on the performance of asymptotic locally risk minimising hedges in the heston stochastic volatility...

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On the performance of asymptotic locally riskminimising hedges in the Heston stochastic volatilitymodelSai Hung Marten Ting a & Christian-Oliver Ewald ba School of Mathematics and Statistics , University of Sydney , Australiab Department of Economics , University of Glasgow , Glasgow, G12 8RT , UKPublished online: 19 Jul 2012.

To cite this article: Sai Hung Marten Ting & Christian-Oliver Ewald (2013) On the performance of asymptoticlocally risk minimising hedges in the Heston stochastic volatility model, Quantitative Finance, 13:6, 939-954, DOI:10.1080/14697688.2012.691987

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On the performance of asymptotic locally risk

minimising hedges in the Heston stochastic

volatility model

SAI HUNG MARTEN TINGy and CHRISTIAN-OLIVER EWALD*z

ySchool of Mathematics and Statistics, University of Sydney, AustraliazDepartment of Economics, University of Glasgow, Glasgow, G12 8RT, UK

(Received 23 September 2010; in final form 26 April 2012)

This paper investigates the use of the asymptotic Heston solution in locally risk minimisinghedging. The asymptotic Heston solution is presented along with issues that are relevant to itsuse. Comparison between the exact and asymptotic Heston hedges are made using bothsimulated and real historical data. The asymptotic Heston hedge is found to be a viablealternative to the exact hedge. It provides a means for faster calculation, while performing aswell as the exact Heston hedge in the locally risk minimising framework.

Keywords: Locally risk-minimising hedging; Stochastic volatility; Asymptotic solutions;Model risk; Empirical hedging performance

JEL Classification: C63, G11, G13

1. Introduction

The addition of stochastic volatility in option pricing

theory has resulted in the development of more sophis-

ticated frameworks for pricing financial derivatives. The

need to incorporate a randomly varying volatility arose

from studies involving the log returns of heavily traded

indices on the major stock exchanges. In the Black–

Scholes (1973) framework, the log returns of assets are

assumed to follow a normal distribution, though empir-

ical studies showed this is not the case. In general,

empirical log return distributions have heavier tails and

higher peaks, which is indicative of a distribution with

differing variances (Gatheral 2006). Further proof, in the

form of implied volatilities, shows that the constant

volatility assumption in the Black–Scholes framework is

rather unrealistic. Plotting implied volatility surfaces

shows that volatility is dependent on both the time to

expiry and the strike price of the option. The shapes of

these surfaces are generally referred to as the volatility

smile, due to the fact that, looking at a fixed time to

expiry, the implied volatilities as the strike price varies are

sometimes reminiscent of a smile shape. Whilst other

shapes such as smirks and frowns do exist, there arecases where no recognisable shapes are observed.The type of smile observed is largely dependent on thetype of asset class considered. These observationsprovide motivation for modelling volatility as a randomvariable.

There is no generally accepted view on which stochasticvolatility model is best used to model option prices. TheHeston model (Heston 1993) is a favourite amongpractitioners due to its tractability and simplicity. Themodel employs the use of a Cox–Ingersoll–Ross (CIR)process to model the square of the volatility, known asvariance, whilst the stock price follows a geometricBrownian motion style process with the diffusion coeffi-cient being the volatility process. The biggest drawcard ofthe Heston model is its closed form solution for Europeanvanilla derivatives along with its ability to fit the impliedvolatility surface. Whilst the issue of pricing has been longexplored, most notably by Heston (1993) in deriving theclosed form solution, what is still unclear is how to hedgesuch options efficiently. It is well known that, under astochastic volatility model, a vanilla option cannot beperfectly hedged with just a combination of stocks andmoney in risk-free bank accounts. This is due to therandomness of the unobservable and untradeable volatil-ity parameter.*Corresponding author. Email: [email protected]

� 201 Taylor & Francis

http://dx.doi.org/10.1080/14697688.2012.691987

Quantitative Finance, 2013

Vol. 13, No. 6, 939–954,

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The conducting of risk minimising hedges in incompletemarkets has intrigued many. This is mainly due to the factthat self-financing hedges do not exist in incompletemarkets. Papers such as Follmer and Schweizer (1990)and Schweizer (1991) have provided general results on thenature of risk minimising hedges in incomplete markets.Risk minimising hedges involve the use of tradingstrategies such that the risk, as measured by predefinedcriteria, is minimised. The concept of locally risk mini-mising hedges, as introduced by Schweizer (1991),requires the minimisation of a quadratic risk function ateach time step. It was shown that this problem is indeedsolvable and in fact is related to variance-optimal hedgingunder a martingale measure. In Heath et al. (2001), locallyrisk minimising hedging is compared to mean-variancehedging. Note however that the latter is performed usingself-financing strategies, which is conceptually a verydifferent approach. For more information on meanvariance and variance-optimal hedging and how it isrelated to locally risk minimising hedging we refer toSchweizer (1995, 1996) and Pham et al. (1998). Otherrelated literature on the aspects of hedging in incompletemarkets includes Alexander and Nogueira (2007a, b) aswell as Bakshi et al. (1997).

El Karuoi et al. (1997) showed how it is possible tocalculate explicitly the locally risk minimising strategy,and in Poulsen et al. (2009) the hedges were derived for ageneral class of stochastic volatility models. Note that thelocally risk minimising framework not only determines ahedging strategy, but also fixates the pricing measure asthe so-called minimal martingale measure. Poulsen et al.(2009) also performed an empirical analysis under theHeston model to evaluate its effectiveness over traditionalhedging methods. As in most of the risk minimisinghedging literature, the goal of this paper will be to hedgederivatives with only primary assets (stocks and bonds).In fact, adding an option in addition to stocks and bondsas a hedging instrument to hedge other options would, inthe case of stochastic volatility, complete the market,allowing for perfect self-financing hedges and hencerendering the locally risk minimising approachmeaningless.

This paper investigates the use of asymptotic solutionsfor the Heston call option, in creating asymptotic locallyrisk minimising hedges. The asymptotic solutions arebased on the assumption that the CIR process that drivesthe volatility possesses fast mean reverting properties. It isasymptotic in the sense that the solution becomes a betterapproximation as the mean reverting rate increases. Theasymptotic solutions are based on modifications to thework of Fouque et al. (1998, 2000) on stochastic volatilitymodels with fast mean reverting Ornstein–Uhlenbeck(OU) processes. The performance of this solution is testedagainst that of the exact Heston solution using locally riskminimising hedge theory. The asymptotic solution will beevaluated based on its accuracy, simplicity and compu-tational speed. These comparisons are made using bothsimulated and historical data. As an addition, the hedgeswill also be compared against that of traditional (butoutdated) Black–Scholes delta hedges.

The paper is organised as follows. Sections 2 and 3review the Heston model, its asymptotic solution and the

locally risk minimising trading strategy, respectively, and

in particular the relevant definitions, models and results.Sections 4 and 5 compare the exact and asymptotic

Heston solution hedges using simulated and historical

data, respectively. For the hedges in the simulationsection, we assume a pricing measure exists, while for

the historical data hedges, calibration is performed on

option data to obtain the minimal martingale measure.

2. Review of the Heston model and its asymptotic

solution

2.1. The Heston model and the exact solution

The Heston model assumes that the stock price process Xt

and variance process Yt satisfy the following set of

stochastic differential equations:

dXt ¼ rXt dtþffiffiffiffiffiYt

pXt dWt, ð1Þ

dYt ¼ �ðm� YtÞ dtþ �ffiffiffiffiffiYt

pdZt, ð2Þ

where r is the interest rate, and a, m and b are the mean

reverting rate, the mean reverting level and the volatility

of the variance process, respectively. To guarantee that Yt

remains strictly positive, we require that 2am4b2, and for

all purposes assume a, b and m are positive. The

Brownian motions Wt and Zt are assumed to be corre-lated with correlation coefficient �.

The price of a European call option with payoff

function, max(XT� k, 0), within this model was firstcomputed by Heston (1993), who solved the following

partial differential equation (PDE):

qPqtþ1

2yx2

q2Pqx2þ ��yx

q2Pqxqy

þ1

2�2y

q2Pqy2þ r x

qPqx� P

� �

þ �ðm� yÞqPqy¼ 0, ð3Þ

with terminal condition

Pðx, y,TÞ ¼ max 0, x� kð Þ: ð4Þ

The solution is given as

Pðx, y, tÞ ¼ xV1 � ke�rðT�tÞV2, ð5Þ

where,

s ¼ ln xð Þ, ð6Þ

Vj ðs, y,T, kÞ ¼1

2þ

1

p

Z 10

Ree�� lnðkÞfj ðs, y,T,�Þ

i�

� �d�,

ð7Þ

fj ðs, y,T,�Þ ¼ exp Bj T� t,�ð Þ þ yDj T� t,�ð Þ þ �s� �

,

ð8Þ

2 S.H.M. Ting and C.-O. Ewald940

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Bj �,�ð Þ ¼ r�i� þa

�2bj � ��iþ dj� �

� � 2 ln1� gje

dj�

1� gj

� �� ,

ð9Þ

Dj �,�ð Þ ¼bj � ��iþ dj

�21� edj�

1� gj

� �, ð10Þ

gj ¼bj � ��iþ djbj � ��i� dj

, ð11Þ

dj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi��i� bj� �2

��2 2uj�i� �2� �

,

qð12Þ

for j¼ 1, 2, u1¼ 1/2, u2¼�1/2, a¼ am, b1¼ a� �b,b2¼ a.

2.2. Asymptotic solution

In the following we provide a brief description of theasymptotic solution to the exact Heston model. Our

derivation shows some analogy to the derivation inFouque et al. (1998, 2000). We only point out the major

differences between the two derivations.The probability density function (PDF) for Yt was first

derived by Feller (1951). However, it is now known that4 aYt/b

2(1� e�at) follows a non-central �2 distribution

(conditional on Y0¼ y) with k¼ 4 am/b2 being the degreesof freedom parameter and �¼ 4 ay/b2(1� e�at) being thenon-centrality parameter. Using this fact, the PDF can be

easily obtained. Furthermore, Yt possesses an invariantdistribution Y which is Gamma distributed with shape

parameter, k¼ 2 am/b2 and slope parameter ¼ b2/2a.This differs slightly from Fouque et al., where the OUprocess and its invariant distribution are both Gaussian in

nature.The PDE (3) can be rewritten with a’s and b’s replaced

by ’s and e’s, where and e are defined as the variance ofY, 2¼mb2/2a and e¼ 1/a, respectively. It then becomes

1

�L0 þ

1ffiffiffi�p L1 þ L2

� �P ¼ 0 ð13Þ

with

L0 ¼2y

m

q2

qy2þ ðm� yÞ

qqy

, ð14Þ

L1 ¼ �ffiffiffi2pffiffiffiffimp xy

q2

qxqy, ð15Þ

L2 ¼qqtþ1

2yx2

q2

qx2þ r x

qqx� �

� �¼ LBSð

ffiffiffiypÞ, ð16Þ

where

(i) aL0 is the infinitesimal generator of the CIRprocess Yt;

(ii) L1 contains the mixed partial derivatives due to

the correlation between the two Brownian

motions;(iii) L2 is the Black–Scholes operator with

ffiffiffiyp

as the

(constant) volatility parameter.

Assume the solution P can be expanded in the form

P ¼ P0 þffiffiffi�p

P1 þ �P2 þ �ffiffiffi�p

P3 þ � � � : ð17Þ

The normal perturbation technique from Fouque et al. is

then applied resulting in an approximation for the Heston

call option as

Pasymp ¼ P0 þ��

ffiffiffiffiffiffi�mp

xe�d21=2

2�ffiffiffiffiffiffi2pp 1�

d1ffiffiffiffiffiffi�mp

� �, ð18Þ

where P0 is the price of a European call option under the

Black–Scholes model withffiffiffiffimp

the constant volatility

parameter, and d1 is defined as

d1 ¼logðx=kÞ þ ðrþ 1

2mÞ�ffiffiffiffiffiffim�p ð19Þ

with �¼T� t. The asymptotic Heston solution is essen-

tially the Black–Scholes solution plus some correcting

terms. The derivation of this asymptotic solution can be

found in appendix A.It can be shown that the asymptotic Heston call price

also satisfies the put–call parity with an asymptotic

Heston put price, and thus one can easily obtain the

asymptotic Heston put option price through this relation.

In deriving the asymptotic solution, it is found that the

parameters a, m and b must satisfy

2�m=�2 4 1, ð20Þ

for the invariant distribution defined by the infinitesimal

generator L0 to be of use. By having this condition, it is

only then possible to make use of the adjoint of L0 to

solve the Poisson equations with ease. This is discussed in

full mathematical rigour in appendix A.2.yThe partial derivative of the asymptotic price with

respect to x, the asymptotic delta, is given as

qPasymp

qx¼Nðd1Þ þ

��ffiffiffiffiffiffi�mp

e�d21=2


2d1ffiffiffiffiffiffi�mp þ

d21m��

1

m�

� �:

ð21Þ

This can be shown by differentiating the asymptotic

solution with respect to x, or alternatively differentiating

equation (3) with respect to x and then applying the

perturbation technique for @P/@x, i.e.

qPasymp

qx¼

qPqx

� �asymp

: ð22Þ

The partial derivative with respect to y is slightly more

difficult to compute. The asymptotic solution is derived

through averaging arguments such that the first two

terms P0 and P1 in equation (17) are independent of y.

yAn expansion up to the 3rd term is discussed in Fouque et al. (2011); the 3rd and 4th terms are derived in an upcoming workingpaper by Ting and Ewald. In this paper, we only concern ourselves with the two-term expansion.

On the performance of asymptotic locally risk minimising hedges 3941

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Taking the next term, eP2, and differentiating it with

respect to y gives an approximation to the required partial

derivative to leading order in terms of e. It can be

shown that

P2ðt, x, yÞ ¼1

2yþ cðt, xÞð Þx2

q2P0

qx2, ð23Þ

where c(t,x) is a function independent of y. The proof of

this can be found in appendix A.3. In our application, the

form of c(t, x) is not important since we are only taking

partial derivatives of P2 with respect to y. This is

approximated as

qPqy�

1

2�x2

q2P0

qx2, ð24Þ

which follows from equation (23), by noting that P0 is

independent of y.

2.3. Model and numerical limitations

The exact Heston solution provides an analytic solution

for the pricing of call options under the CIR stochastic

volatility model. However, the solution is in the form of

integrals with complicated integrands. The most common

techniques used in evaluating these integrals are numer-

ical quadrature or fast Fourier transforms (FFTs). When

using either technique, there is usually a trade-off between

its accuracy and its efficiency in terms of speed. The

numerical results that follow for the exact Heston

solution will be calculated using code provided by Janek

and Weron (2010).y Any partial derivatives of the option

price with respect to stock price or the square of volatility

will be approximated using the finite difference method.The asymptotic Heston solution provides a faster

method for calculating call option prices in the Heston

stochastic volatility model. Naturally, there are issues

concerning the accuracy of the resulting option prices.

The following parameters are all important in determin-

ing the accuracy of the solution: the initial variance and

its mean reverting level, the mean reverting rate and the

time to expiry of the option. In particular, it is important

that a, m and b satisfy condition (20).The accuracy for extreme out-of-the-money call

options is quite poor. In particular, the asymptotic

solution may become negative, due to the addition of

the correction term. This is problematic for extremely out-

of-the-money options, since the Black–Scholes solution is

already quite small and, if the correcting term is negative

and has magnitude greater than the Black–Scholes

solution, then the asymptotic solution will be negative too.

As such, all negative option prices for the asymptoticsolution will be replaced by the value zero.

2.3.1. Initial variance and mean reverting level. Theasymptotic solution does not contain the initial varianceestimate in its parameter set. However, the asymptoticsolution is found to be a good approximation to the exactHeston solution when the initial variance is close to itsmean reverting level. The reason for being a goodapproximation in this situation is that the mean revertinglevel is used as a proxy for the variance process in theasymptotic solution. If the initial variance is close to thatmean reverting level, then that proxy will be a goodchoice. Figure 1 illustrates the relative errors for at-the-money call options, using parameters x0¼ 100, �¼ 1,and other parameters listed in table 1. The dotted line isthe reference line at zero and the blue line represents therelative errors. From these graphs, it is evident thatthe general rule of thumb regarding the sensitivity of theinitial variance for at-the-money call options is that, if theinitial variance is less (greater) than the mean revertinglevel, the asymptotic price tends to over (under) price theoption. Thus, any further comparison of the two solutionswill be made with the assumption that the initial varianceand the mean reverting level are equal, unless statedotherwise.

2.3.2. Mean reverting rate. Another parameter to con-sider when determining the accuracy of the asymptoticsolution is the mean reverting speed. Figure 2 shows that,as a increases, the asymptotic solution improves inaccuracy across a range of initial stock prices. The a’stested are a¼ 1, 2, 3, 4 and 5. The other parameters in thistest are m¼ 0.1, b¼ 0.3, �¼ 0.7, r¼ 0.05, y0¼ 0.1, �¼ 1and x02 [80, 120]. Importantly we have the initialvariance equaling the mean reverting rate in this case.For a¼ 4 and 5, the relative errors for out-of-the-moneyand in-the-money options are no larger (in absolute value)than 0.04 and 0.01, respectively.

From equation (18), it can be observed that, as aincreases to infinity, the asymptotic solution tendstowards the Black–Scholes solution. Under this limit,the Black–Scholes solution is a good proxy for the Hestonsolution. This can be reasoned by the fact that, for aninfinite mean reverting speed, the Yt process will hoverabout its mean reverting level indefinitely. As such, thiseffectively makes Yt constant, which implies the Black–Scholes solution should be close to the exact Hestonsolution.z

For completeness, figure 3 shows the relative errors ofthe asymptotic and Black–Scholes solutions when

yThe code provided both solutions using the Gauss–Kronrod quadrature and FFT. Speed tests showed that the Gauss–Kronrodquadrature solution was quicker than the FFT method. Both solutions were tested against other freely available codes and it wasfound that the Gauss–Kronrod quadrature provided better results. The Gauss–Kronrod quadrature method is implemented usingthe MATLAB� quadgk routine.zConlon and Sullivan (2005) showed that within an OU variance driven model, as in the original work of Fouque et al., the exactoption price does not converge to the Black–Scholes solution unless �¼ 0. As such, we do not expect that the exact Heston solutionconverges (in a mathematical sense) to the Black–Scholes solution. However, the asymptotic solution still presents a goodapproximation of the exact solution over the range of a’s considered.


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compared to the exact Heston solution for large a. Theparameter set is the same as for figure 2, except a is now

set to 500. Noting the scales on the y-axis, figure 3 shows

that the Black–Scholes solution is not such a poor

estimator for the exact Heston solution at this level ofa, although the asymptotic solution performs significantlybetter.

2.3.3. Time to expiry. The last of the major sources oferror in the asymptotic solution is its time to expiry. Theasymptotic solution is derived from time averagingarguments, assuming that, on average, the instantaneousvariance stays at or around its mean reverting level moreoften than not. The issue of whether the initial variance

0 0.05 0.1 0.15 0.2

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Initial Variance

Rel

ativ

e E

rror

m = 0.1

0.1 0.15 0.2 0.25 0.3

−0.02

0

0.02

0.04

Initial Variance

Rel

ativ

e E

rror

m = 0.2

0.05 0.1 0.15 0.2 0.25

−0.02

−0.01

0

0.01

0.02

Initial Variance

Rel

ativ

e er

ror

m = 0.15

0.15 0.2 0.25 0.3 0.35−6

−4

−2

0

2

4

6x10−3

Initial Variance

Rel

ativ

e er

ror

m = 0.25

Figure 1. Sensitivity analysis for initial variance; the asymptotic solution has the smallest absolute relative error when the initialvariance equals the mean reverting rate.

Table 1. Parameters for the sensitivity analysis for the initialvariance.

Position a m b � r y0

Top left 2 0.1 0.3 �0.7 0.05 [0, 0.2]Top right 5 0.2 0.2 0.4 0.1 [0.1, 0.3]Bottom left 10 0.15 0.4 0.7 0.15 [0.05, 0.25]Bottom right 100 0.25 0.1 �0.4 0.2 [0.15, 0.35]

80 85 90 95 100 105 110 115 120−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Stock Price

Rel

ativ

e er

ror

Reference Lineα = 1α = 2α = 3α = 4α = 5

Figure 2. Sensitivity analysis for a; the relative error of theasymptotic solution decreases as the mean reverting rateincreases.

80 85 90 95 100 105 110 115 120−10

−8

−6

−4

−2

0

2x10−4

Stock Price

Rel

ativ

e er

rors

Reference LineAsymptoticBlack−Scholes

Figure 3. Relative errors for a¼ 500; the relative error of theasymptotic solution is negligible in this case; the Black–Scholesrelative error is also very small.


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starts at its mean reverting level or not has beenaddressed. What needs to be considered is the length oftime given for this assumption to realise. This length oftime is in fact the time to expiry of an option, since pricingonly takes place while the option is still alive. Up until thispoint, all the graphs have been calculated using times toexpiry set at one year. Figure 4 shows varying times toexpiry for a fixed parameter set. The parameters are thesame as the ones listed in the sensitivity analysis for themean reverting rate, with the only difference being that ais set to 5, and �¼ 0.2, 0.4, 0.6, 0.8 and 1. Much like themean reverting rate, increasing time to expiry improvesthe asymptotic approximation, with the best result infigure 4 obtained for �¼ 1. For smaller times to expiry,the approximation can be quite poor. This poses aproblem for when the asymptotic solution is used in atrading strategy for times up until the expiry date.

2.3.4. Parameter condition for a, m and b. The conditionthat a, m and b must satisfy in equation (20) comes fromsolving the Poisson equation, see appendix A. Whilst,from a theoretical point of view, this condition must bemet in order to obtain a valid asymptotic solution, thiscondition is actually necessary for the CIR processdriving the volatility to remain positive at all times andfor the invariant distribution to take on a Gamma form.

3. Review of locally risk minimising hedging

The concept of reducing risk in option hedging has beenexplored for a long time. Self-financing perfect hedges arehedges whose cost process is constant and replicates thederivative perfectly, thus attracting no risk. However,such hedges are only available in complete marketmodels. As real world markets show clear signs ofincompleteness, the need to develop trading strategiesfor incomplete market models, to minimise the riskinvolved in hedging strategies, arose. One method of

minimising the risk involves a criterion proposed byFollmer and Schweizer (1990), which is to minimise a riskfunction, defined as the conditional variance process ofthe cost process involved in conducting the hedge.However, this leads to a dynamic optimisation problemwhich may not have any solutions.

Schweizer (1991) explored the concept of locally riskminimising hedges for incomplete markets. The generalidea is to minimise the conditional variance of instanta-neous cost increments sequentially over time. While thisproblem is solved in theory, many computational aspectsof practical implementation still deserve attention. ElKaruoi et al. (1997) showed how it is possible to obtainlocally risk minimising hedges by first completing themarket introducing a new tradable asset, then calculatinga hedging strategy for this complete market, and finallyprojecting the hedging strategy back onto the originalincomplete market. In Poulsen et al. (2009), the locallyrisk minimising hedges for a general class ofstochastic volatility model are derived in this way inexplicit form. A brief review of their results is outlinedbelow.

3.1. Cost function of trading strategy

Define a trading strategy u(t)¼ (u0(t), u1(t)) such that thecomponents indicate the holding amounts (in units of)risk-free asset Bt (bank account) and risky asset Xt

(stock), respectively. The cost function is defined as thedifference between the holdings of the trading strategy attime t and the cumulative gains or losses up to time t.Mathematically, the cost associated with a trading strat-egy u(t) at time t is calculated as

Cost’ðtÞ ¼ V’ðtÞ �

Z t

0

’0ðsÞ dBs �

Z t

0

’1ðsÞ dXs, ð25Þ

where Vu(t)¼u0(t)Btþ u1(t)Xt} denotes the value of thetrading strategy at time t. If the cost associated with atrading strategy is constant, the trading strategy is said tobe self-financing. Stochastic volatility models describeincomplete markets; the volatility cannot be traded.This means that not all contingent claims can beperfectly hedged using self-financing hedging strategiesand thus the need to develop other forms of hedgingevolves.

3.2. Locally risk minimising strategy

In Poulsen et al. (2009), the locally risk minimisingstrategy for a general class of stochastic volatility modelshas been presented. In particular, this paper is concernedwith the hedging strategy for when the stochastic volatil-ity model is that of the benchmark tested Heston modelunder a minimal martingale measure. For further detailson derivation, please see Poulsen et al. (2009).

Using the notation for the Heston model as introducedin subsection 2.1, we define the value of a European calloption at time t, with terminal payoff max(0, XT� k) asP(Xt,Yt,T ). It follows from Poulsen et al. (2009), for the

80 85 90 95 100 105 110 115 120−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Stock Price

Rel

ativ

e E

rror

Reference Lineτ = 0.2τ = 0.4τ = 0.6τ = 0.8τ = 1

Figure 4. Sensitivity analysis for time to expiry; the relativeerrors of the asymptotic solution decrease as the time to expiryincreases.


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Heston model, that the locally risk minimising hedgingstrategy is given by

’0minðtÞ ¼ e�rt PðXt,Yt, tÞ � ’1minðtÞXt

h i, ð26Þ

’1minðtÞ ¼ PX þ ��PY

Xt, ð27Þ

where PX and PY denote the partial derivatives of theoption price with respect to stock price and variance,respectively.

Using the explicit solutions for the asymptotic callprice, and its partial derivatives, the locally risk minimis-ing hedging strategy is given as

’0minðtÞ � e�rt P0 � xNðd1Þ þ��xd1e

�d21=2


d1ffiffiffiffiffiffi�mp

� �" #,

ð28Þ

’1minðtÞ � N ðd1Þ þ��

ffiffiffiffiffiffi�mp

e�d21=2


d1ffiffiffiffiffiffim�p

� �2

: ð29Þ

Immediately, it is clear that the amount invested in therisky asset, ’1minðtÞ, is always more (less) as compared tothe standard Black–Scholes delta hedge, when the sign of� is positive (negative). The absolute difference isproportional to the absolute values of � and b, whilebeing inversely proportional to a. Its dependence on m isless obvious as m not only appears in the expression but isalso hidden within the constant d1.

The relationship between the CIR parameters and theamount invested in the risk-free asset, ’0minðtÞ, is less obvi-ous. For the special case when the option is at-the-money,’0minðtÞ is more than the standard Black–Scholes deltahedge, if �40 and m42r, or �50 and m52r. If one ofthe two inequalities from either of the last two pairs is notsatisfied, then ’0minðtÞ will be less than the standardBlack–Scholes delta hedge. The difference, in terms ofmagnitude, is again proportional to the absolute value of� and b, inversely proportional to a and again difficult todetermine in terms of m.

4. Asymptotic hedge on simulated data

In this section we evaluate the use of the asymptoticsolution in locally risk minimising hedging usingsimulated data. The design of this analysis is similar toPoulsen et al. (2009) and is as follows: asset prices andinstantaneous variances paths are simulated according toset parameters. Different trading strategies are applied tothese simulations and the cost associated with eachrecorded. This is repeated 10 000 times in order tocompute the hedging error, which is defined to be

Hedging Error ¼ 100�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivarðCost’ðTÞÞ

pPðX0,Y0, 0Þ

, ð30Þ

i.e. the standard deviation of the cost process at expirydivided by the initial cost of the option value, ascalculated using the exact Heston formula, as apercentage. Smaller hedging errors indicate less varianceassociated with the final cost of the trading strategy.

4.1. Hedger types

In this analysis, we consider the following four differenttypes of hedger.

. Hedger 1. The exact Heston locally risk mini-mising hedger who uses the full parameter set,the Heston solution and its partial derivatives.

. Hedger 2. The asymptotic Heston locally riskminimising hedger who uses the full parameterset, the asymptotic Heston solutions and itspartial derivatives.

. Hedger 3. The Black–Scholes locally risk mini-mising hedger who uses the Black–Scholessolution and its partial derivatives in place ofthe exact Heston solution and its partial deriv-atives. This hedger also takes the square root ofthe instantaneous variance to be the Black–Scholes volatility parameter.

. Hedger 4. The Black–Scholes delta hedger, whouses the Black–Scholes solution and delta, tocreate a standard delta hedge. This hedger takesthe square root of the instantaneous variance tobe the Black–Scholes volatility parameter anddoes not use any stochastic volatility model.

The main difference between these simulations and thatof Poulsen et al. (2009) is the inclusion of Hedger 2, theasymptotic Heston hedger. The options to be hedgedagainst are all one-year European call options. Theportfolio is to be re-hedged daily, assuming a 250 dayper year calendar.

4.2. Simulation parameters

The simulated data are generated using the two differentparameter sets listed in table 2.

The hedging errors within each parameter set will becompared across different values of �, in [�0.575, 0.575],spaced in intervals of 0.05.

These two parameter sets were chosen for variousreasons. Set 1 is the same parameter set used by Poulsenet al. (2009) and in fact are parameters calibrated tohistorical data by Eraker (2004).y Using the sameparameter set allows for a direct comparison betweenthese results and the one previously obtained. In addition,the mean reverting rate is slightly less than the initialvariance. Set 2 contains parameters with relatively large a,the initial variance is equal to the mean reverting level.

It is of interest to see the impact of the asymptotichedging strategy in these two test cases, given that set 1has a few conditions that are not ideal for the asymptotic

yHistorical option data are calibrated by fitting a joint posterior density, then Markov chain Monte Carlo sampling is used toobtain a sample of the model parameters.


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solution, while in set 2 the scenario satisfies all therequirements for using the asymptotic solution. Theinterest rate parameters are chosen as plausible values,while the strike prices k are chosen so that the options areeither at-the-money or forward at-the-money. Thesechoices of strikes are usually popular in the markets,and are also where the asymptotic solutions excel.

Figure 5 shows the option price and partial derivativesfor particular simulated paths using parameter set 1. Theplots are generated with a � value of �0.575. These figuresshow that the asymptotic solution, as it tracks a simulatedstock price, performs reasonably well. As expected, mostof the inaccuracies arise from when the times to expiry areshort as expected. In most cases, with large times toexpiry, the asymptotic solution outperforms the pureBlack–Scholes solution. From graph 3 of figure 5, it isexpected that Hedger 3 will perform very poorly due tothe bad estimates of the partial derivative with respect tovariance. The plots for parameter set 2 are similar to theone in figure 5, and are thus omitted.

4.3. Hedging errors and costs for different correlations

The hedging errors across different correlations areshown in figures 6 and 7 for parameter sets 1 and 2,respectively. The general observations that can be madeabout the performance of these hedges are that Hedger 3performs the worst in all situations as expected andHedger 1 performs strongly when the magnitude of thecorrelation is high. The asymptotic hedge performs betterthan the standard delta hedge for negative �, and roughlythe same for positive �.

Since parameter set 1 is the same as in Poulsen et al.(2009), a direct comparison can be made. The asymptotichedging errors are found to lie in between that of theexact Heston hedging errors and the standard deltahedging errors for negative �. For positive �, theasymptotic hedge seems to return a hedging error that isslightly higher than the standard delta hedge, but it doesnot seem significant. Given that the parameter set isactually derived from calibrated data, and in the cali-brated parameter set � was �0.569, this gives hope inusing the asymptotic solution as a fast replacement for theHeston solution, at least in locally risk minimising hedges.

The asymptotic solution obtained using parameter set 2are theoretically the best, out of the two parameter sets, atapproximating the exact Heston solution. This is becausethe parameter set satisfies many of the conditions in orderfor the asymptotic solution to provide good approxi-mates. In terms of the hedging error, the same results wereobserved as in the parameter set 1 simulations. Again, the

slight drop in performance, in terms of the hedging error,in the asymptotic hedge for positive �.

It must be reiterated that the hedging errors are onlyindicative of the variances involved in the cost associatedwith performing these hedges and by no means relay theactual cost associated with these hedges. Whilst it may beuseful to know the variances involved in performing thesehedges, practically, the expected cost involved is equallyas important. Since the hedges are supposed to replicatethe call option, on average, the costs associated with thesehedges should in fact be close to the cost of the initialHeston price. For the exact Heston locally risk minimis-ing hedge it can be concluded from Follmer andSchweizer (1990), that the hedge is in fact mean self-financing, i.e. subtracting the initial cost (option price) theexpectation of the cost process is zero at all times. This atleast is the theory. In practice, due to the discretisation,the hedges may not be mean self-financing. Figures 8and 9 show the mean average costs subtracted by theinitial Heston option price, associated with these hedges.These figures show that Hedgers 1, 2 and 3 have averagecost prices which are roughly near the initial Heston price.Furthermore, these average costs are roughly equal to oneanother with none being significantly more or less thanthe others. The absolute difference between the averagecost of Hedger 4 and the initial Heston price increases asthe absolute value of � increases.

In summary, Hedger 1 performs the best in terms ofhaving a lower hedging error, and being able to maintainan average cost close to the initial Heston price. Hedger 3,while having an average cost close to the initial Hestonprice, also has a much greater hedging error, and thus isnot using a suitable hedging method. Hedger 4, has astable hedging error across various values of � but theaverage costs of the hedges are much greater (in absolutevalue) than the initial Heston price, i.e. not mean self-financing. The implications of this are that, depending onwhether he/she is buying (as in our simulation tests) orselling the hedge, there are chances where on average, thecost associated with Hedger 4’s hedge is significantly morethan the initial price of the option. Ideally, a hedge shouldbe mean self-financing, irrespective of whether one buysor sells it. Thus Hedger 4’s overall performance is ratherpoor. Hedger 2, has a lower hedging error than the twoBlack–Scholes type hedges for negative �, while forpositive �, the difference in hedging error betweenHedger 2 and Hedger 4 is not very significant. However,the average cost for Hedger 2 is much closer to the initialHeston price than Hedger 4, across all values of �, andthus exhibits mean self-financing features. Thus, theasymptotic hedge is a viable alternative to the Black–Scholes hedges.

Table 2. Simulation parameters.

Set no. a m b � r � k x0 y0

1 4.75 0.0483 0.550 [�0.575, 0.575] 0.04 1 100er 100 0.052 5.00 0.0500 0.300 [�0.575, 0.575] 0.10 1 100 100 0.05


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Under ideal conditions,y the results show that the

asymptotic hedge is a viable alternative to traditional

Black–Scholes hedging methods. However, it must be

noted that, in practice, model parameters are recalibrated

to actual option data at every rehedge, instead of

assuming they are fixed for the life of the hedge.

Of course, in these simulations we only simulate the

stock price and variance paths so we do not have any

option data to recalibrate against. A study that is more

in line with industry practices, i.e. recalibration to

00.10.20.30.40.50.60.70.80.910

10

20

30

Time to ExpiryO

ptio

n P

rice

HestonAsymptoticBS

00.10.20.30.40.50.60.70.80.910

0.5

1

1.5

Time to Expiry

Del

ta

HestonAsymptoticBS

00.10.20.30.40.50.60.70.80.91−100

0

100

200

300

Time to Expiry

Veg

a

HestonAsymptoticBS

Figure 5. Parameter set 1: the asymptotic solution and its partial derivatives track the exact Heston solution closely over time.The difference is smaller when the time to expiry is large. The Black–Scholes solution and its derivatives do not do as well.

−0.4 −0.2 0 0.2 0.4 0.6

20

25

30

35

40

ρ

Hed

ging

Err

or

Hedger 1Hedger 2Hedger 3Hedger 4

Figure 6. Hedging error for parameter set 1: the asymptotichedging strategy is still robust enough to provide an adequatehedge. Performance for positive � is similar to a standard Black–Scholes delta hedge. Note that the bump for Hedger 1 at�¼� 0.425 is the result of one large cost price, this resulting inthis anomaly.

−0.4 −0.2 0 0.2 0.4 0.6

7

8

9

10

11

12

13

ρ

Hed

ging

Err

or


Figure 7. Hedging error for parameter set 2: when the param-eter set satisfies many conditions in using the asymptoticsolution, the asymptotic hedge performs well. Again, perfor-mance for positive � is similar to that for the standard deltahedge.

yLarge mean reversion rate and mean reverting level approximately equal to the initial variance.


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option data at every rehedge, is undertaken in the nextsection.

5. Asymptotic hedge on real data

In the following analysis, the asymptotic Heston hedge iscompared to that of the exact Heston hedge using realhistorical data. In this comparison the calibration, theaccuracy, the performance, and the computational time ofthe hedge will be considered.

The setup of the hedges is slightly different from thesetup used in the simulation section of the previoussection. When the portfolio is rehedged, the modelparameters are recalibrated using option price dataavailable at that time. The motivation for this is so that,at each rehedge, the trading strategy is more in line withmarket data than data from the past. Furthermore, itcannot be expected that the option being hedged against

will be priced in the market by the parameters calibratedwhen initialising the hedge, but more so by the parameterscalibrated at the time of the rehedge. As such, the hedgesare more like a Heston hedge that has been set in motionin the spirit of Carmona and Nadtochiy (2009) than atraditional Heston hedge with fixed parameters.

5.1. Dataset

The asymptotic Heston hedge is applied to two datasets.They are the S&P 500 and EUROSTOXX 50 indexspanning from 7 January 2004 to 4 June 2008. The dataare collected weekly with 231 weeks in total. For eachweek, there are 15 implied volatility values correspondingto the three lengths of expiry times, those being one year,six months and three months. Furthermore, each expirydate has a strike price at 110, 105, 100, 95 and 90% of thecurrent spot price. In addition, there are three interestrates, i.e. the one-year, the six-months and the three-months rate.

5.2. Calibration

Calibration was performed using both the exact Hestonsolution and the asymptotic Heston solution on the twodatasets. For each week’s implied volatilities, the calibra-tion process yields a set of model parameters. Theseparameters are obtained using the minimisation of leastsquares method through the MATLAB� routinelsqnonlin.

Figures 10 and 11 show the root mean square errors(RMSEs) of the calibrated implied volatilities of both theexact and asymptotic solutions. The exact Heston cali-bration is shown to be a better overall fit than theasymptotic Heston calibration, which is expected giventhe exact Heston solution has the additional initialvariance as an additional parameter. However, thecalibration using the asymptotic solution seems to providea reasonable approximation.

We have also tested the use of the asymptoticcalibrated parameters as a starting point for the

−0.4 −0.2 0 0.2 0.4 0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

ρ

Ave

rage

Cos

t − In

itial

Hes

ton

Pric

e


Figure 8. Difference between average cost and initial Hestonprice for parameter set 1: the differences show that Hedgers 1, 2and 3 are roughly mean self-financing, whereas Hedger 4 is not.Also, the average costs for the asymptotic hedge are similar tothose for the exact Heston hedge.

−0.4 −0.2 0 0.2 0.4 0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

ρ

Ave

rage

Cos

t − In

itial

Hes

ton

Pric

e


Figure 9. Difference between average cost and initial Hestonprice for parameter set 2: the mean self-financing result is similarto that for parameter set 1.

0 50 100 150 2000.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Week Index

RM

SE

ExactAsymp

Figure 10. Root mean square errors for the S&P 500: theRMSEs are generally higher for the calibration using theasymptotic solution than for the exact Heston solution.


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exact Heston calibration. However, it was found that the

two calibrated parameter sets were quite different from

each other. Furthermore, using the asymptotic calibrated

parameters as starting points was of no benefit in terms of

accuracy or computational time. This can be explained by

the fact that there are four parameters in the asymptotic

calibration, while in the exact Heston calibration there are

five (additional starting volatility). Thus, the parameters

from the asymptotic calibration must make up this

shortfall, which is why the two parameter sets exhibit

very different values.

5.3. Hedge portfolio

The performance of the hedge is tested on one-year call

options with forward at-the-money strikes. The hedging is

performed as follows: at time t, u1(t) units of stocks are

held and u0(t) units of money are invested in the money

market. u1 and u0 are calculated using both the exact and

asymptotic Heston solutions, together with their respec-

tive calibrated parameters. The quality of the hedges are

measured via the weekly profit and loss ratios (PLRs)

defined as

P&Lðtþ dtÞ ¼’1ðtÞXtþdt þ ’

0ðtÞer dt � Cðtþ dtÞ

CðtÞ, ð31Þ

where dt¼ 1/52, r is the one-year interest rate, and C(t) is

the call option price at time t, calculated using the exact

Heston solution and its calibrated parameters. Ideally,

C(t) would be the actual observed option prices; however,

they may not exist with the parameters as specified, thus it

is best to use market calibrated theoretical prices. Also,

when calculating the PLRs for the asymptotic Heston

hedge, the exact Heston option price will still be used.y

With 231 weeks of data, we can start a total of 179

hedges, with the last hedge starting on week 179. This is

because each hedge takes 52 weeks to complete, and thus

for hedges starting on weeks after 180, we do not haveenough data to complete the hedge.

A problem that has been brieflymentioned before occurswhen the option price is extremely out-of-the-money.While, in simulated cases, treating these prices and theirpartial derivatives as zero may not be too impactful, in thereal world this has many more ramifications. One way todeal with this is by using a hybrid hedging process. Toimplement such a scheme, we could designate the asymp-totic solution to be the default formula to use in calculatingthe trading strategy.Whenever any of the asymptotic pricesor their partial derivatives are zero, we can replace all thecalculations at that point with the exact Heston solution.We can do so because we know that while the option and itspartial derivatives are small, they are not exactly zeroeither. By treating them as zero, we may in fact be ruiningthe whole hedge. The parameters to be used in the exactHeston solution will be the ones calibrated using theasymptotic solution. This is justified because the calibratedparameters from the asymptotic solution are able to fit thegeneral shape of the observed market prices. In terms ofcomputation, it is expected that the exact Heston solutionwill not be invoked much. From here onwards, referencesto the asymptotic hedge will actually refer to the hybridhedging method.

5.4. Results

The performance of the PLRs for exact and asymptotichedges will be compared over the life of the option, andalso for options starting on different dates. Figure 12shows the PLRs over time, on options starting on thedates 18 February 2004, 20 October 2004 and 15 June2005 for the S&P 500 dataset. The figure shows that thePLRs for the asymptotic hedge have roughly the sameperformance as the exact Heston hedge.

Figure 13 shows in more detail the performances of the179 hedges for the S&P 500 dataset. The top part ofthe figure shows the difference in the mean PLRs, while thebottom part shows the difference in the standard deviationof the PLRs, for hedges starting onweeks given by theweekindex. For example, the hedge starting on 18February 2004has week index 7, and the corresponding differences in themean and standard derivation for the life of that hedge canbe found in figure 13 by looking at week index 7.

The results show that the asymptotic hedge is able totrack the exact Heston hedge quite well for most of thehedges. However, after week index 160, the asymptoticand exact Heston hedge seem to experience somedifficulties. This can be explained by noting that, ataround week index 211, the spot price of the S&P 500index began falling. This meant that many of the optionsstarting on and after week index 160 finished out-of-the-money. This poses two problems, the first of which is onlyfor the asymptotic hedge. This has already been dealtwith using the hybrid scheme. The second problem is that,as these options approach the expiry date, they begin to

0 50 100 150 2000

0.005

0.01

0.015

0.02

0.025

Week Index

RM

SE

ExactAsymp

Figure 11. Root mean square errors for the EUROSTOXX 50:the RMSEs are generally higher for the calibration using theasymptotic solution than for the exact Heston solution.

yThe assumption is that other market participants would only be calculating their Heston prices using the exact Heston solution.


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lose a large portion of their value. This translates to the

PLRs (for both the asymptotic and exact Heston hedge)

close to the expiry date being less meaningful, because a

decreasing denominator in the ratio makes the ratio

unnecessarily large in absolute value. This, in turn,

distorts the mean and standard deviation of the PLRs

over the life of that hedge. Figure 14 shows the mean for

week index 160 to 179, by taking the first 40 PLRs instead

of the full 52. It is clear that, for week index 160 to 172,

the asymptotic hedge still tracks the exact Heston hedge

well for the first 40 weeks of the respective hedges. The

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

Weeks Since StartP

&L

Rat

io

ExactAsymp

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

Weeks Since Start

P&

L R

atio

ExactAsymp

0 5 10 15 20 25 30 35 40 45 500

0.5

1

Weeks Since Start

P&

L R

atio

ExactAsymp

Figure 12. Three profit and loss ratios over the life of the hedge. These are just three samples out of the 160 hedges performed.They correspond to week index 7, 42 and 76, respectively, from top to bottom.

0 20 40 60 80 100 120 140 160−0.1

−0.05

0

0.05

0.1

Week Index

Diff

eren

ce in

Mea

n P

LR

Exact − AsympReference

0 20 40 60 80 100 120 140 160 180−0.1

−0.05

0

0.05

0.1

Week Index

Diff

eren

ce in

Sta

ndar

d D

evia

tion

of P

LR


Figure 13. Differences of the mean and standard deviation of the PLRs for the S&P 500. Each week index corresponds to one of the179 hedges performed. They show that the asymptotic hedge performs as well as the exact Heston hedge. Note that the largedifferences between week index 160 and onwards, compared to the previous week indices, are explained in subsection 5.4.

160 165 170 175 1800.05

0.1

0.15

0.2

0.25

Week Index

Mea

n P

LR

ExactAsymp

Figure 14. Mean PLRs for week index 160 to 179. The meansare calculated using the first 40 PLRs for each of the hedges.They show that, for the early stages of the hedges, theasymptotic solution still tracks the exact Heston hedge.


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effects of the falling S&P 500 index can still be seen inweek index 173 to 179, even though we restrict ourselvesto only the first 40 PLRs.

Figure 15 shows the differences in the mean andstandard deviation of the PLRs for the EUROSTOXX 50dataset. Many of the phenomena observed in the S&P 500dataset are again observed here. The asymptotic hedgetracks the exact Heston hedge well until about week index160. Again, a decrease in the spot price of theEUROSTOXX 50 index explains the results observed.

In the hybrid scheme, the exact Heston solution wasnever once invoked for the first 157 hedges for the S&P500 dataset, or for the first 158 hedges for theEUROSTOXX 50 dataset. This further highlights that,when all is ‘well’ in the market, the asymptotic hedge is agood replacement for the exact Heston hedge.

5.5. Computational time

In assessing the worthiness of using the asymptotic hedge,the most important factor to consider is the computa-tional time associated with the whole process. While thetrading strategy in these case studies is only rehedged on aweekly basis, it stands to reason that the time taken tocalibrate the model parameters and to compute thetrading strategies scales with the number of rehedges.

Table 3 lists the computational time required for eachstep of the process in one particular run. It is important tonote that these run times differ depending on thecomputer used. Furthermore, the calibration is alsodependent on the initial guess.

The calibration process involves all 231 weeks of dataand the profit and loss ratio process involves thecalculation of the ratio and the trading strategies of the179 weeks. The run times show that the calibrationprocess and the profit and loss calculations using theasymptotic solution has reduced the computational timeby factors of about 6 and 1.6, respectively. The reason forthe differences between run times in the calibration is due

to the fact that, computationally, the asymptotic solutionis a much simpler expression to evaluate than the exactsolution. This was largely covered in subsection 2.3. Thus,in general, it is expected that any computations involvingthe asymptotic solution will be much faster compared tothose involving the exact solution.

The PLR run times do not differ as significantly, onlybecause the PLR for the asymptotic hedge required thecalculation of the exact Heston call price, since it wasassumed that the observed market price of these optionswas based on exact Heston calculations. When the Hestonprices are given, the run times for the PLR for theasymptotic hedge are only 6.15 and 7.05 seconds, respec-tively, for the two datasets, which is a reduction incomputational time by a factor of about 10. Thesecomputational times show that asymptotic hedges aresignificantly quicker to compute than exact hedges.

Note that the calculations (including those from thesimulation study) were all performed using an Intel Core2 Quad 3.6GHz PC with 8GB of RAM. As such, runtimes may differ from PC to PC, but the magnitude of thedifferences in the run times should remain.

6. Conclusion

This paper investigates the use of the asymptoticHeston solution in locally risk minimising hedging.

0 20 40 60 80 100 120 140 160−0.1

−0.05

0

0.05

0.1

Week IndexD

iffer

ence

in M

ean

PLR Exact − Asymp

Reference

0 20 40 60 80 100 120 140 160−0.1

−0.05

0

0.05

0.1

Week Index

Diff

eren

ce in

Sta

ndar

d D

evia

tion

of P

LR


Figure 15. Mean and standard deviation of the PLRs for the EUROSTOXX 50. Each week index corresponds to one of the179 hedges performed. The results are similar to those for the S&P 500 hedges. Again, note that the large differences between weekindex 160 and onwards, compared to the previous week indices, are explained in subsection 5.4.

Table 3. Computational time required for the calculations.

Asymptotic hedger Exact hedger

S&P 500Calibration 1min 22 sec 8min 13 secProfit and loss ratio 39.59 sec 65.28 sec

EUROSTOXX 50Calibration 1min 9 sec 8min 6 secProfit and loss ratio 40.69 sec 66.32 sec


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The asymptotic solution is found to be a good approx-imation when the initial variance of the Heston model isequal to its mean reverting level, the mean reverting rate isgreater than 4, and the time to expiry of the option isgreater than one year. The solution is also found to betechnically valid only under condition (20), which is alsothe requirement that the CIR process remains positiveat all times, and that a Gamma distributed invariantdistribution exists. The asymptotic hedges for simulateddata results in hedging errors that do not differ too muchfrom the exact Heston hedge, whilst also on average notincurring any additional cost. Under real historical data,the asymptotic hedge is found to require less time incalibration than the exact solution. However the trade-offis accuracy, with the exact Heston solution returning alower RMSE error than that of the asymptotic solution.The PLRs of these asymptotic hedges are comparable tothe ones obtained using the exact Heston hedge. Thecomputational time of the asymptotic hedges is alsosignificantly faster than that of the exact hedge. Theseadvantages provide enough evidence to show that theasymptotic Heston solution is robust enough to be aviable alternative to the exact Heston solution, in thecontext of locally risk minimising hedges.

Acknowledgements

Both authors gratefully acknowledge funding through theAustralian Research Council’s Discovery GrantDP1095969. We further acknowledge useful suggestionsand support from Neville Weber, David Ivers and twoanonymous referees in preparing this article.

References

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Alexander, C. and Nogueira, L., Model-free hedge ratios forhomogeneous claims on tradable assets. Quant. Finan., 2007b,7, 473–479.

Bakshi, G., Cao, C. and Chen, Z., Empirical performance ofalternative option pricing models. J. Finan., 1997, 52,2003–2049.

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El Karuoi, N., Peng, S. and Quenez, M., Backward stochasticdifferential equations in finance. Math. Finan., 1997, 7, 1–77.

Eraker, B., Do stock prices and volatility jump? Reconcilingevidence from spot and option prices. J. Finan., 2004, 59,1367–1403.

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Heath, D., Platen, E. and Schweizer, M., A comparison of twoquadratic approaches to hedging in incomplete markets.Math. Finan., 2001, 11, 385–413.

Heston, S.L., A closed-form solution for options with stochasticvolatility with applications to bond and currency options.Rev. Finan. Stud., 1993, 6, 327–343.

Janek, A. and Weron, R., HESTONFFTVANILLA: MATLABfunction to evaluate European FX option prices in the Heston(1993) model using the FFT approach of Carr and Madan.Statistical Software Components M430002, Department ofEconomics, Boston College, MA, 2010.

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Poulsen, R., Schenk-Hoppe, K.R. and Ewald, C.O., Riskminimization in stochastic volatility models: Model risk andempirical performance. Quant. Finan., 2009, 9, 693–704.

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Appendix A: Derivation of the asymptotic Heston

solution

In this appendix, we provide details for the derivation ofthe asymptotic solution as given in equation (18). Fromequation (17), let the terminal conditions of the Pi’s besuch that P0(�, T )¼ (x� k)þ, while for i40, Pi(�, T )¼ 0.The normal perturbation techniques from Fouque et al.involve the substitution of equation (17) into equation(13), upon which coefficients of various orders of

ffiffiffi�p

areequated to zero. These equations form restrictions on theform which the expanded solution can take.

Also note that the derivation presented below is for theHeston model. If the volatility process is a generalfunction f(Yt), instead of

ffiffiffiffiffiYt

p, then the asymptotic

technique can still be applied. However, closed formsolutions for the asymptotic price will ultimately dependon the functional form of f.

A.1. Orders 1/e and 1=ffiffiffi�p

Firstly, order 1/� implies that P0 is independent of y dueto the form of equation (14). Similarly, order 1=

ffiffiffi�p

implies that P1 must also be independent of y due to P0

being independent of y and the form of equation (15).


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A.2. Poisson equation review

In going to the next order, a brief review of the Poisson

equation is first provided. Consider the equation

L0�þ g ¼ 0,

which is known as a Poisson equation for �(y) withrespect to the operator L0. In this case, L0 is defined by

equation (14). A solution for �(y) exists if and only if thefunction g(y) is centred with respect to the invariant

distribution whose infinitesimal generator is given by L0.If a general CIR process has mean reverting rate and

volatility as �� and ��, respectively, then L0 is theinfinitesimal generator of a CIR process with �� ¼ 1 and�� ¼

ffiffiffi2p=ffiffiffiffimp

. This follows directly from equation (14).Thus, the invariant distribution is Gamma distributed

with shape k¼m2/2 and scale ¼ 2/m. Converting back in terms of a and b gives the parameters as k¼ 2a m

/b2 and ¼ b2/2a. The centring condition requires

g

:¼

Z 10

gð yÞ�invð yÞdy ¼ 0,

where �inv(y) is the probability density function of ~Y.

This is given as

�invð yÞ ¼e�y=yk�1

�ðkÞk: ðA1Þ

The centring condition is shown to hold by using thePoisson equation, integration by parts, and the adjoint

operators L?0 and its property that L?0�inv ¼ 0. We furtherassume that both g and �, and all their partial derivatives,

have bounded growth. The proof of the centring condi-tion is as follows: we have

g ¼ � L0�h i

¼ �

Z 10

L0�ð yÞð Þ�invð yÞdy

¼ �

Z 10

�invð yÞ ðm� yÞq�qyþ�2

2�yq2�qy2

� �dy

¼ � �invð yÞ ðm� yÞ�ð yÞ þ�2

2�yq�qy

� �� 10

þ

Z 10

�ð yÞqqyðm� yÞ�invð yÞ½ �dy

þ

Z 10

q�qy

qqy

�2

2�y�invð yÞ

� �dy: ðA2Þ

From the form of �inv(y) in equation (A1), theboundaries at y¼ 0 and y!1 of the following functions

are

�invð0Þ ¼ 0, if 2�m=�2 4 1,

�invð yÞ ! 0, if y!1,

y�invð yÞ ¼ 0, at y ¼ 0 , if 2�m=�2 4 0,

y�invð yÞ ! 0, if y!1:

Thus, if the condition

2�m

�24 1

is satisfied, then the first term in equation (A2) is zero

provided �(y) does not grow to infinity faster than �inv(y)

goes to zero at the end points. Continuing on,

g ¼

Z 10

�ð yÞqqyðm� yÞ�invð yÞ½ �dy

þ

Z 10

q�qy

qqy

�2

2�y�invð yÞ

� �dy

¼ � �ð yÞqqy

�2

2�y�invð yÞ

� �� 10

�

Z 10

�ð yÞL?0�invð yÞdy,

ðA3Þ

with the adjoint operator L?0 defined as

L?0 ¼ �

qqy

m� yð Þ�½ � þ�2

2�

q2

qy2y�ð Þ:

Using the boundary conditions at y¼ 0 and y!1 again,

the first term in equation (A3) is zero because

qqy

y�invð yÞð Þ ¼ �invð yÞ þ y�0invð yÞ

¼2�m

�2�invð yÞ �

2�

�2y�invð yÞ:

This gives

g ¼ �

Z 10

�ð yÞ L?0�invð yÞ� �

dy

¼ 0,

which completes the proof. In addition, the solution for

�(y) is given by

�ð yÞ ¼

Z 10

E gðYtÞjY0 ¼ yð Þdt, ðA4Þ

where the expectation in equation (A4) is calculated using

the transitional probability density function of the CIR

function in Feller (1951) given by

�ð yt, tj y0Þ ¼2�

�2ð1� e��tÞ

yty0e��t

� �ðm�=�2Þ�ð1=2Þ

� exp �2�ð y0e

��t þ ytÞ

�2ð1� e��tÞ

� �

� Ið2m�=�2Þ�14�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy0yte��tp

�2ð1� e��tÞ

� �,

where Ik(�) is the modified Bessel function of the first kind

of order k. Note that we have presented the general

transitional probability density function here, and that,

for our application, we set � ¼ �� and � ¼ ��.

A.3. Order 1

In continuing to equate the orders, order 1 requires

L0P2 þ L1P1 þ L2P0 ¼ 0,

and by using the results of order 1=ffiffiffi�p

, this becomes

L0P2 þ L2P0 ¼ 0: ðA5Þ


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This is exactly in the form of a Poisson equation, if weonly consider the dependence from y alone. The centringcondition requires

L2P0h i ¼ 0, ðA6Þ

and, using h y i¼m, gives the PDE for P0 to satisfy as

qP0

qtþ1

2mx2

q2P0

qx2þ r x

qP0

qx� P0

� �:

Together with the terminal condition, it is easy toconclude that P0 is the Black–Scholes solution with

ffiffiffiffimp

as the volatility parameter.From equation (A5) and using equation (A6),

L0P2 ¼ �L2P0

¼ � L2P0 � L2P0h ið Þ

¼ �1

2y�mð Þx2

q2P0

qx2:

This gives P2 as

P2ðt, x, yÞ ¼ �1

2L�10 y�mð Þx2

q2P0

qx2

¼ �1

2�ð yÞ þ ~cðt, xÞð Þx2

q2P0

qx2, ðA7Þ

where ~cðt, xÞ is a function independent of y, and �(y) is thesolution to the Poisson equation

L0� ¼ y�m

with solution given as

�ð yÞ ¼

Z 10

E m� YtjY0 ¼ yð Þdt

¼

Z 10

m� yð Þe�tdt

¼ m� y,

where we have used

E YtjY0 ¼ y0ð Þ ¼ y0e��t þm 1� e��t

� �,

which is valid for a general a, and not just a¼ 1.The solution for P2(t, x, y), is thus

P2ðt, x, yÞ ¼1

2y�m� ~cðt, xÞð Þx2

q2P0

qx2

¼1

2yþ cðt, xÞð Þx2

q2P0

qx2,

where cðt, xÞ ¼ �ð ~cðt, xÞ þmÞ, is a function

independent of y.

A.4. Orderffiffiffi�p

The equation required for orderffiffiffi�p

is

L0P3 þ L1P2 þ L2P1 ¼ 0,

with the centring condition

L1P2 þ L2P1h i ¼ 0:

Using this and the form for P2 in equation (A7), we have

L2ffiffiffiffimp� �

P1 ¼ L2P1h i

¼ � L1P2h i

¼1

2L1�ð yÞx

2 q2P0

qx2

�,

using the fact that L1 takes derivatives with respect to y,

and that ~cðt, xÞ is independent of y. It can be shown that,

for any u(t, x),

L1�ð yÞuðt, xÞ

¼ �

ffiffiffiffi2

m

ry�0ð yÞ

xquqx¼ ��

ffiffiffiffiffiffiffi2mp

xquqx:

Applying this to u(t, x)¼x2 @2P0/@x2 gives

L2P1h i ¼ �1ffiffiffi2p �

ffiffiffiffimp

2x2q2P0

qx2þ x3

q3P0

qx3

� �:

Multiply the above byffiffiffi�p

and define ~P1 ¼ffiffiffi�p

P1. Further

convert the parameters and � back into a and b.This results in

L2ffiffiffiffimp� �

~P1 ¼ ��m

2�2x2

q2P0

qx2þ x3

q3P0

qx3

� �:

A solution for ~P1 is given by

~P1ðt,xÞ ¼��m

2�T� tð Þ 2x2

q2P0

qx2þ x3

q3P0

qx3

� �:

To obtain the solution given in equation (18), substitute

the Black–Scholes partial derivatives in the above, and

write the asymptotic solution containing only the first two

terms.


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on the performance of asymptotic locally risk minimising hedges in the heston stochastic volatility...

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