

2V2q�1t

2Vt

dtþ �Vq�1t dZt, ð56Þ

which can be solved by the aid of a simple Euler scheme.The major drawback with this approach is that, whilst theEuler scheme applied to the transformed stochastic differ-ential equation (56) preserves positivity, it is also likely tobecome unstable for suitable time steps (Andersen andBrotherton-Ratcliffe 2001). These instabilities are a directconsequence of the divergence of both the drift and thediffusion terms near zero. For that reason Andersen and

yFor most schemes, spurious negative values incurred as an undesirable side effect of the numerical method disappear as the stepsize �t is decreased. For a negative variance to appear at any one step, the drawn normal variate generating the step typically has toexceed a certain threshold. This threshold tends to grow as step size decreases. Thus, with decreasing step size, eventually, thethreshold exceeds the maximum standard normal random number attainable on the finite representation computer system used.zFor q ¼ 1=2 the numerator becomes zero. Despite this, positivity can be preserved with d BMM

0 ¼ �.


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Brotherton-Ratcliffe suggested a moment matchedlog-normal approximation

Vnþ1 ¼ � þ Vn � �ð Þe��tn� �

e�1=2�2nþ�n ~z, ð57Þ

�n ¼ ln 1þ1=2�2V2p

n ��1 1� e�2��tn� �

� þ Vn � �ð Þe��tn� �2

!ð58Þ

with ~z � N 0, 1ð Þ. We will refer to this integration schemeas moment matched Log-Euler in the following.This method is at its most effective for the Brennan-Schwartz model (67) as we can see in figure 6(b) sincefor p¼ 1 the logarithmic transformation leads to an addi-tive diffusion (56) term. However, even in that case, it isoutperformed by the bespoke method we call PathwiseAdapted Linearization which is explained in section 3.1,as well as the BMM (51). For the Heston case, where thestochastic volatility is given by the Cox–Ingersoll–Rossequation with q ¼ 1=2 which is shown in figures 4and 5, the moment matched log-Euler method has prac-tically no convergence advantage over straightforwardexplicit Euler integration as long as the size of � is reason-ably small. Contrarily, the approximation quality of allintegration schemes is decisively reduced when dealingwith large � as we can see in figure 4(b). Making matterseven worse, one can observe that schemes of Milstein typeare losing their strong convergence order of 1. Theexplanation for this behaviour is rather simple: theMilstein method is not even guaranteed to converge atall for the mean-reverting CEV process (2)! Having acloser look at the diffusion bðxÞ ¼ �xq, we recognizethat for q<1 this function is not continuously differenti-able on R which is necessary for the application ofstochastic Taylor expansion techniques. Nonetheless, aslong as the stochastic process is analytically positive, i.e.x>0 there exists a local stochastic Taylor expansionpreserving strong convergence of the Milstein method.However, when zero is attainable, the discontinuity ofthe first derivative of the diffusion b(x) reduces the strongconvergence order to 1=2.

In figures 4, 5 and 6 we present examples for theconvergence behaviour of the different methods incomparison. For the standard Milstein (A5) and theMilsteinþ scheme (A15), for some of the parameterconfigurations, it was necessary to floor the simulatedvariance values at zero since those schemes do notpreserve positivity by construction.

The depicted strong approximation convergencemeasure is given by the L2 norm of the difference betweenthe simulated terminal value, and the terminal value ofthe reference calculation, averaged over all M paths, i.e.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

M

XMi¼1

XðnstepsÞi ðTÞ � X

ðnreferenceÞi ðTÞ

� 2vuut : ð59Þ

This quantity is shown as a function of average CPU time

per path. This was done because the ultimate criterion for

the choice of any integration method in applications is the

cost of accuracy in terms of calculation time since calcula-

tion time directly translates into the amount of required

hardware for large scale computations such as overnight

risk reports, or into user downtime when interactive

valuations are needed. This does, of course, make the

results dependent on the used hardware,y not only in

absolute terms but also in relative terms since different

processor models require different numbers of CPU

clock cycles for all the involved basic floating point

operations. Nevertheless, the pathwise error as a function

of average CPU time is probably the most

yThroughout this article, all calculations shown were carried out on a processor from the Intel Pentium series (Family 6, Model 9,Stepping 5, Brand id 6, CPU frequency 1700MHz).

0.0001

0.001

0.01

1 10 100

Euler

Milstein

Milstein+

Balanced Implicit Method

Balanced Milstein Method

Moment matched log-Euler

Pathwise Adapted Linearisation Quadratic

Pathwise Adapted Linearisation Quartic

0.01

0.1

1 10 100

Euler

Milstein

Milstein+





(a)

(b)

Figure 4. Strong convergence measured by expression (59) as afunction of CPU time [in ms] averaged over 32 767 paths for themean reverting CEV model (2) for q ¼ 1=2, �¼ 1,V0 ¼ � ¼ 1=16, T¼ 1, cBIM0 ¼ 1, cBIM1 ¼ �=

ffiffiffix

p, d BMM

0 ¼ �,d BMM1 ¼ 0. The number generator was the Sobol’ method.

(a) � ¼ 0:2, �2 � 2�� ¼ �0:085; zero is unattainable.(b) � ¼ 0:8, �2 � 2�� ¼ 0:515; zero is attainable.


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significant criterion for the quality of any integrationmethod. Examples for this consideration are the fact

that in figure 6 the nominal advantage of the moment

matched Log-Euler is almost precisely offset by the

additional calculation time it requires compared to

the Euler scheme, and also that in figure 6(b) the

relative performance of the BMM is compatible with

the scheme denoted as Pathwise Adapted Linearization

which is explained in section 3.1.1.The curves in figures 4, 5 and 6 have been constructed

by repeated simulation with increased numbers of steps inthe Brownian bridge Wiener path generation in powers oftwo from 1 to 128:

nsteps 2 f1, 2, 4, 8, 16, 32, 64, 128g: ð60Þ

The reference solution was always computed with 215

steps. The number generation mechanism used was theSobol’ algorithm (Jackel 2002) throughout apart from

figure 5(b) where we also show the results from usingthe Mersenne Twister (Matsumoto and Nishimura 1998)in comparison. Note that the results are fairly insensitiveto the choice of number generator. In addition to themethods discussed above, we also included the resultsfrom bespoke schemes denoted as Pathwise AdaptedLinearization. These schemes are carefully adapted tothe respective equation and we introduce them in thefollowing section.

3.1. Pathwise approximations for specific cases

Yet another approach for the numerical integration ofstochastic differential equations of the form

dX ¼ aðXÞdtþ bðXÞdZ, ð61Þ

0.001

0.01

0.1

1 10 100

Euler

Milstein

Milstein+






0.001

0.01

0.1

1 10 100

Euler

Milstein

Milstein+






(a)

(b)

Figure 5. Strong convergence measured by expression (59) as afunction of CPU time [in ms] averaged over 32 767 paths for themean reverting CEV model (2) for q ¼ 1=2, �¼ 1,V0 ¼ � ¼ 1=16, �¼ 0.5, �2 � 2�� ¼ 0:125, zero is attainable,T¼ 1, cBIM0 ¼ 1, cBIM1 ¼ �=

ffiffiffix

p, dBMM

0 ¼ �, dBMM1 ¼ 0. The num-

ber generator method was (a) Sobol’s and (b) the MersenneTwister.

0.0001

0.001

0.01

1 10 100

Euler

Milstein

Milstein+




0.0001

0.001

0.01

1 10 100

Euler

Milstein

Milstein+




Pathwise Adapted Linearisation

(a)

(b)

Figure 6. Strong convergence measured by expression (59) asa function of CPU time [in ms] averaged over 32 767 pathsfor the mean reverting CEV model (2) for �¼ 1,V0 ¼ � ¼ 0:0625 ¼ 1=16, T¼ 1. The number generator was theSobol’ method. (a) q ¼ 3=4, c BIM

0 ¼ 1, c BIM1 ¼ �jxj�1=4,

d BMM0 ¼ ð�=2Þþ 3=8ð�2=

ffiffiffix

pÞ, d BMM

1 ¼ 0. (b) q¼ 1,c BIM0 ¼ 1,c BIM

1 ¼ �, d BMM0 ¼ 1=2 �þ �2

� �, d BMM

1 ¼ 0.


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as is the case for (2), is to apply Doss’s (1977) method ofconstructing pathwise solutions first used in the context ofnumerical integration schemes by Pardoux and Talay(1985). The formal derivation of Doss’s pathwise solutioncan be found in Karatzas and Shreve (1991, pp. 295–296).

In practice, Doss’s method can hardly ever be applieddirectly since it is essentially just an existence theoremthat states that any process for which there is a uniquestrong solution can be seen as a transformation of thesolution to an ordinary differential equation with astochastic inhomogeneity, i.e. a solution of the form

X ¼ f ðY,ZÞ with boundary condition f ðY,Z0Þ ¼ Y

ð62Þ

with

dY ¼ gðY,ZÞdt ð63Þ

implying

X0 ¼ Y0, ð64Þ

whereby the functions f and g can be derived construc-tively from the stochastic differential equation for X:

@Yf ðY,ZÞ ¼ exp

Z Z

Z0

b0 f ðy, zÞð Þdz

� �, ð65Þ

gðY,ZÞ ¼ a f ðY,ZÞð Þ �1

2b f ðY,ZÞð Þb0 f ðY,ZÞð Þ

� �

� exp �

Z Z

Z0

b0ð f ðY, zÞÞdz

� �: ð66Þ

Even though one can rarely use Doss’s method in its fullanalyticity, one can often devise a powerful bespokeapproximate discretization scheme for the stochastic dif-ferential equation at hand based on Doss’s pathwise exis-tence theorem by the aid of some simple approximativeassumptions without the need to go through the Dossformalism itself.

3.1.1. Pathwise approximation of the Brennan-Schwartz

SDE. For q¼ 1, the mean-reverting CEV process (2)becomes

dX ¼ �ð� � XÞdtþ �XdZ: ð67Þ

Assuming

� > 0, � > 0, � > 0, and Xð0Þ > 0, ð68Þ

we must have

Xt � 0 for all t > 0: ð69Þ

Using equation (65), we obtain

f ðY,ZÞ ¼ Ye�Z: ð70Þ

and by the aid of (66), we have

dY ¼ ��e��Z � �þ1

2�2

� �Y

� �dt: ð71Þ

We cannot solve this equation directly. Also, a directlyapplied explicit Euler scheme would permit Y to crossover to the negative half of the real axis and thusX ¼ f ðY,ZÞ ¼ Ye�Z would leave the domain of (67).What’s more, an explicit Euler scheme applied toequation (71) would mean that, within the scheme,we interpret Zt as a piecewise constant function. Notsurprisingly, it turns out below that we can do betterthan that!

Recall that, for the given time discretization, we explic-itly construct the Wiener process values Z(ti) and thus, forthe purpose of numerical integration of equation (67),they are known along any one given path. If we nowapproximate Zt as a piecewise linear function in betweenthe known values at tn and tnþ1, i.e.

Zt � ntþ �n for t 2 ½tn, tnþ1� ð72Þ

with

�n ¼ ZðtnÞ � ntn and n ¼Zðtnþ1Þ � ZðtnÞ

tnþ1 � tn, ð73Þ

then we have the approximating ordinary differentialequation

dY ¼ ��e��ð ntþ�nÞ � �þ1

2�2

� �Y

� �dt: ð74Þ

Using the abbreviations

�n :¼ �þ1

2�2 � � n, �tn :¼ tnþ1 � tn, and

Znþ1 :¼ Zðtnþ1Þ

we can write the solution to equation (74) as

Ynþ1 ¼ Yne� �þ1

2�2ð Þ�tn þ �� e��Znþ1

1� e��n�tn

�n

� �, ð75Þ

which gives us

Xnþ1 ¼ Xne��n�tn þ ��

1� e��n�tn

�n

� �: ð76Þ

This scheme is unconditionally stable. We refer to it asPathwise Adapted Linearization in the following. Apartfrom its stability, this scheme has the additional desirableproperty that, in the limits � ! 0 and/or �! 0, i.e. in thelimit of equation (67) resembling a standard geometricBrownian motion, it is free of any approximation.Since in practice � and/or � tend to be not too large, thescheme’s proximity to exactness translates into aremarkable accuracy when used in applications.

It is interesting to note that a similar approach basedon replacing the term dZ directly in the stochastic differ-ential equation

dX ¼ �ð� � XÞdtþ �XdZ ð77Þ

by a linear approximation dZ � dt gives rise to a schemethat does not converge in the limit �t ! 0 as first observedby Wong and Zakai (1965). However, if we make the


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same replacement in the Milstein scheme and drop termsof order Oðdt2Þ and higher, which for (67) means

�X � �ð� � XÞ�tþ �X�Zþ1

2�2X �Z2

��t� �

, ð78Þ

�X � �ð� � XÞ�tþ �X �tþ1

2�2X 2�t2 ��t

� �, ð79Þ

dX

dt� �ð� � XÞ �

1

2�2Xþ � X, ð80Þ

and integrate, we arrive at exactly the same scheme (76)as if we had gone through the full Doss formalism.The reason for this is that the lowest order schemethat includes explicitly all terms that are individuallyin expectation of order dt is the Milstein scheme,not the Euler scheme, and the difference terms arecrucial to preserve strong convergence when weintroduce piecewise linearization of the discretizedWiener process.

3.1.2. Pathwise approximation of the Cox–Ingersoll–Ross/

Heston SDE. The special case q ¼ 1=2 of (2) represents

the stochastic differential equation of the variance processin the Heston (1993) model, as well as the short rateprocess in the Cox–Ingersoll–Ross (1985) model

dV ¼ �ð� � VÞdtþ �ffiffiffiffiV

pdZ: ð81Þ

In this case, an explicit solution of the Doss formal-ism (65) is not obvious. However, by conditioning onone specific path in Z we can bypass this difficulty bydirectly approximating Zt as a piecewise linear functionin between the known values as given in equations (72)and (73). Using the resulting dependency dZ ¼ ndt in theMilstein scheme applied to (81)

dV � �ð� � VÞdtþ �ffiffiffiffiV

pdZþ

1

4�2 dZ2

� dt� �

, ð82Þ

i.e.

dV � �ð� � VÞdtþ �ffiffiffiffiV

p ndtþ

1

4�2 2ndt

2� dt

� �, ð83Þ

and dropping terms of order dt2, we obtain the approx-imate ordinary differential equation

dV

dt� �ð� � VÞ �

1

4�2 þ � n

ffiffiffiffiV

pð84Þ

which has the implicit solution

t� tn ¼ TðVt, nÞ � TðVtn , nÞ ð85Þ

with

Tðv, Þ :¼2�

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 2þ4��2��2

p atanh2�

ffiffiffiv

p�� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2þ4��2��2p

!

�1

�ln � v� �ð Þþ

1

4�2��

ffiffiffiv

p� �

:

ð86Þ

Equation (86) can be solved numerically comparativelyreadily since we know that, given n, over the time stepfrom tn to tnþ1, Vt will move monotonically, and that forall �tn :¼ ðtnþ1 � tnÞ we have

Vtnþ1>

� n�� 2�

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� n2�

� �2

þ� ��2

4�

s0@

1A2

, ð87Þ

which can be shown by setting the argument of the loga-rithm in the right-hand side of equation (86) to zero.Alternatively, an inverse series expansion can be derived.Up to order Oð�t4nÞ, we find

with

~� :¼ � ��2

4�: ð89Þ

The shape of the curves generated by (86) and its fourthorder inverse expansion (88) is shown in figure 7 wherevalues for directly represent the standard normal devia-tion equivalent of the drawn Gaussian random number.In the following, we denoted the expansion (88) asPathwise Adapted Linearization Quartic, and its second-order truncation

Vnþ1 ¼ Vn þ �ð ~� � VnÞ þ � nffiffiffiffiffiffiVn

p� �tn

� 1þ� n � 2�

ffiffiffiffiffiffiVn

p

4ffiffiffiffiffiffiVn

p �tn

� �þOð�t3nÞ

ð90Þ

as Pathwise Adapted Linearization Quadratic. We onlyshow results for expansions of even order for reasons ofnumerical stability since all odd order expansion canreach zero which is undesirable. For small values of �as in figure 4(a) both schemes are remarkably effective.Unfortunately, these schemes are inappropriate for largevalues of � due to numerical instabilities.

4. Approximation of stochastic volatility models

In this section, we discuss the numerical treatment of thefull two-dimensional stochastic volatility model.

Vnþ1 ¼Vn þ �ð ~� � VnÞ þ � nffiffiffiffiffiffiVn

p� ��tn

�

"1þ

� n � 2�ffiffiffiffiffiffiVn

p

4ffiffiffiffiffiffiVn

p �tn þ� Vn 4�


p� 3� n

� �� n ~�

� �24


p 3�t2n

þ� 3� n� ~�

2þ �V2

n 7� n � 8�ffiffiffiffiffiffiVn

p� �þ 2� n ~�


p� n þ �


p� �� 192


p 5�t3n

#þOð�t5nÞ

ð88Þ


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Irrespective of the volatility or variance process, thedynamics of the financial underlying are given byequation (1). As for the stochasticity of volatility/var-iance, both the transformed Ornstein–Uhlenbeck processas well as the mean-reverting CEV process (2) can be castin the form

dVt ¼ aðVtÞdtþ bðVtÞdZt: ð91Þ

For the mean-reverting CEV process, the functionalforms for a and b are directly given. For the exponentiallyand hyperbolically transformed Ornstein–Uhlenbeckprocess, they can be obtained from (35) and (36),respectively.

In logarithmic coordinates, the process equation for thefinancial underlying is given by

lnSt ¼ lnSt0 þ

Z t

t0

�ðsÞds�1

2

Z t

t0

V2ps dsþ

Z t

t0

VpsdWs:

ð92Þ

The easiest approach for the numerical integration of (92)is the Euler–Maruyama scheme

lnStnþ1¼ lnStn þ ��tn �

1

2V

2ptn�tn þ V

ptn�Wn: ð93Þ

This scheme has strong convergence order 1=2, is veryeasy to implement, and will be our benchmark for allother methods discussed in the following.

An alternative is of course the two-dimensionalMilstein scheme (see Appendix A.3) which has strongconvergence order 1. It requires the simulation of thedouble Wiener integral

~Ið2, 1Þðt0, tÞ ¼

Z t

t0

Z s

t0

d ~W2ðuÞd ~W1ðsÞ ð94Þ

for two uncorrelated standard Wiener processes ~W1 and~W2. The standard approximation for this cross-termrequires several additional random numbers which weconsider undesirable for the same reasons we gave toexclude the adaptive Milstein scheme (50). There are,however, approaches (Gaines and Lyons 1994,Abe 2004) to avoid the drawing of many extra randomnumbers by using the relation of this integral to theLevy area (Levy 1951)

Að1, 2Þðt0, tÞ ¼

Z t

t0

Z s

t0

d ~W1ðuÞd ~W2ðsÞ � d ~W2ðuÞd ~W1ðsÞ� �

:

ð95Þ

The idea is to employZ t

t0

Z s

t0

d ~W1ðuÞd ~W2ðsÞ þ d ~W2ðuÞd ~W1ðsÞ� �

¼ � ~Wðt0, tÞ1 � ~W

ðt0, tÞ2

ð96Þ

to obtain

~Ið2, 1Þðt0, tÞ ¼1

2� ~W

ðt0, tÞ1 � ~W

ðt0, tÞ2 � Að1, 2Þðt0, tÞ

� : ð97Þ

The joint density of the Levy area is known semi-analytically

�ða, b, cÞ ¼1

22

Z 1

0

x

sinhðxÞexp

�ðb2 þ c2Þx

2 tanhðxÞ

" #cosðaxÞdx

ð98Þ

with a ¼ Að1, 2Þð0, 1Þ, b ¼ � ~Wð0, 1Þ1 and c ¼ � ~W

ð0, 1Þ2 . Hence,

the simulation of the double integral (94) is reduced to thedrawing of one additional random number (conditionalon � ~W1 and � ~W2) from this distribution. Gainesand Lyons (1994) used a modification of Marsaglia’s rec-tangle-wedge-tail method (see Marsaglia et al. 1964, 1976)

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

0 0.2 0.4 0.6 0.8 1

σ (t

)

t

β = 3

4th order expansion for β = 3

β = 2


β = 1


β = 0


β = -1

4th order expansion for β = -1

β = -2


β = -3


Figure 7. Approximation (86) and its quartic expansion (88) for the CIR/Heston volatility process for �ð0Þ ¼ffiffiffiffiffiffiffiffiffiffiVð0Þ

p¼ 20%,

� ¼ Vð0Þ, � ¼ 20%, � ¼ 1 over a unit time step for different levels of the variate ¼ Zð1Þ � Zð0Þ.


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to draw from (98) which works well for small step sizes�tn. We are, however, interested in methods that alsowork well for moderately large step sizes, and are simplein their evaluation analytics in order to be sufficiently fastto be useful for industrial purposes.

In essence, all of the above means that we would like toconstruct a fast numerical integration scheme without theneed for auxiliary random numbers. The formal solu-tion (92) requires that we handle two stochastic integralterms. First, we need to approximate the stochastic partof the drift Z t

t0

V 2ps ds, ð99Þ

and second, we have to simulate the diffusion termZ t

t0

V psdWs: ð100Þ

For both parts we make intensive use of the Ito–Taylorexpansion of the process followed by the mth powerof Vs,

Vms ¼ Vm

t0 þ

Z s

t0

mVm�1u bðVuÞdZu

þ

Z s

t0

mVm�1u aðVuÞ þ

1

2mðm� 1ÞVm�2

u b2ðVuÞ

!du,

ð101Þ

with positive exponent m, for any s 2 ½t0, t�. The term thatdominates the overall scheme’s convergence is the Wienerintegral over dZu.

4.1. Interpolation of the drift term (99)

A simple way to improve the approximation of the driftintegral somewhat isZ tnþ1

tn

V2ps ds �

1

2V

2ptnþ V

2ptnþ1

� �tn, �tn ¼ ðtnþ1 � tnÞ

ð102Þ

which gives us


1

4V

2ptnþ V

2ptnþ1

� �tn þ V

ptn�Wn:

ð103Þ

This Drift interpolation scheme comprises practicallyno additional numerical effort due to the fact that wealready know the whole path of the volatility Vti

.Unfortunately, a pure drift interpolation has only aminor impact on the strong approximation quality.Moreover, having a closer look at figure 8, it seems thatthe Drift interpolation method is inferior to the standardlog-Euler scheme (93). Nevertheless, this approximationhas some side effects of benefit for applications that

are not fully strongly path dependent whence we discussit in more detail.

In order to analyse the Drift interpolation scheme (103),we start with the Ito–Taylor expansion of the integral ofthe 2pth power of stochastic volatility by setting m¼ 2pin equation (101) to obtain

Z tnþ1

tn

V2ps ds � V

2ptn

Z tnþ1

tn

ds|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}Euler

þ 2pV2p�1tn

bn

� Z tnþ1

tn

Z s

tn

dZuds|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}First remainder term: R1

þ 2pV2p�1tn

an þ pð2p� 1ÞV2p�2tn

b2n

� �Z tnþ1

tn

Z s

tn

du ds|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Second remainder term: R2

ð104Þ

1

10

0.01 0.1 1

log-Euler

Drift interpolation

Diffusion interpolation

Drift + Diffusion interpolation

Drift + Diffusion interpolation + decorrelation

IJK

1

10

0.01 0.1 1

log-Euler

Drift interpolation




IJK

(a)

(b)

Figure 8. Strong convergence of the financial underlying mea-sured by expression (59) averaged over 32 767 paths as a func-tion of scheme step size for T¼ 1, � ¼ 0:05, S0 ¼ 100, �¼ 0. Thevolatility dynamics were given by the (a) exponentially (24) and(b) the hyperbolically (25) transformed Ornstein–Uhlenbeckprocess (4) with y0¼ 0, �0 ¼ 1=4, �¼ 1 and � ¼ 7=20.


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with �tn :¼ ðtnþ1 � tnÞ, an :¼ aðVtnÞ, and bn :¼ bðVtn Þ.In comparison, the Ito–Taylor expansion of thedrift-interpolation scheme (102) leads to

1

2�tn V

2ptnþV

2ptnþ1

� �1

2�tn

V

2ptnþV

2ptnþ 2pV

2p�1tn

bn

� �Zn

þ 2pV2p�1tn

anþpð2p�1ÞV2p�2tn

b2n

h i�tn

!:

ð105Þ

This means that the leading order terms of the localapproximation error incurred by the drift interpolationscheme are

ftn :¼

Z tnþ1

tn

V2ps ds�

1

2V

2ptnþ V

2ptnþ1

� �tn ð106Þ

¼ 2pV2p�1tn

bn

Z tnþ1

tn

Z s

tn

dZu �1

2

Z tnþ1

tn

dZu

� �ds

¼ 2pV2p�1tn

bn

Z tnþ1

tn

1

2

Z tnþ1

tn

du�

Z s

tn

du

� �dZs: ð107Þ

Thus, by interpolating the drift, the term on the secondline of (104) involving the double integral

Ið0, 0Þðtn, tnþ1Þ ¼

Z tnþ1

tn

Z s

tn

du ds ð108Þ

is catered for. In expectation, we have the unconditionallocal mean-approximation error

Eftn jF 0

�¼ O �t3n

� �: ð109Þ

In order to analyse the relation between local and globalconvergence properties, we assume that the integrationinterval [0,T] is discretized in N steps,0 < t1 < � � � < tN�1 < tN ¼ T with step size �t ¼ T=N.Let Xti, xðtiþ1Þ be the numerical approximation at tiþ1

starting at time ti at point x and let Yti, xðtiþ1Þ be theanalytical solution of the stochastic differential equationstarting at (ti, x). Furthermore, we already know the localmean-approximation errors for i ¼ 0, � � � ,N� 1,

EXti,Yi

ðtiþ1Þ � Yti,Yiðtiþ1Þ

�� F ti

�¼ O �t3n

� �: ð110Þ

Next we consider the global mean-approximation error

EX0,X0

ðTÞ�� E

Y0,X0

ðTÞ��

¼ jEX0,X0

ðTÞ � Y0,X0ðTÞ�j ð111Þ

¼ jEX0,X0

ðtN�1Þ � Y0,X0ðtN�1Þ þ O �t3

� ��j

¼ EX0,X0

ðt1Þ � Y0,X0ðt1Þ þ ðN� 1Þ � O �t3

� �� ¼ N � O �t3

� �¼ Oð�t2Þ: ð112Þ

This means, the use of the drift interpolation term12 ðV

2ptnþ V

2ptnþ1

Þ�tn instead of the straightforwardEuler scheme term V

2ptn�tn improves the global mean-

approximation order of convergence. Alas, it is notpossible to improve the global weaky order ofconvergence in the two-dimensional case without generat-ing additional random numbers. Nevertheless, theinterpolation of the drift leads to a higher globalmean-convergence order (112) which may be of benefitwhen the simulation target is the valuation ofplain-vanilla or weakly path-dependent options, and thisissue will be the subject of future research.

Having analysed the approximation quality of the termgoverned by Ið0, 0Þ in (103), we now turn our attention tothe local estimation error induced by the handling of thedouble Wiener integral

Ið2, 0Þðtn, tnþ1Þ ¼

Z tnþ1

tn

Z s

tn

dZuds, ð113Þ

which can be simulated by the aid of our knowledge of thedistribution Ið2, 0Þðtn, tnþ1Þ:

Ið2,0Þðtn, tnþ1Þ �1

2�Zn�tn þ

1

2ffiffiffi3

p ��tn, with ��Nð0,�tnÞ:

ð114Þ

Sampling Ið2, 0Þ exactly would thus require the generationof an additional random number � for each step.In analogy to the reasoning leading up to theapproximation (A14) which is at the basis of theMilsteinþ scheme in Appendix A.2, we argue that

Ið2, 0Þðtn, tnþ1Þ;1

2�Zn�tn ð115Þ

is, conditional on our knowledge of the simulatedWiener path that drives the volatility process, or, moreformally, conditional on the �-algebra P

2N generated by

the increments

�Z0 ¼ Z1 � Z0, �Z1 ¼ Z2 � Z1, . . . ,

�ZN�1 ¼ ZN � ZN�1,ð116Þ

the best approximation attainable without resorting toadditional sources of (pseudo-)randomness. Applyingthe approximation (115) to the term R1 in (104) leads usto precisely the corresponding term in the expansion (105)(last term on the first line) of the drift interpolationscheme, and hence the scheme (103) also aids with respectto the influences of the term Ið2, 0Þðtn, tnþ1Þ.

The conditional expectation of the local approximationerror (106) of the scheme (103) conditional on knowing thefull path for Z is thus of order

Eftn jP

2N

�¼ O �Z2

n�tn� �

þO �t3n� �

: ð117Þ

yThe global weak order of convergence is defined by EgðX0;X0

ðTÞÞ�� E

gðY0;X0

ðTÞÞ�� with g being a sufficiently smooth test-

function. One can find the multidimensional second-order weak Taylor approximation scheme in section 14.2 of Kloeden and Platen(1992, 1995, 1999).


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The quality of this path-conditional local approximationerror is not visible in error measures designed to show thestrong convergence behaviour of the integrationscheme. However, it is likely to be of benefit for thecalculation of expectations that do not depend stronglyon the fine structure of simulated paths, but onthe approximation quality of the distribution of theunderlying variable at the terminal time horizon of thesimulation.

Another aspect of the drift interpolationscheme (103) is that it reduces the local mean-square error

Ef2tn jF tn

�¼ 2pV

2p�1tn

bn

� 2E

Z tnþ1

tn

�tn2

� s� tnð Þ

� �dZs

� �2" #

ð118Þ

¼ 2pV2p�1tn

bn

� 2Z tnþ1

tn

�tn2

� s� tnð Þ

� �2

ds ð119Þ

¼ 2pV2p�1tn

bn

� 2 112

�t3n ð120Þ

compared with the mean-square error of the first remain-der term R1 in (104) of the Euler scheme

ER1ð Þ

2jF tn

�¼ 2pV

2p�1tn

bn

� 2E

Z tnþ1

tn

s dZs

� �2" #

ð121Þ

¼ 2pV2p�1tn

bn

� 213�t3n: ð122Þ

In summary, the interpolation of the drift given by thescheme (103) effectively improves the numerical integra-tion by fully representing terms governed by Ið0, 0Þ in theIto–Taylor expansion of the formal solution, and byimproving the approximation for the term governed byIð2, 0Þ. We could not really expect to enhance theglobal strong convergence order induced by thedrift term (99) without the drawing of additional randomnumbers. Still, with nearly no extra computationaleffort one can improve, at least theoretically, over theconventional Euler scheme. Specifically, we are notcompletely erasing the leading error term of the Eulerscheme which is of order Oð�Z�tÞ. However, byusing approximation (105), conditional on any onegiven path in Z, we are able to remove the leadingorder bias term which is of order Oð�Z�tÞ. Effectively,the drift interpolation scheme (103) simply reducesthe absolute value of the coefficient of the lowest strongconvergence order error term.

4.2. Mixed interpolation of the diffusion term (100)

A suitable approximation of the diffusion is a little bitmore difficult than the integration of the drift. The firstidea might be to useZ tnþ1

tn

VpsdWs �

1

2V

ptnþ V

ptnþ1

� �Wn, ð123Þ

resulting in


1

2V

2ptn�tn þ

1

2V

ptnþ V

ptnþ1

� �Wn

ð124Þ

which was a simple interpolation for the drift approxima-tion. Furthermore combining the drift and diffusioninterpolation leads to


1

4V

2ptnþ V

2ptnþ1

� �tn

þ1

2V

ptnþ V

ptnþ1

� �Wn:

ð125Þ

We will denote these schemes as Diffusion interpola-tion (124) and DriftþDiffusion interpolation (125).Considering figure 8 we recognize that these integrationschemes are remarkably effective in the case of no correla-tion between the underlying and the stochastic volatility.In contrast, convergence is lost altogether for the diffu-sion interpolation scheme (124) when correlation isnon-zero as we can see in figures 9 and 10.

In order to understand the loss of convergence we takea closer look at the diffusion interpolation. The first stepis to decompose the correlated Wiener processes intoindependent components by the aid of the Choleskydecomposition

dW ¼ �d ~Zþ �0d ~W, ð126Þ

dZ ¼ d ~Z, ð127Þ

where ~W and ~Z are uncorrelated, and

�0 :¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

p: ð128Þ

This gives usZ tnþ1

tn

VpsdWs ¼ �0

Z tnþ1

tn

Vpsd ~Ws þ �

Z tnþ1

tn

Vpsd ~Zs: ð129Þ

The reason for the loss of convergence is that the vola-tility process Vs is driven itself by the Wiener process ~Zs.Thus by using the trapezoidal rule we are not interpretingthe stochastic integral in the Ito but in the Stratonovichsense. As a consequence, we are overestimating the influ-ence of the Wiener process ~Zs. We can circumvent thisproblem by applying the trapezoidal rule only on theuncorrelated part of the diffusion

�0Z tnþ1

tn

Vpsd ~Ws þ �

Z tnþ1

tn

Vpsd ~Zs

�1

2�0 V

ptnþ V

ptnþ1

� � ~Wn þ �V

ptn� ~Zn,

ð130Þ

which gives us in combination with (125)


1

4V

2ptnþ V

2ptnþ1

� �tn

þ1

2V

ptnþ V

ptnþ1

� �Wn þ

1

2V

ptn� V

ptnþ1

� ��Zn,

ð131Þ


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which we shall refer to as the Drift þ Diffusion interpola-tionþ decorrelation scheme. This scheme not only restoresconvergence but also improves the approximation

quality when correlation is non-zero. To verifythese statements we analyse the local approximationerror again

1

10

0.01 0.1 1

log-Euler

Drift interpolation




IJK

IJK no Drift interpolation

1

10

0.01 0.1 1

log-Euler

Drift interpolation




IJK


(a)

(b)

Figure 9. Strong convergence of the financial underlyingmeasured by expression (59) averaged over 32 767 paths as afunction of scheme step size for T¼ 1, � ¼ 0:05, S0 ¼ 100,� ¼ �ð2=5Þ. The volatility dynamics were given by the (a)exponentially (24) and (b) the hyperbolically (25) transformedOrnstein–Uhlenbeck process (4) with y0¼ 0, �0 ¼ 1=4, �¼ 1 and� ¼ 7=20.

1

10

0.01 0.1 1

log-Euler

Drift interpolation




IJK


1

10

0.01 0.1 1

log-Euler

Drift interpolation




IJK


(a)

(b)

Figure 10. Strong convergence of the financial underlyingmeasured by expression (59) averaged over 32 767 paths as afunction of scheme step size for T¼ 1, � ¼ 0:05, S0 ¼ 100,� ¼ �ð4=5Þ. The volatility dynamics were given by the (a)exponentially (24) and (b) the hyperbolically (25) transformedOrnstein–Uhlenbeck process (4) with y0¼ 0, �0 ¼ 1=4, �¼ 1, and� ¼ 7=20.

ftn : ¼ �0Z tnþ1

tn

Vpsd ~Ws þ �

Z tnþ1

tn

Vpsd ~Zs �

1

2�0 Vp

tnþ Vp

tnþ1

� � ~Wn þ �V

ptn� ~Zn

� �ð132Þ

¼ �0 pVp�1tn

bn

Z tnþ1

tn

Z s

tn

d ~Zud ~Ws þ pVp�1tn

an þ1

2pðp� 1ÞVp�2

tnb2n

� � Z tnþ1

tn

Z s

tn

du d ~Ws

� �þ � pVp�1

tnbn

Z tnþ1

tn

Z s

tn

d ~Zud ~Zs

�

þ pVp�1tn

an þ1

2pðp� 1ÞV

p�2tn

b2n

� � Z tnþ1

tn

Z s

tn

du d ~Zs

��1

2�0 pV

p�1tn

bn� ~Zn þ pVp�1tn

an þ1

2pðp� 1ÞV

p�2tn

b2n

� ��tn

� �� ~Wn

¼ �0 pVp�1tn

bn

Z tnþ1

tn

ð ~Zs �~Ztn Þ �

1

2� ~Ztn

� �d ~Ws|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ftn , 1

þ pVp�1tn

an þ1

2pðp� 1ÞVp�2

tnb2n

� �Z tnþ1

tn

ðs� tnÞ �1

2�tn

� �d ~Ws|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ftn , 2

0BBB@

1CCCA

þ � pVp�1tn

bn

Z tnþ1

tn

Z s

tn

d ~Zud ~Zs|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ftn , 3

þ pVp�1tn

an þ1

2pðp� 1ÞVp�2

tnb2n

� � Z tnþ1

tn

Z s

tn

du d ~Zs

0BBB@

1CCCA: ð133Þ


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In analogy to the interpolation of the drift, the trape-zoidal integration rule applied to the uncorrelated part ofthe Ito integral in (130) leads to a reduced variance forthe local truncation errors ftn, 1 and ftn, 2. Taking theconditional expectation based on the knowledge of ourWiener paths W and Z we obtain

Eftn, 1j

~P1N, ~P

2N

�¼ 0 and E

ftn, 2j

~P1N

�¼ 0, ð134Þ

where the �-algebras ~P1N and ~P2

N are generated by theincrements of the Wiener processes ~W and ~Z. Onceagain the interpolation is the best estimate based on theknowledge of the paths of ~W and ~Z. Especially in the caseof low correlation this scheme is remarkably effective.Taking a closer look at the local approximationerror (133) of the correlated part we recognize that theleading error term is

ftn, 3 ¼ pVp�1tn

bn

Z tnþ1

tn

Z s

tn

d ~Zud ~Zs ¼ pVp�1tn

bn1

2� ~Z2

n ��tn� �

:

ð135Þ

To make matters even better, we can improve the integra-tion by including this term in our integration scheme.Luckily the double Ito integral Ið2, 2Þ does not requireadditional random numbers. The importance of the inclu-sion of this term grows with increasing correlation coeffi-cient �, unlike the benefit from the (de-correlated)diffusion interpolation (131) which diminishes withincreasing correlation. For the sake of brevity, we willcall this integration scheme based on interpolation ofthe drift, interpolation of the diffusion term, considera-tion of decorrelation of the diffusion term, and inclusionof a higher order Milstein term, simply the IJK scheme inthe following. Its explicit propagation equation is givenby


1

4V

2ptnþ V

2ptnþ1

� �tn þ �V

ptn� ~Zn

þ1

2�0 V

ptnþ V

ptnþ1

� � ~Wn þ

1

2�pVp�1

tnbn

� � ~Z2n ��tn

� �ð136Þ

or, equivalently,


1

4V

2ptnþ V

2ptnþ1

� �tn þ �V

ptn�Zn

þ1

2V

ptnþ V

ptnþ1

� ð�Wn � ��ZnÞ

þ1

2�pVp�1

tnbn �Z2

n ��tn� �

:

ð137Þ

Since the Drift interpolation (103) scheme was not able toincrease the strong approximation quality of the standardlog-Euler we also tried the IJK method without using adrift interpolation which we denote as IJK no Drift inter-polation.

In figures 8, 9 and 10 we show all consideredapproximation procedures in comparison and we seethat a combination of drift interpolation, diffusion

interpolation allowing for (de-)correlation as givenin (130), and the addition of the higher order term (135)outperforms any of the other approximation schemes.The advantage of the IJK scheme is that we get goodapproximation results for low and high correlations dueto the fact that we cover both the dominant error termsfor low correlation (130) and for high correlation (135),and that comparatively little extra computational effort isrequired. In addition, one can observe that in the caseof high correlation, as given in figures 10 and 13, thedrift-interpolation is a small but valuable enhancementfor the IJK scheme particularly for large step sizes.

Until this point we have only compared integrationschemes by looking at the approximation quality as afunction of step size. In financial applications, however,a scheme is considered better if it is more accurate andfaster. It is thus of paramount interest to comparethe residual error as a function of calculation time.

1

10

0.001 0.01 0.1

log-Euler

Drift interpolation




IJK

1

10

0.001 0.01 0.1

log-Euler

Drift interpolation




IJK

(a)

(b)

Figure 11. Strong convergence of the financial underlyingmeasured by expression (59) averaged over 32 767 paths as afunction of CPU time [in ms] for T¼ 1, � ¼ 0:05, S0 ¼ 100,�¼ 0. The volatility dynamics were given by the (a)exponentially (24) and (b) the hyperbolically (25) transformedOrnstein–Uhlenbeck process (4) with y0¼ 0, �0 ¼ 1=4, �¼ 1, and� ¼ 7=20.


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In figures 11, 12 and 13 we can see that the combinationof all speed-ups (drift and diffusion interpolation, decor-relation and additional term) does not affect the compu-tational effort significantly. We also notice that, whensimulation of the stochastic volatility process itself is tri-vial as is the case for the exponential and the hyperbolicvolatility processes discussed in section 2.2, the use of theIJK scheme provides on average a speed-up of approxi-mately a factor 2 for low correlation and as much as afactor 4 for pronounced negative correlation (see table 1).In this context, it is noteworthy to recall that most marketcalibrations require a strong negative correlation toreproduce the observable implied volatility skews, whichmakes the use of the IJK scheme particularly attractive.

In section 5, we present the results of further numericaltests for the case when the volatility process itself requiresa numerical integration scheme. In particular, weconsider the situation where the volatility is given by

the mean-reverting CEV process (2). For this case, wewish to find the best combination of integration schemesfor the stochastic volatility as well as for the financialunderlying.

5. Numerical results for mean-reverting

CEV volatility processes

In this section we go one step further as we consider atwo-dimensional stochastic volatility model where thestochastic volatility process is given by the mean-revertingCEV process (2). We already recognized in section 3that the numerical results for the integration of a mean-reverting CEV process are sensitive to the size ofthe diffusion exponent q 2 1=2, 1½ �. Hence we focus onthe two extreme choices the Brennan–Schwartz (67) andthe Cox–Ingersoll–Ross (81) equations. In the following

1

10

0.001 0.01 0.1

log-Euler

Drift interpolation




IJK


1

10

0.001 0.01 0.1

log-Euler

Drift interpolation




IJK


(a)

(b)

Figure 12. Strong convergence of the financial underlying mea-sured by expression (59) averaged over 32 767 paths as a func-tion of CPU time [in ms] for T¼ 1, � ¼ 0:05, S0 ¼ 100,� ¼ �ð2=5Þ. The volatility dynamics were given by the (a) expo-nentially (24) and (b) the hyperbolically (25) transformedOrnstein–Uhlenbeck process (4) with y0¼ 0, �0 ¼ 1=4, �¼ 1,and � ¼ 7=20.

1

10

0.001 0.01 0.1

log-Euler

Drift interpolation




IJK


1

10

0.001 0.01 0.1

log-Euler

Drift interpolation




IJK


(a)

(b)

Figure 13. Strong convergence of the financial underlyingmeasured by expression (59) averaged over 32 767 paths as afunction of CPU time [in ms] for T¼ 1, � ¼ 0:05, S0 ¼ 100,� ¼ �4=5. The volatility dynamics were given by the (a)exponentially (24) and (b) the hyperbolically (25) transformedOrnstein–Uhlenbeck process (4) with y0¼ 0, �0 ¼ 1=4, �¼ 1, and� ¼ 7=20.


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we consider four schemes for the integration of stochasticvolatility or variance:

(1) Euler — (45),(2) Milstein — (A4),(3) BMM — (51),(4) Pathwise Adapted Linearization — (76) for Brennan-

Schwartz and (88) for CIR.

We combine these with suitable integration schemes forthe whole system. Specifically, we consider

(1) Euler–Maruyama — (93),(2) IJK — (137).

The Euler scheme was already the benchmark insection 4 where we developed the IJK scheme. It is ofinterest to see if the IJK scheme can preserve itsadvantage even if we have to integrate the stochasticvolatility process numerically.

In the following we concentrate on two different testcases. The first one is based on the Brennan-Schwartzequation (67) for the modelling of the stochastic volatil-ity. This equation is directly coupled to the underlyingwith exponent p ¼ 1=2

dSt ¼ �StdtþffiffiffiffiffiVt

pStdWt,

dVt ¼ � � � Vtð Þdtþ �VtdZt:ð138Þ

The parameter configuration is chosen as followsSt0 ¼ 100, � ¼ 0:05, V0 ¼ � ¼ 1=16, � ¼ 1, � ¼ 0:5where we present results for different levels of correlation� 2 f0:0, � 0:4, � 0:8g.

As a second benchmark we consider the Heston model

dSt ¼ �StdtþffiffiffiffiffiVt

pStdWt,

dVt ¼ � � � Vtð Þdtþ �ffiffiffiffiffiVt

pdZt,

ð139Þ

where the parameters are given by St0 ¼ 100, � ¼ 0:05,V0 ¼ � ¼ 1

16, � ¼ 1, � ¼ 0:5. Again we show results fordecreasing correlation � 2 f0:0, �0:4, �0:8g.

We see in figures 14–16 that the decisive point for thestrong approximation quality is the choice of theintegration scheme IJK as the integration of the stochasticvolatility has just a minor impact on the numerical results.In accordance with the numerical results of the lastsection, we can observe that the IJK scheme is at itsmost impressive when dealing with high correlation asin figure 16. In fact for high negative correlation,the approximation efficiency of the IJK scheme incomparison to conventional Euler & log-Euler methodsappears to be even greater in the case when the volatility

process itself requires numerical integration consideringthat the speed gain appears to be approximately a factor5 in figure 16 (see also table 2). Nonetheless, even if wecan neglect the influence of the numerical integration ofthe stochastic volatility process on the strong convergencebehaviour of the underlying, the details of the integrationof the stochastic volatility process become importantwhen pricing derivatives that are sensitive to the dynamicsof the volatility. In that case, the results of section 3 cangive guidance in the selection of the integration schemefor the mean-reverting CEV process. In any case, oneshould be aware of the fact that an unstable integrationof the stochastic volatility can crash the integration of thewhole system in the sense that the occurrence of spuriouspaths where variance crosses over to the negative domaincan spoil the convergence behaviour irrecoverably as wesaw in figure 4(b) for the Milsteinþ scheme. On that note,we have a closer look at figure 16(b) where we see that theconvergence behaviour of the Milstein–IJK scheme seemssomewhat unexpected as the approximation error does

Table 1. Average speed-up IJK (137) compared withlog-Euler (93).

Figure � Exp. Hyp.

11 �0.0 2.0 1.912 �0.4 2.2 2.113 �0.8 4.4 4.2

1

0.001 0.01 0.1

Euler & log-Euler

Milstein & log-Euler

Balanced Milstein Method & log-Euler

Pathwise Adapted Linearisation & log-Euler

Euler & IJK

Milstein & IJK

Balanced Milstein Method & IJK

Pathwise Adapted Linearisation & IJK

1

10

0.001 0.01 0.1

Euler & log-Euler




Euler & IJK

Milstein & IJK



(a)

(b)

Figure 14. Strong convergence measured by expression (59) asa function of CPU time [in ms] averaged over 32 767 paths for(a) model (138) and (b) for model (139). The number generatormethod was Sobol’s. Correlation: � ¼ 0.


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not decrease when halving the step size from �t ¼ 1 to�t ¼ 1=2 even though this integration scheme iscompetitive to the BMM–IJK scheme for small stepsizes. The explanation for this is surprisingly simple.In table 3 we compare the percentage of non-positivepaths for the integration of the stochastic volatilitywhere we do not count those paths becoming non-posi-tive in the final integration step.y With this counting con-vention, no non-positive paths occur for�t ¼ 1 as we onlyhave to take a single step. In comparison, for�t ¼ 1=2 weobtain the highest number of non-positive paths for theMilstein scheme which explains the bump in the conver-gence plot of Milstein–IJK in figure 16(b). Thus, even inthis very simple case of estimating the strong convergenceerror of the financial underlying, an appropriate integra-tion scheme for the stochastic volatility process is key toguaranteeing a stable approximation.

6. Conclusion

In this article, we discussed various Monte Carloapproximation schemes for stochastic volatility diffusionmodels. Our main focus was on the strong convergencebehaviour as an indicator for the valuation of path-dependent derivatives. In order to maintain the ability

yIt is of minor importance if one path becomes negative or zero in the last integration step as we do not have to use the final value asa starting point for the next integration step.

1

0.001 0.01 0.1

Euler & log-Euler




Euler & IJK

Milstein & IJK



1

10

0.001 0.01 0.1

Euler & log-Euler




Euler & IJK

Milstein & IJK



(a)

(b)

Figure 15. Strong convergence measured by expression (59) asa function of CPU time [in ms] averaged over 32 767 paths for(a) model (138) and (b) for model (139). The number generatormethod was Sobol’s. Correlation: � ¼ �0:4.

1

0.001 0.01 0.1

Euler & log-Euler




Euler & IJK

Milstein & IJK



1

10

0.001 0.01 0.1

Euler & log-Euler




Euler & IJK

Milstein & IJK



(a)

(b)

Figure 16. Strong convergence measured by expression (59) asa function of CPU time [in ms] averaged over 32 767 paths for(a) model (138) and (b) for model (139). The number generatormethod was Sobol’s. Correlation: � ¼ �0:8

Table 2. Average speed-up BMM & IJK (137) compared withEuler & log-Euler (93).

Figure � (138) (139)

14 �0.0 2.1 2.615 �0.4 2.5 2.916 �0.8 4.6 4.5


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to apply exogenous variance reduction techniques such aslow discrepancy numbers, importance sampling, andothers (Jackel 2002, Glasserman 2003), with ease, werestricted our research to methods that effectively requireonly two simulated uniform variates per step in the timediscretization of the volatility-underlying evolution pair.Given this self-imposed constraint, we attemptedto exploit all information available from the simulatedprimary standard Wiener diffusion process pair, and toadapt the integration scheme of the volatility andfinancial underlying process as much as possible to eachsimulated primary process path pair. Whilst we had torealize that within our scope we are limited to improvingsimulation results not by increasing the convergenceorder, but mainly by decreasing the magnitude of theleading order error term, we found that significantspeed gains can be accomplished at surprisingly littleexpense in numerical effort. In fact, for the stochasticvolatility models we examined, the observed accelerationin comparison to the standard Euler and log-Euler variesfrom a factor two for zero correlation to as much as afactor five when correlation is significantly negative asusually required for calibration to market-observableimplied volatility profiles.

As part of our investigations, we also introduced a newvariation of stochastic volatility models, namely thehyperbolically transformed Ornstein–Uhlenbeck processmodel given by equation (25). This model inherits thebenefits of Scott’s (1987) model such as mean reversionand the fact that zero is not attainable, but, like Scott’smodel, does not provide us with (semi-)closed formanalytical solutions for the density or characteristicfunction of the distribution of the underlying, or plain-vanilla option prices. It does, however, avoid the issuesraised in Andersen and Piterbarg (2004) regarding theexplosion of moments etc. due to the fact that the tailsof its distribution, both towards low and towards highvolatility levels, are significantly thinner than those ofthe Scott model. We examined the behaviour of thismodel as part of our research because we believe thatadvanced Monte Carlo integration schemes, in combina-tion with modern variance reduction methods, inconjunction with ever increasing computer power, willmake the use of stochastic volatility models that are notreadily amenable for convenient plain-vanilla optionpricing formulæ an industrially viable possibility, evenat the point where model parameters or parameter termstructures are calibrated to market observable plain

vanilla option prices. This latter conjecture will be thesubject of future research.

As for currently favoured stochastic volatility modelssuch as Heston’s (1993), whose instantaneous variance isdriven by the Cox–Ingersoll–Ross process, we found inour numerical experiments that the use of PathwiseAdapted Linearizations of the driving Wiener processintroduced in section 3.1, with approximate analyticalsolutions along the path, can provide numerical integra-tion improvements for small to moderate values of �.However, from our set of investigated methods,the clear overall favourite for the integration of theCox–Ingersoll–Ross process is the Balanced MilsteinMethod since it performs as efficiently as the CIR-specificpathwise adapted linearization-based expansions, butremains stable for all parameter values. The BalancedMilstein Method is thus our method of choice for thenumerical integration of the CIR process, both insidestochastic volatility model applications and otherwise.

Finally, it remains to be stated that the main focus of ourresearch, namely the investigation of integration schemesfor stochastic volatility models that combine an essentiallygeometric Brownian motion process with a secondarysource of noise influencing the volatility term of the for-mer, resulted in the method we named the IJK schemegiven by equation (137). The purpose of this method isnot to show the highest possible convergence order as afunction of the step size. Instead, we endeavoured to find asimple and robust method that, without the need for adap-tive refinement, or additional random numbers per step, isas efficient as possible. In other words, we were looking forthe most efficient method that is robust and yet essentiallyas simple as the standard Euler–Maruyama algorithm.The IJK scheme is, as a consequence, not superior in con-vergence order, but excels over other equally simple meth-ods by showing a superior convergence behaviour that isfaster by a multiplicative factor. The IJKmethod is as easyto implement as the standard (log-) Euler scheme, and canthus be used as a so-called drop-in replacement for the(log-)Euler scheme since it requires no additional randomnumbers or other convergence acceleration aids.

Acknowledgments

The authors thank Vladimir Piterbarg and an anonymousreferee for helpful comments and suggestions.

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�t Euler Milstein BMM

20 0% 0% 0%2�1 23.9% 33.2% 0%2�2 37.5% 15.7% 0%2�3 43.8% 0.1% 0%2�4 46.7% 0% 0%


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Appendix A: Milstein schemes

A.1 The one-dimensional Milstein method

The Milstein (1974) scheme for the stochastic differentialequation

dx ¼ aðxÞdtþ bðxÞdW ðA1Þ


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can be derived from the Ito–Taylor expansion (Kloedenand Platen 1992, 1995, 1999, equation 5.5.4)

xt ¼x0 þ a0

Z t

0

dsþ b0

Z t

0

dWs þ b00b0

Z t

0

Z s

0

dWu dWs

þ a00b0

Z t

0

Z s

0

dWu dsþ a0b00 þ

1

2b000b

20

� �Z t

0

Z s

0

du dWs

þ b0b002þb000b

20

� Z t

0

Z s

0

Z u

0

dWr dWu dWs þOðt2Þ

ðA2Þ

with a0 ¼ aðx0Þ, etc. Retaining terms up to order OðtÞ,and evaluating the integral

Ið1, 1ÞðtÞ :¼

Z t

0

Z s

0

dWu dWs ¼1

2W2

t � t� �

, ðA3Þ

assuming (without loss of generality) that W0¼ 0, weobtain the one-dimensional Milstein scheme

xtnþ1¼ xtn þ aðxtnÞ�tn þ bðxtn Þ�Wn

þ1

2b0ðxtnÞbðxtnÞ �W2

n ��tn� �

:

ðA4Þ

Applied to equation (2), we obtain

VMilsteintnþ1

¼ Vtn þ �ð� � VtnÞ�tn þ �Vqtn�Zn

þ1

2�2qV2q�1

tn�Z2

n ��tn� �

:

ðA5Þ

A.2 The MilsteinQ scheme

All of the explicitly given integral terms on the second lineof (A2)

Ið1, 0ÞðtÞ ¼

Z t

0

Z s

0

dWu ds, ðA6Þ

Ið0, 1ÞðtÞ ¼

Z t

0

Z s

0

du dWs ¼ Wtt� Ið1, 0ÞðtÞ, ðA7Þ

Ið1, 1, 1ÞðtÞ ¼

Z t

0

Z s

0

Z u

0

dWr dWu dWs ¼1

6W3

t �1

2Wtt,

ðA8Þ

can be expressed in terms of the primaryconstituents Wt and Ið1, 0ÞðtÞ. The two-dimensionaldistribution of the random numbers Wt and Ið1, 0ÞðtÞis given by a bivariate Gaussian law with mean (0, 0)and covariance matrix

t t2�2

t2�2 t3

�3

0@

1A: ðA9Þ

This means,

Ið1, 0ÞðtÞ �1

2Wttþ

1

2ffiffiffi3

p t3=2y ðA10Þ

with y � Nð0, 1Þ, and thus the distribution of xt isgiven by

xt � x0 þ a0tþ b0Wt þ1

2b00b0 W2

t � t� �

þ1

6b0b

002þb000b

20

� W3

t þ a0b00 �

1

2b0b

002

� �Wtt

þ a00b0 � a0b00 �

1

2b000b

20

� �

�1

2Wttþ

1

2ffiffiffi3

p t3=2y

� �þOðt2Þ: ðA11Þ

Hence, in order to fully cater for all terms to orderOðt3=2Þ, an extra source of randomness is required.It can also be seen that, conditional on a given valuefor Wt, the terms of order Oðt3=2Þ have non-zero expecta-tion. This means, that, not accounting for the terms onthe second and third line of (A11), i.e. the terms notpresent in the standard Milstein scheme, introduces aconditional bias given by

1

6b0b

002þb000b

20

h iW3

t þ1

2a00b0 þ a0b

00 � b0b

002�1

2b000b

20

� �Wtt:

ðA12Þ

This bias can be corrected by simply adding these terms tothe Milstein scheme which gives us

xtnþ1¼ xtn þ aðxtnÞ�tn þ bðxtn Þ�Wn þ

1

2b0ðxtn ÞbðxtnÞ

� �W2n ��tn

� �þ1

6bðxtn Þb

0ðxtn Þ

2þb00ðxtnÞbðxtn Þ

2h i

�W3n

þ1

2a0ðxtn Þbðxtn Þ þ aðxtnÞb

0ðxtnÞ � bðxtnÞb

0ðxtn Þ

2h

�1

2b00ðxtnÞbðxtn Þ

2

��Wn�tn: ðA13Þ

For the sake of brevity in the main text, we refer to thismethod as Milsteinþ. An alternative way to arrive at theapproximation

Ið1, 0ÞðtÞ;1

2Wtt ðA14Þ

at the core of this scheme is to look for the most likelyor the expected value of Ið1, 0ÞðtÞ conditional on thediscretized path for Wt, i.e. in the filtration generatedby the increments �Wn. Newton (1991, 1994)introduced the idea of strong asymptotically efficientschemes following a similar line of reasoning and resultingin the approximation on (A14). Specifically for the termIð1, 0ÞðtÞ, it so happens that the most likely and theexpected value conditional on the discretized pathf�Wng are both given by (A14). Still, the advantage ofthe Milsteinþ scheme over the original Milstein scheme isnot that it has a higher convergence order but that it


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reduces the magnitude coefficient of the leading ordererror terms.

Applied to equation (2), the Milsteinþ scheme reads

VMilsteinþtnþ1

¼Vtn þ �ð� � VtnÞ�tn þ �Vqtn�Zn

þ1

2�2qV2q�1

tn�Z2

n ��tn� �

þ1

6�3qð2q� 1ÞV

3q�2tn

�Z3n

þ1

2��qVq�1

tn� ��ðqþ 1ÞV

qtn

h�1

2�3q 3q� 1ð ÞV

3q�2tn

��Zn�tn:

ðA15Þ

A.3 The Milstein scheme for stochastic volatility systems

For the N-dimensional system of stochastic differentialequations

dx ¼ aðxÞdtþ BðxÞd ~W, ðA16Þ

with a 2 C1ðR

N,RNÞ and B 2 C2

ðRN,RN�N

Þ,8i ¼ 1, . . . ,M and uncorrelated Brownian motions~W 2 R

N, the ith component of the multidimensionalMilstein scheme is

xiðtþ�tÞ ¼ xiðtÞ þ ai�tþXMj¼1

bij� ~Wj

þXM

j, k, l¼1

bjkð@xjbilÞ~Iðk, lÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Milstein term

,ðA17Þ

wherein all of the coefficient functions aið�Þ and bijð�Þ, etc.,are to be evaluated with xðtÞ. We define the double Itointegral ~Iðk, lÞ as

~Iðk, lÞ ¼

Z tþ�t

s¼t

Z s

u¼t

d ~WkðuÞ d ~WlðsÞ: ðA18Þ

For k¼ l, it simplifies to

~Iðk, kÞ ¼1

2� ~W2

k ��t� �

: ðA19Þ

For the stochastic volatility system (1) and (2), we use aCholesky decomposition of the correlated Brownianmotions

dW ¼ �0d ~W1 þ � d ~W2, ðA20Þ

dZ ¼ d ~W2 ðA21Þ

with �0 :¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

pand set x1 :¼ lnS and x2 :¼ V to

obtain from (1) and (2) the coupled stochastic differentialequations

dx1dx2

� �¼

�� 12 x

2p2

�ð� � x2Þ

� �dtþ

�0xp2 �xp20 �xq2

� �d ~W1

d ~W2

� �:

ðA22Þ

Since the volatility process x2 is not influenced directlyby the dynamics for x1, the Milstein scheme for x2 isgiven by the standard one-dimensional formula (A5).For x1, the fact that b21 ¼ 0 and @x1BðxÞ ¼ 0 simplifiesthe calculation of the Milstein term:

XMj, k, l¼1

bjkð@xjb1lÞ~Iðk, lÞ ¼ b22ð@x2b11Þ

~Ið2, 1Þ þ b22ð@x2b12Þ~Ið2, 2Þ

¼ �pxpþq�12 �0 ~Ið2, 1Þ þ � ~Ið2, 2Þ

� �: ðA23Þ


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Fast strong approximation Monte Carlo schemes for stochastic volatility models