model uncertainty and the robustness of hedging … · model uncertainty and the robustness of...

MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS

DR TILL C. SCHROTER AND DR MICHAEL MONOYIOS

University of OxfordMathematical Institute

24-29 St GilesOxford

OX1 3LB,United Kingdom

Tel.: +44 (0) 1865 280617

DR MARIO ROMETSCH AND PROF KARSTEN URBAN

University of UlmInstitute of Numerical Mathematics

Helmholtzstr. 2089069 UlmGermany

Tel.: +49 (0) 731 5023535

Abstract. In this paper, we study the robustness of popular hedging models (the Black-Scholes,the Heston, and the SABR model) to model uncertainty and the ensuing risk that arises fromhedging a payoff on the basis of an incorrect hedging model. The study is performed by hedgingan Asian option in simulated real-world financial markets. These real-world markets are modelledby a stochastic volatility model with stochastic correlation, by a stochastic volatility model withjumps in the asset and the volatility, and by a jump-dominated Levy model. The results of thestudy show that the use of more sophisticated hedging models is not in general warranted by abetter hedging performance or by a higher robustness of these models to structural changes in thefinancial market.

E-mail addresses: [email protected], [email protected], [email protected],

[email protected].

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1. Introduction

This paper is a contribution to the understanding of the robustness of classical option hedgingschemes to model error. It is generally accepted that no parametric model can ever hope to captureperfectly the risk-neutral dynamics of underlying factors which generate the observed dynamicimplied volatility surface in traded options markets. This means that an agent who dynamicallyhedges an option on the basis of some parametric model will almost inevitably face model risk: therisk of losses due to the mis-specification of the hedging model.

As pointed out by Davis [Dav04], one is not unduly exposed to model risk in pricing a non-exchange-traded option, since an agent seeking to price such an option will choose parameters in amodel (the so-called hedging model) which, at the time of trade, result in the model correctly pricingrelated exchange-traded options. Sometimes, for pricing purposes, no model at all is needed, asany sensible ‘formula’ mapping parameters to prices can be used (see Figlewski [Fig02]). But forclassical hedging, one is concerned with the full distribution of the underlying security until thederivative’s expiry date and does indeed need a model in some shape or form. Hence, model riskbecomes a much more serious issue. In this paper we are concerned with the model risk that arisesfrom using a mis-specified hedging model, for an agent that is dynamically hedging a position inan exotic, non-exchange-traded option.

One way to avoid model risk is to use completely model-free, static hedges, which occasionallycan be found for contracts such as European vanilla options, barrier options or Asian options (seeCarr and Wu [CW09], Cox and Ob loj [CO11] and Albrecher et al. [ADGS05], for example). Thesemethods are promising but often give only wide bounds for super-hedges, or else (at present) arelimited to a narrow class of claims. For these reasons, dynamic hedges based on a specific modelformula are still very popular amongst financial practitioners and model risk is a serious issue fora wide class of claims. Naturally, market participants are interested in minimising their exposureto model risk and will therefore aim to use hedging models that are relatively robust with respectto different possible structures of the underlying market.

To give a good hedging performance, a hedging model does not necessarily have to constitutea realistic model of the actual dynamics of the underlying asset as long as the model parameterssensibly capture parts of the market dynamics. For example, the Black-Scholes model, if used witha judiciously chosen volatility parameter, is known to be able to provide an effective dynamic hedgein a diffusion environment, even one involving stochastic volatility and other factors, as shown in ElKaroui et al. [EKJPS98]. In general, it is not implausible that a relatively simple model, involvingfew free parameters, can perform well. The philosophy behind this idea is that since all modelsare wrong, and since all models need to be calibrated (and usually re-calibrated) to observed pricedata, a model in which only a few parameters need to be adjusted may work better than one inwhich a large number of parameters must be fixed in an ill-posed way.

In this paper, we study the hedging performance of three popular hedging models by assessing thehedging error distributions that result from dynamically hedging a path-dependent claim using theBlack-Scholes [BS73], Heston [Hes93] and SABR [HKLW02] models. The hedges are performed inthree simulated ‘true’ market environments (which we refer to as market models) that are based on:(i) a three-dimensional Ito diffusion featuring stochastic volatility and correlation, (ii) a stochasticvolatility model with jumps in both the stock price and the volatility and (iii) a jump-dominatedLevy process. In particular, we analyse the robustness of the hedging performance of the threehedging models to changes in the underlying market regime, and the effectiveness of the hedgingmodels to local and global calibration approaches.

We evaluate the hedging models by hedging a path-dependent claim, an arithmetic Asian option,which we call the ‘target option’. The relevance of path-dependent payoffs in testing hedging modelshas been discussed by Hull and Suo [HS02] and An and Suo [AS09]. As few liquid markets forexotic options exist, model risk is a particular concern to the hedger of exotic, path-dependentinstruments, who cannot simply hedge away the risk by trading a similar exotic in the market.Further, a well-performing hedging model for a path-dependent option gives some clues about a

MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 3

model’s capability of approximating the full distribution of the underlying asset process. Thismotivates our choice of a path-dependent, arithmetic Asian option as the ‘target option’ which weaim to hedge.

Simulation studies provide a fruitful approach to assess the performance of hedging models,since they give us control over the underlying simulated market data. Thereby, we can obtain acomprehensive picture of the robustness of hedging models in very distinct market environments.Of course, a theoretical analysis of model risk would give a deeper understanding of the robustnessof hedging models, but this subject is hard to tackle, in particular when the true or the mis-specifiedasset price process are not modelled by diffusions. When the true and the mis-specified asset priceprocesses are diffusions, bounds on the hedging error arising from dynamic hedging schemes arecomputed by Corielli [Cor06] and [EKJPS98]. Boyle and Emanuel [BE80] study errors arising fromdiscrete hedging in a Black-Scholes market, Anagnou-Basioudis and Hodges [ABH03] decomposethe hedging error to investigate the effects of discretisation and volatility forecasting errors on thehedging performance, and Ahn et al. [AMS99] minimise hedging error in a model with misspecifiedvolatility via a worst-case scenario approach. Gibson et al. [GLPT99] define model risk and give amore qualitatively focused account of its sources.

Given the advantages of simulation studies, it is not surprising that these studies have beenconducted frequently to study model risk in equity markets. Among the first, Figlewski [Fig89] in-vestigates the effects that a misspecified volatility and market frictions have on hedges of Europeanvanilla options in a market generated by the Black-Scholes model. Jiang and Oomen [JO01] areconcerned with misspecified volatility parameters and study the impact of a wrongly selected hedg-ing model on hedging performance. Specifically, they ask how well the Black-Scholes and Hestonmodels hedge standard European options in stochastic volatility environments (with and withoutjumps). Hull and Suo [HS02] test different local volatility models by pricing and hedging exoticoptions in a stochastic volatility market. Coleman et al. [CKLV01] investigate how well local andimplied volatility models hedge European vanilla options when the market data is generated by anon mean-reverting Ornstein-Uhlenbeck process. The question of whether volatility derivatives aresuited to hedge European vanilla options in a stochastic volatility environment is examined in Psy-choyios and Skiadopoulos [PS06]. Poulson et al. [PSHE09] simulate a stochastic volatility marketto test the effectiveness of locally risk minimising strategies. The performance of semi-static hedgesfor European vanilla options in different jump and stochastic volatility environments is studied in[CW09]. Finally, Branger et al. [BKSS12] use a simulation approach to test different hedgingstrategies for European vanilla options in a stochastic volatility market that contains jumps in theunderlying. These studies indicate that simple models such as the Black-Scholes model can some-times be effective, and our study goes much further in confirming this broad conclusion. It wouldbe of great interest to theoretically determine conditions under which a low-dimensional model canbe an effective hedging tool in a higher-dimensional ‘true’ market environment, and this would bean important future research challenge.

The simulation study of this paper is based on a two-step procedure, outlined in [HS02]. In thefirst step, we generate 50, 000 realisations of the market data in each of the market models over athree month time-span. The market data comprises the stock price trajectories, the trajectories ofall other stochastic factors that drive the market (e.g. stochastic volatility), and the option pricesof 15 European vanilla options. These option prices are calculated along every trajectory and forevery day.

In a second step, we evaluate the hedging performance of the hedging models on the basis of thegenerated market data. This is accomplished by setting up a portfolio consisting of a stock and atmost two European vanilla options to dynamically hedge the Asian claim. The hedging portfoliois rebalanced daily, using sensitivity parameters derived from the different hedging models. Thesehedging models are, at every rebalancing time, re-fitted (or re-calibrated) to the available marketdata. At the end of the three month period, the difference between the value of the hedging portfolioand the price of the Asian claim in the market model gives us the terminal hedging error along a

4 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS

single trajectory. The hedging error distribution is obtained from the hedging error along each ofthe 50, 000 trajectories.

Our simulation study is distinct from the related literature in several ways. (i) We model themarket environments by a three-factor Markovian model, by a stochastic volatility model withjumps in the underlying and the volatility and by a Levy process model. These models have notbeen used in previous studies of the same type and they span a wide range of potential marketenvironments that fit various features of the empirical data well (see the discussion in Section 3).(ii) We evaluate the hedging error in terms of several statistics of its distribution as opposedto investigating moments of first or second order only. (iii) We investigate the impact of local(using only a subset of the available vanilla options) and global (using all available vanilla options)calibration methods on the hedging performance. (iv) We perform the study on the basis of 50, 000market data simulations and therefore obtain a higher accuracy than most previous simulations.(v) We test the performance of three-asset hedges throughout. Moreover, the hedging study isperformed on the basis of how financial practitioners hedge, that is on the basis of frequentlyrebalanced portfolios and frequently recalibrated hedging models. The importance of this procedurefor reasonable estimates of the hedging error has been pointed out in [HS02]. In particular, thefrequent recalibration is computationally demanding. All computations have been performed on theUZWR cluster [UZW] consisting mostly of processors with a clock speed of 2.6 GHz. To completethe simulation we required a run time of 2 months using 18 of the available processors.

Our ultimate contribution is two-fold: from a model risk perspective, we show that the seem-ingly simple and easy to implement Black-Scholes model achieves a hedging performance that iscomparable to or better than any other hedging model in terms of minimal variance and near zeroshortfall of the hedging error. The more sophisticated Heston model performs relatively badly,though it sometimes achieves a better expectation of the hedging error.1 We also find that, ingeneral, the best performing hedging models do not yield better results when they are used on thebasis of globally calibrated parameters. This latter finding is, of course, a reflection of the factthat any model is bound to be mis-specified, and attempting to make it match all observed tradedoption prices will render it ineffective as a hedging tool. The best one can do is to make sure amodel locally fits traded options of similar maturity and moneyness to the target option, and thismaximises its hedging effectiveness.

These findings illustrate that a parsimonious model, where the calibration issues are condensedinto a small number of parameters, and where re-calibration is local, can be successful in hedging.More complex models do not necessarily lead to a greater robustness to model error. Also, giventhe sometimes more favourable expectation of the hedging errors in the Heston model, it seems thatthe Heston model should be used if many hedges of the same type are to be performed. In this casewe can hope to realise the favourable expected error. However, when the goal is to conduct onehedge as accurately as possible, the Black-Scholes model is the preferred model. Finally, we studythe robustness of the previous findings. In particular, we find that the Heston model performs morestrongly than the Black-Scholes model in environments that are characterised by a high kurtosis ofthe market returns.

From a numerical perspective, our contribution is to show how the efficient simulation of an equitymarket with associated traded derivatives and the hedging of an exotic claim can be accomplished.In particular, we face the very challenging task of solving the pricing PDE of the Asian optionmore than 3×106 times to simulate just one single hedging error distribution. To achieve this withreasonable computational effort, we make use of a reduced model by employing a proper orthogonaldecomposition in the framework of finite element methods to solve the pricing PDE.

The work is organised as follows. In Section 2, we fix the notation and outline the probabilis-tic setting. Section 3 contains a discussion of the market models that describe the underlying

1This finding illustrates the importance of assessing the full distribution of the hedging error. If this study, likemost previous studies, were based on the expected error or the variance only, the Heston model or the Black-Scholesmodel respectively would have delivered the best hedging performance. Both results would be slightly misleading.


‘true’market environments. The different hedging models are reviewed in Section 4. The method-ology is explained in Section 5. Section 6 contains a discussion of the results.

2. Preliminaries

We model the market on a filtered probability space (Ω,F ,F, P ), where P stands for the physicalmeasure. The market exists on a finite interval [0, T ] for some T ∈ R+ and, for notational simplicity,we write T to refer to the time interval [0, T ]. The flow of information is modelled by a filtrationF = (Ft)t∈T , such that FT ⊆ F . F is assumed to satisfy the usual conditions of right-continuityand completeness. By WP = (WP,1, . . . ,WP,n), we denote an n-dimensional (n ∈ N) standard P -Brownian motion on the filtered probability space and N = (Nt)t∈T stands for a one-dimensionalPoisson process on that stochastic basis. The risk-neutral pricing measure is denoted by Q. Thismeasure is not necessarily unique and will be fixed in Section 3 for different market models. ABrownian motion under Q is written as WQ = (WQ,1, . . . ,WQ,n). For a local martingale M ,M0 = 0, with quadratic variation [M ], the stochastic exponential is introduced via

E(M)st = eMt−Ms− 12

([M ]t−[M ]s

), 0 ≤ s ≤ t ≤ T ,

and we write E(M)t = E(M)0t. For any given process (Ut)t∈T we frequently write U = (Ut)t∈T todenote the entire process. The stock that underlies all derivatives in the market is modelled by theF-adapted process S. Finally, the interest rate is constant and set to r = 0.04.

Throughout this study, we encounter two sorts of European derivatives. First, a fixed strikeAsian option that depends on the average price of the underlying stock during its lifespan andwhose price at t ∈ T is given by

(1) CAt = e−r(TA−t)EQ

[(1

TAYTA −KA

)+

|Ft

],

where TA ∈ T denotes the maturity, KA ∈ R+ the strike price and

Yt =

∫ t

0Sudu.

Second, European vanilla calls that serve as hedging instruments and are needed for calibrationpurposes. Their prices at t ∈ T are given by

Ct = e−r(T−t)EQ[(ST −K)+ |Ft

],

for strike levels K ∈ R+ and exercise times T ∈ T .For certain options we may want to specify whether the price has been calculated on the basis

of a market model or based on a hedging model. In that case, we use the superscript M to denotethe former situation and the superscript H to identify the latter scenario, for example, we writeCM for the price of a European vanilla call in the market model. The only exception to this is theprice of an Asian option, where we write CAM and CAH .

3. The Market Models

The market models have two distinct functions in this study: (i) they are used to simulate thetrajectories of the factors that drive the market and (ii) they provide the prices of the Europeanvanilla and Asian options which constitute the derivative data in the market. For this reason, themarket models are specified under the physical and the risk-neutral measure and their parametersare chosen to reflect pertinent time series and option price data.

The equity market and equity market models have been studied widely in the literature. There-fore, several approaches exist to analyse the structure of this market. One popular approach toobtain information on the factors that drive the equity market is to study the properties of Eu-ropean vanilla option prices via the related implied volatility surface. On the basis of a principalcomponent analysis of the implied volatility surface, such studies have been performed, for example,


by Skiadopoulos et al. [SHC99], Cont and da Fonseca [CdF02] and Fengler et al. [FHS02]. Depend-ing on the focus and the methodology, these studies show that 70–90% of the surface movementscan be explained by only two to three independent stochastic driving factors. These findings aresomewhat reflected in our choice of a market model that is driven by three independent diffusionfactors (Subsection 3.1).

Alternatively, the fit of different models has been examined in terms of their pricing and hedgingperformance in the equity market and in terms of their potential to replicate the statistical proper-ties of financial time series data. Several studies document the importance of jumps and stochasticvolatility for a good pricing and hedging performance. By evaluating out-of-sample pricing andhedging errors of S&P 500 options, Bakshi et al. [BCC97] demonstrate that, within the frameworkof their study, stochastic volatility models with jumps in the underlying perform best when it comesto pricing options and that stochastic volatility models are of first-order importance for minimisingthe hedging error. Comparable results are also given in Pan [Pan02] and [Bat00], who additionallypoint out the importance of jumps in the asset price for realistic diffusion parameter estimates.

A frequent extension of jump models is to consider jumps not only in the asset price processbut also as part of the stochastic volatility. Models with jumps in stochastic volatility seem toperform best in mimicking market dynamics. This has been attributed to the greater persistenceof jumps in the volatility process which, contrary to transient jumps in the underlying, activelychange the distribution of the underlying over periods of time, see Eraker et al. [EJP03], Chernovet al. [CGGT03] and Eraker [Era04]. However, [Era04] finds that the improved time series fit ofmodels with jumps in the volatility does not carry on to a better out-of-sample pricing performanceof these models for European vanilla options. The importance of jumps in the underlying and thevolatility is reflected in our choice of the SVJJ model (Subsection 3.2) as a market environment.

Price processes are never fully continuous and any model with (mostly) continuous trajectoriesmust be regarded as an approximation to the true price process. To account for that behaviour, wehave included the jump-based CGMYe model of Carr et al. [CGMY02] as third choice of a marketmodel. Below we discuss the market models in greater detail.

3.1. The Three-Factor Model. The three-factor model (3F model, 3FM) is a stochastic volatilitymodel whose correlation is driven by an additional stochastic variable. It therefore consists of threeindependent factors driving the market. Das and Sundaram [DS99] show that a negative correlationbetween an asset and its volatility leads to the volatility smile being tilted. From that point ofview, a stochastic correlation effectively models a stochastic skew in the implied volatility surface.

In the three-factor model, the asset S follows an Ito diffusion, the variance v is described by aCox-Ingersoll-Ross process and the correlation ρ is modelled by the translation of a Jacobi2 process.For M ∈ P,Q, the dynamics of the three-factor model are

dSt = µMStdt+√vtStdW

M,1t ,

dvt = αM (βM − vt)dt+ σv√vtdB

Mt ,

dρt = κM (λM − ρt)dt+ σρ

√1− ρ2

tdWM,3t ,

(2)

where dBMt = ρtdW

M,1t +

√1− ρ2

tdWM,2t . The correlation ρ is a mean-reverting process that takes

values in [−1, 1]. This process is related to a Jacobi processes, J , by the linear transformation

ρt = 2Jt − 1,

as pointed out in Veraart and Veraart [VV10].In (2), we specify the factor dynamics under the measures P and Q. This specification entails

that the structure of the SDEs is the same under both measures. The parameters that describethe SDEs under the physical and risk-neutral measures are given in Table 1.

2The Jacobi process, J , is a mean-reverting process that solves the SDE dJt = a(b−Jt)dt+ c√Jt(1 − Jt)dWt, for

a, b, c > 0 and Brownian motion W .


P - Physical Measure

µP αP βP σv κP λP σρ0.10190 0.93086 0.01537 0.06150 1.50000 -0.1830 1.00000

Q - Risk Neutral Measure

µQ αQ βQ σv κQ λQ σρ0.04000 0.69011 0.00956 0.06150 1.50000 -0.1830 1.00000

Table 1. The annualised values of the modelling parameters under P and Q in thethree-factor model.

The asset and volatility parameters are taken from Chernov and Ghysels [CG00]. The correlationparameters are chosen such that the correlation dynamics do not not change under the measuresP and Q. This choice is somewhat arbitrary but feasible as we are only interested in a reasonablesetting for the market dynamics.

Under the current parameter choices, an equivalent martingale measure Q exists such that stock,variance and correlation have the Q-dynamics of (2). However, it has been observed that parameterestimates from market prices do not always satisfy the theoretical conditions that guarantee theexistence of an equivalent martingale measure and hence a set of arbitrage free option prices (e.g.in [BCC97]). In Appendix A, we discuss the existence of an equivalent martingale measure in thethree-factor model in greater detail.

3.2. The Stochastic Volatility Model with Jumps in Stock and Volatility. Market modelswith jumps in volatility have been suggested in [Bat00], [Pan02] and others after it became clearthat jumps in the asset price alone are not sufficient to model a market correctly. The rationalefor introducing jumps in the volatility process, in the words of [EJP03], is that

“Jumps in returns can generate large movements such as the crash of 1987, butthe impact of a jump is transient: a jump in returns today has no impact on thefuture distribution of returns. On the other hand, diffusive volatility is highly per-sistent, but its dynamics are driven by a Brownian motion. For this reason, diffusivestochastic volatility can only increase gradually via a sequence of small normally dis-tributed increments. Jumps in volatility fill the gap between jumps in returns anddiffusive volatility by providing a rapidly moving but persistent factor that drivesthe conditional volatility of returns.”

In the stochastic volatility model with jumps in stock and volatility (SVJJ model), the marketis driven by the log-stock price L, L = lnS, and the variance v. For M ∈ P,Q, the dynamics ofthe factors in the SVJJ model are

dLt =

(µM − 1

2vt−

)dt+

√vt−dW

M,1t + ξr,MdNt,

dvt = αM (β − vt−)dt+ σv√vt−dB

Mt + ξvdNt,

(3)

where dBMt = ρdWM,1

t +√

1− ρ2dWM,2t for some constant ρ ∈ [−1, 1]. In (3), N is a Poisson

process with constant jump arrival intensity λN > 0, identical under any measure. The parametersthat describe the dynamics of L and v under the physical and the risk-neutral measures are givenin Tables 2 and 3. Their orders of magnitude are taken from [Era04].

The jumps in stock and volatility are modelled by an exponentially distributed random variable(with expectation µv) and by a normally distributed random variable, that is

ξv ∼ exp (µv),

ξr,M |ξv ∼ N(µMr + ρJξv, σ2

r ).


P - Diffusion Parameters

µP αP β σv ρ0.06552 5.79600 0.03410 0.41076 -0.58200

P - Jump Parameters

µPr σr ρJ µv λN-0.06062 0.03630 -0.27500 0.04128 0.50400

Table 2. The annualised values of the modelling parameters under P in the SVJJmodel.

This jump specification accounts for two empirical observations. First, volatility does not becomenegative. This is accounted for by the exponential distribution. Second, jumps in volatility andthe underlying often occur contemporaneously, as a crash in the underlying usually leads to asubsequent increase in volatility. This observation is reflected in the choice of a single jump processN that drives both factors. In the SVJJ model, a semi-closed form formula exists for the price ofa European vanilla option. This attractive feature has been developed in Duffie et al. [DPS00].

Q - Diffusion Parameters

µQ αQ β σv ρ0.08140 2.77200 0.03410 0.41076 -0.58200

Q - Jump Parameters

µQr σr ρJ µv λN-0.07508 0.03630 -0.27500 0.04128 0.50400

Table 3. The annualised values of the modelling parameters under Q in the SVJJmodel. Note that the risk-neutral drift parameter is given by µQ = r−λN (θ(1, 0)−1)(for details and notation see (4.5) in [DPS00]).

3.3. The CGMYe Model. To account for the diffusion-like behaviour of small, daily price vari-ations, we previously introduced a diffusion and a jump-diffusion model. Trading, however, takesplace in discrete time and any continuous process only aims to approximate market behaviour.Although jump-diffusion processes are not necessarily Gaussian distributed (see, for example,Dragulescu and Yakovenko [DY02]) and may contain some jumps, other classes of processes of-fer greater flexibility in modelling the probability distributions and the path properties of stockprices. Popular, jump-dominated processes introduced to model the behaviour of stocks are theCGMY and CGMYe processes of [CGMY02]. The CGMY process is an extension of the Variance-Gamma (VG) process of Madan et al. [MCC98] and allows for price trajectories of infinite variation.Like the Variance-Gamma process, the CGMY process belongs to the class of Levy processes. TheLevy density of the distribution that underlies the CGMY process is given by

kCGMY (x) =

CM

exp (−GM |x|)|x|1+YM

for x ≤ 0 ,

CMexp (−MM |x|)|x|1+YM

for x ≥ 0 ,

for parameters CM > 0, GM ≥ 0, MM ≥ 0 and YM < 2. The first three of these parametersinfluence the skewness and kurtosis of the distribution of the CGMY process and the parameterYM controls the jump behaviour of that process. For a CGMY process X and corresponding


density kCGMY (x), the characteristic function is explicitly known and given by

E[eiuXt

]= exp

(t

∫ ∞−∞

eiux − 1

kCGMY (x)dx

)= exp

(tCMΓ(−YM )

[(MM − iu

)YM − (MM )YM

+ (GM + iu)YM − (GM )Y

M

])=: φCGMY

(u, t;CM , GM ,MM , YM

),

where Γ denotes the Γ-function.The CGMYe process extends the CGMY process by a Brownian motion. Therefore, in the

CGMYe model, the stock price under the measure M ∈ P,Q is modelled by the exponential ofthe CGMY process X and an additional Brownian motion WM,1 independent of X, i.e.

(4) St = S0 exp

((µM + ωM − 1

2ηM)t+ ηMWM,1

t +Xt(CM , GM ,MM , YM )

),

with ωM defined by

e−ωM t = φCGMY

(−i, t;CM , GM ,MM , YM

).

The parameters that describe the dynamics of the asset price under the physical and risk-neutralmeasures are given in Table 4. These parameter choices have been made on the basis of [CGMY02].

P - CGMYe

µP CP GP MP Y P ηP

0.1019 1.5000 22.1800 27.1200 0.7836 0.0201

Q - CGMYe

µQ CQ GQ MQ Y Q ηQ

0.0400 9.6100 09.9700 16.5100 0.1430 0.0458

Table 4. The values of the modelling parameters under P and Q in the CGMYe model.

4. The Hedging Models

In this study, the hedging models are used to set up a replicating portfolio for the Asian option.As the structure of the real-world market is unknown, the portfolio is set up to replicate thedynamics of CAH , the price of the Asian option in the hedging model. Specifically, for m, d ∈ N, let

us assume that the hedging model H is based on an m-dimensional Ito diffusion X and that themarket contains d traded assets. Then, the price of any asset in the hedging model at time t ∈ Tcan be written as AH

lt = AHl (t, Xt), l ∈ 1, . . . , d, where AH

l (t, x), x ∈ Rm, denotes the pricingfunction of the l-th asset in the hedging model. By Ito’s formula, the asset dynamics in the hedgingmodel are

(5) dAHlt = · · · dt+∇AH

ltdXt,

while those of the Asian option can be described by

(6) dCAHt = · · · dt+∇CAHtdXt.

Also, the market contains a bank account, B, that satisfies the differential equation

dBt = rBtdt.


In (5) and (6), we omit the locally risk-free components of the asset dynamics as they play no rolein setting up a replicating portfolio. By convention, ∇ denotes the gradient of the pricing function.Hence, for the l-th asset in the market we would have

∇AHlt =

(∂x1A

Hl (t, Xt), . . . , ∂xmA

Hl (t, Xt)

).

The hedging portfolio Π consists of the assets A and the bank account B,

(7) Πt = ΨBt Bt +

d∑l=1

ΨltAlt,

where ΨB denotes the amount of money in the bank account. The number of shares held in asset lis recorded in the l-th entry of the vector Ψ ∈ R1×d. As we are only concerned with self-financingstrategies, the portfolio Π has the dynamics

dΠt = rΨBt dt+

d∑l=1

ΨltdAlt,

for ΨB = Π−∑d

l=1 ΨlAl. By (5), the portfolio evolution can be written as

(8) dΠt = · · · dt+ Ψt

∇AH1t

...∇AH

dt

dXt.

To replicate the dynamics of CAH with the portfolio Π, a comparison of (8) with (6) shows that thevector of portfolio weights, Ψ, must solve the linear system

(9) ∇CAHt = Ψt

∇AH1t

...∇AH

dt

.

This system of equations is solvable, whenever the matrix in (9) is invertible. The general solvabilityof (9) is a non-trivial question. Romano and Touzi [RT97] discuss this question for a stochasticvolatility model. Some results for more general m-factor Ito diffusions are obtained in [Dav04].

The hedging instruments in our simulation are the stock S and two European vanilla optionsC1 and C2. This corresponds to setting A1 = S, A2 = C1 and A3 = C2. Then, the portfolio (7)becomes

Π = Ψ1S + Ψ2C1 + Ψ3C2 + ΨBB.

The market models are based on a maximum of three stochastic driving factors. Therefore, in thehedging models, we always select the underlying asset as the first factor, the volatility/variance as

the second factor and the correlation as third factor, hence we have X1 = S, X2 = v and X3 = ρ.With that choice, (9) becomes explicitly3 ∂SC

AHt

∂vCAHt

∂ρCAHt

=

1 ∂SCH1t ∂SC

H2t

0 ∂vCH1t ∂vC

H2t

0 ∂ρCH1t ∂ρC

H2t

Ψ1t

Ψ2t

Ψ3t

.

Clearly, if this equation has a solution it is

(10)

Ψ1t

Ψ2t

Ψ3t

=

∂SC

AHt −Ψ2

t∂SCH1t −Ψ3

t∂SCH2t

∂vCAHt−∂vCH2tΨ

3t

∂vCH1t

∂ρCAHt∂vCH1t−∂vCAHt∂ρC

H1t

∂vCH1t∂ρC

H2t−∂ρCH

1t∂vCH2t

.

3Sometimes, the first derivatives of a pricing formula with respect to the underlying asset and the volatility arecalled ‘delta’ and ‘vega’.


If we hedge with less than three instruments, the corresponding portfolio is obtained by settingthe redundant portfolio weights to zero. For instance, if we hedge with the stock only, we setΨ2 = Ψ3 = 0 and therefore have Ψ1 = ∂SC

AH .

Although the Black-Scholes model is driven by only one stochastic factor and the Heston andSABR models by two stochastic factors , the portfolio (10) has been set up to always accommodatea possible second and third factor. The reason for this is the widespread practice of ‘out-of-model’hedging. If the real-world market were to coincide with the market outlined by the hedging model,it would, in theory, suffice to hedge the risk resulting from the underlying stochastic factors toeliminate all risk in the market. It is, however, difficult to exactly isolate the risk factors that drivea stock price or a derivative. As a result, no hedging model accurately reflects the dynamics of thereal-world market and the hedging models must be frequently calibrated to match market prices.The frequent calibration introduces a dynamic behaviour to parameters which are conceived staticin the original hedging model. This gives rise to an extra source of risk. To hedge against that risk,the dynamic parameters are treated as additional stochastic processes and the portfolio weights arecalculated as outlined in (10). This hedge would not be necessary if the hedging model were to be agood proxy for the real-world market. Therefore, it is called an ‘out-of-model’ hedge. In this study,we include a two-factor out-of-model hedge with respect to the (theoretically) constant volatilityparameter in the Black-Scholes model and two three-factor out-of-model hedges with respect to the(theoretically) constant correlation parameter in the Heston and SABR models.

Even in the framework of out-of-model hedging, a hedging model does only take into account thesources of risk it recognises. For example, even in the framework of the Black-Scholes out-of-modelhedge, the hedger only takes into account price changes of the target option that can be explainedby changes in the underlying stock and changes in the implied volatility surface. If the price ofthe target option is also affected by other sources of risk, say a change in the curvature of theimplied volatility surface, the hedger will fail to hedge against these risk sources (see, for example,Chapter 11 of Rebonato et al. [RMW09]).

In the remainder of this section, we introduce the hedging models. These models are used forcalibration purposes and to set up the hedging portfolios. It is therefore sufficient to specify themunder the risk-neutral measure Q.

4.1. The Black-Scholes Model. In the model of Black and Scholes [BS73], the risk-neutraldynamics of the stock price are given by

(11) dSt = rStdt+ σStdWQ,1t , S0 ∈ R+,

for two constants r, σ > 0.The SDE (11) contains only one source of randomness and therefore a hedge with respect to

the underlying asset S is (in theory) adequate to eliminate all risk in the Black-Scholes market.Empirically, however, the implied volatility surface is not constant, but a function of the option’sstrike level and maturity. Moreover, it is subject to daily stochastic changes (see, for example,[CdF02]). Therefore an out-of-model hedge with respect to the volatility parameter, a so-called‘vega-hedge’, is frequently performed. We follow this practice by including a vega-hedge in ourstudy.

To set up a hedging portfolio, as outlined in (10), we need the partial derivatives of the pricingformulae for Asian and European vanilla options with respect to the underlying factors. For aEuropean vanilla option, [BS73] establish a closed-form pricing formula. Hence, the respectivepartial derivatives can be easily calculated. No closed-form formula exists for the price of anAsian option in the Black-Scholes model. Therefore, prices and partial derivatives are obtained assolutions to a suitable PDE. We briefly present this PDE in Subsection 5.3.


4.2. The Heston Model. The model of Heston [Hes93] is a two-factor model of the stock, S, andits variance, v. In this model, the risk-neutral dynamics of the underlying factors are given by

dSt = rStdt+√vtStdW

Q,1t , S0 ∈ R+,(12)

dvt = ν(φ− vt)dt+ ϕ√vtdB

Qt , v0 ∈ R+,(13)

where dBQt = ρdWQ,1

t +√

1− ρ2dWQ,2t and ρ ∈ [−1, 1] as well as r, ν, φ, ϕ > 0. To make sure, that

a positive solution to the variance SDE (13) exists, we also impose the condition

(14) 2νφ ≥ ϕ2.

For European vanilla options in the Heston model, [Hes93] establishes the existence of a semiclosed-form pricing formula. This allows for simple calculation of the European vanilla option pricesand of the related partial derivatives. No closed-form pricing formula exists for an Asian optionin the Heston model. In Subsection 5.3, we therefore present a PDE that allows us to obtain theprices of the Asian options in the Heston model numerically.

4.3. The SABR Model. The SABR model of Hagan et al. [HKLW02] is a two-factor model of the

forward price, F , and its volatility, v. At time t ∈ T , the forward price is given by Ft = Ster(T−t).

Under the risk-neutral measure, the factors in the SABR model solve the SDEs

dFt = vtFβt dW

Q,1t , F0 ∈ R+,(15)

dvt = ϕvtdBQt , v0 ∈ R+,(16)

where dBQt = ρdWQ,1

t +√

1− ρ2dWQ,2t and ρ ∈ [−1, 1] as well as β, ϕ > 0. We fix β = 1.

Stochastic volatility in the SABR model does not have a mean-reversion property and can there-fore become very high. This unrealistic property highlights in particular that the SABR model isnot intended as a realistic model of the market dynamics but aims to provide a useful parametri-sation that relates observed prices to hedging parameters. The SABR model has been designed toprovide a good fit to market data and to allow for convenient pricing formulae on the basis of afunctional relationship (see [HKLW02] or Ob loj [Ob l08]) between the SABR parameters and theimplied volatility surface. In fact, in terms of the forward price Ft, the volatility level vt and thetime-to-maturity τ at t ∈ T , the implied volatility surface can be approximated through the SABRparameters via

σIV (K, τ ;Ft, vt) ≈(17)

ϕx(K;Ft)

ln

(√1−2ρz(K;Ft)+z2(K;Ft)+z(K;Ft)−ρ

1−ρ

) (1 +

ρϕvt

4+

2− 3ρ2

24ϕ2

τ

),

where x = ln (Ft/K) denotes the log-moneyness and z = xϕ/vt. On the basis of (17), the optionprices and corresponding partial derivatives in the SABR model can be calculated via the pricingformulae for Asian and European vanilla options in the Black-Scholes model.

5. Methodology

In this study, we examine the hedging models of Section 4 by testing their performance inhedging Asian options. The hedging performance is evaluated over the hedging interval [t0, TH ],with t0 ≤ TH ≤ TA, and measured in terms of the hedging error. At any given time t ∈ T , thehedging error, ε, is defined as the difference between the market price of the Asian option CAM, andthe value of the hedging portfolio Π, that is

(18) εt = Πt − CAMt.


The terminal hedging error, εTH , is made up of a sequence of one-step hedging errors, εi, thatmeasure the contribution of the period (ti−1, ti] (a daily interval in our study) to the terminalerror, i.e.

εi = εti − εti−1 =(Πti − CAMti

)−(Πti−1 − CAMti−1

).

In terms of the εi, the terminal hedging error that results from revising the portfolio N ∈ N timescan be written as

(19) εTH = ΠTH − CAMTH =

N∑i=1

εi + Πt0 − CAMt0 .

By (19), the hedging error εTH depends on the one-step errors εi, i = 1, . . . , N , and on Πt0− CAMt0 ,the difference between the market price of the Asian option and the initial value of the hedgingportfolio. Since this difference is Ft0-measurable, its value only impacts the expected hedging errorand leaves all higher central moments of the hedging error distribution unchanged. When thehedging portfolio is started at Πt0 = CAMt0 , it follows from (19) that the terminal hedging error canbe written as the sum of the one-step hedging errors only. Instead, when the initial value of theportfolio is Πt0 = CAHt0 , the terminal hedging error is composed of the sum of the one-step hedging

errors and the additional term CAHt0 −CAMt0 . This difference reflects the degree to which the Asian

option is mispriced by the given hedging model. Therefore, when Πt0 = CAMt0 , the terminal hedgingerror can be seen as a measure of the hedging model’s pure hedging performance. Otherwise, whenΠt0 = CAHt0 , the terminal hedging error must be understood as a measure of the hedging model’sjoint pricing and hedging performance. For example, situations in which the hedging portfolio isstarted at Πt0 = CAHt0 occur when a bank sells a non-traded Asian derivative. In this case, the banktypically prices the non-traded claim with some hedging model and subsequently uses the samehedging model and the proceeds from the sale to set up a portfolio to hedge the resulting liability.

A simulation study is well-suited to separate the effects of hedging and pricing on the hedgingerror. As the price of the Asian option is known in the market models (by means of Monte Carlosimulations), we are able to consider (i) hedges started on the basis of the true market price and(ii) hedges started on the basis of the hedging model price. In the Appendix B, we list the expectederrors of the hedges performed in the different hedging models. There, the expected errors of hedgesstarted on the basis of the hedging model price of the Asian claim are recorded in brackets.

In the literature, different metrics are used to evaluate the hedging error. Frequently, averagehedging errors of different hedging models are compared to rank these models. For instance,[BCC97] evaluate the average of M ∈ N one-step hedging errors, 1

M

∑Mi=1 εi, and the same measure

in absolute terms, 1M

∑Mi=1 |εi|. However, the average (or expected) hedging error as the sole

measure of a model’s hedging performance is somewhat misleading, as different hedging models cangive rise to hedging errors with similar expectations but strongly different variations around themean. Instead of the expectation, [BKSS12] suggest to evaluate the variance of the one-step hedgingerror. While the variance has some favourable properties in measuring the quality of hedges (see[BKSS12] for details), the underlying problem remains unchanged: when different hedging modelsgive rise to hedging errors with similar variances we lack the information to compare these modelsfurther. Therefore, the hedging error should be evaluated on the basis of several moments of thehedging error distribution. In this study, we evaluate the hedging error in terms of expectation,variance, skewness, (excess) kurtosis and expected shortfall. The latter is defined as the expectationof the hedging error conditional on a negative hedging performance, i.e. E [εTH |εTH < 0].

Throughout the study, all time periods and times are recorded in days, if not stated otherwise,and the expiry dates of all options are given with respect to t0 = 0, the initial day of the hedge.That means, for example, that an option expiring on day 21 is going to expire 21 days after t0. Wealso assume that a year consists of 252 working days.

Some hedging models are better suited than others to price and hedge options of longer orshorter maturities. For example, it has been observed that the Heston model tends to underprice


Option Characteristics

Asian Calls

TA/KA 21/50.0 126/50.0 189/50.0

European Vanilla Calls

T/K 21/47.0 126/47.0 189/47.0T/K 21/48.5 126/48.5 189/48.5T/K 21/50.0 126/50.0 189/50.0T/K 21/51.5 126/51.5 189/51.5T/K 21/53.0 126/53.0 189/53.0

Table 5. Characteristics of the Asian and the European vanilla options.

short term options, but achieves good results in valuing longer term options (see Gatheral [Gat06]).Therefore, to obtain a good understanding of the hedging models, it is crucial to evaluate thesemodels by hedging Asian options of different maturities. For this reason, we hedge Asian calls whofit in one of the three categories: short term (days to expiration < 60), medium term (days toexpiration ∈ [60, 180)) and long term (days to expiration ≥ 180).

In this study, we hedge Asian calls that mature after 21, 126 and 189 days and have a strikelevel of KA = 50. The European vanilla calls that constitute the option market data also matureafter 21, 126 and 189 days and comprise five different strike levels. Table 5 contains the details.

We perform three kinds of hedges. First, we hedge with only one hedging instrument, the stock,against changes in the value of the underlying. Second, we hedge with two hedging instruments,the stock and a European vanilla option, against changes in the values of the underlying and thestochastic volatility. Third, we hedge with three hedging instruments, the stock and two Europeanvanilla options, against changes in the values of the underlying, the stochastic volatility and thecorrelation. The two-instrument hedge is performed in all hedging models and, when performed inthe Black-Scholes model, is an out-of-model hedge. The three-instrument hedge is only performedin the Heston and SABR models and always an out-of-model hedge. Whenever we hedge withone European vanilla option, the 126/504-European vanilla call is used as the hedging instrument.Whenever we hedge with two European vanilla options, the 126/50-European vanilla call is usedas the first and the 189/50-European vanilla call is used as the second hedging instrument. Thehedging period consists of three months, i.e. TH = 63 days, for the medium and long term Asianoption. The short term Asian call is hedged until its expiry, hence TH = 21 days. We do nothedge the medium term and the long term Asian option until their expiry dates, as this allowsus to perform the simulations considerably faster. Also, since we are comparing hedging modelsin this study, it is mainly the difference between the hedging performances of the hedging modelsthat interests us. By the previous choice of the hedging periods, we evaluate this difference after21 days for hedges of the short term Asian option and after 63 days for hedges of the medium andlong term Asian options.

The basis study evaluates the performance of the hedging models when calibration is conductedlocally. Local calibration means that the parameters of the hedging models are calibrated againstthe prices of the five European vanilla options that have the same expiry date as the Asian call weare hedging. The calibration is performed on every day of the hedge.

4This notation should always be understood as (maturity date)/(strike level). In the sequel, we also refer to the‘T -day Asian call’ when we mean the Asian call that matures on day T .


To examine whether the calibration method has an impact on the performance of the hedgingmodels, we also study the hedging error on the basis of globally calibrated parameters. Globalcalibration means, that the parameters of the hedging models are calibrated against the prices ofall European vanilla options in the market. The calibration is also performed daily. In general,a global calibration approach seems sensible if the payoff that is being hedged depends on thedistribution of the underlying asset at several time points of the hedging interval.

In this study, we set up the market models on the basis of parameters which are given in theTables 1 - 4. These parameters are taken from various publications (see discussion in Section 3)to reflect standard market behaviour. In periods of market turmoil, the markets do not show thestandard behaviour but instead exhibit frequent and strong up or downward movements whichare mainly reflected in a considerably higher kurtosis of the market returns. To test whether theresults of our hedging study are robust with respect to different market parameters, we reperformparts of our analysis with different parameters. In the 3FM market, a higher kurtosis can beobtained by increasing the volatility constants σv and σρ while keeping αM , κM , |βM | and |λM |relatively low. Specifically, we posit σv = 3, λP = λQ = −0.3 and leave all other parameters inTable 1 unchanged. By this choice, we increase the kurtosis and also ensure a somewhat strongerimpact of the correlation on the dynamics of the 3FM market. In the SVJJ market, we tweak thekurtosis by increasing the frequency and magnitude of the jumps. This leads to the following new

parameter values: λN = 3, µv = 0.12, σr = 0.494, µPr = −0.15 and µQr = −0.186. All other entriesin the Tables 2 and 3 remain unchanged. In the CGMYe market, the hedging errors are of highmagnitude and all hedging models are more or less equally suited (or unsuited) for hedges in thismarket. Therefore, subtle differences in the performance of hedging models under different marketparameters are comparatively meaningless given the overall magnitude of the hedging error. Forthis reason, we do not test the robustness of our results in the CGMYe market.

We conclude this section with a discussion of some aspects of the numerical implementation. Inprinciple, the implementation is divided into three steps. In a first step, we generate the marketdata. In a second step, we calibrate the hedging models and in a third step, we calculate the hedgeratios and the hedging errors.

5.1. Market Data Generation. For each market model, we first simulate the daily values ofthe physical trajectories5 of the underlying risk factors over the hedging horizon and generate theEuropean vanilla and Asian call prices. In total, we simulate 50000 trajectories. The trajectories ofthe underlying risk factors in the 3FM model are started at the values St0 = 50, vt0 = 0.01537 andρt0 = −0.183. The starting values in the SVJJ model are given by St0 = 50 and vt0 = 0.0341 andin the CGMYe model we use St0 = 50. The prices of the European vanilla options are calculatedalong every physical trajectory for every day. The prices of the Asian calls are also calculated alongevery physical trajectory, but only on the initial and final day of the hedging horizon. The Asiancall prices are needed to set up the initial portfolio and to calculate the terminal hedging error.

In the three-factor model, all SDEs are simulated by a first-order Euler scheme. In the SVJJmodel, we also simulate the drift and diffusion terms of all SDEs by a first-order Euler scheme.The Poisson process in the SVJJ model is simulated by standard methods. These methods are,for example, outlined in Glasserman [Gla04]. In the CGMYe model, we simulate the trajectorieson the basis of an algorithm by Tankov [Tan10]. One of the difficulties in simulating the varianceSDEs is to ensure that they do not become negative. In theory, this can be achieved by imposingthe second condition of (24) on the drift and diffusion parameters of the variance SDEs. However,even with this condition in place, the discretisation error can still lead to negative variances. If weobserve a negative variance value, this value is replaced by zero. How discretised variance SDEsshould be treated at their boundaries without distorting the distributional properties too much hasbeen studied in Lord et al. [LKvD09].

5By this, we mean the trajectories under the physical measure P .


To obtain the prices of the European vanilla calls in the 3F model, we rely on Monte Carlo sim-ulations. In the SVJJ model, the European vanilla call prices are given in terms of semi-analyticalpricing formulae. These formulae require us to perform a Fourier inversion of the characteristicfunction of the stock price. All integrals resulting from the Fourier inversion are evaluated nu-merically by using Gaussian quadrature routines which are provided by the Gnu Scientific Library(GSL). To calculate the prices of vanilla calls in the CGMYe model, we solve the correspondingpartial integro differential equations numerically. The prices of the Asian options on the initial andfinal day of the hedge are obtained by Monte Carlo simulations in all market models.

5.2. Calibration. The hedging models are calibrated daily against the prices of the Europeanvanilla options to infer the implied risk-neutral parameters. The calibration is performed by min-imising the squared sum of the pricing errors. Therefore, when calibrating a hedging model on thebasis of p ∈ N options on day t ∈ T , we minimise the sum

p∑i=1

∣∣∣CMit − CH

it

∣∣∣2 ,over the parameters of the hedging model. In general, this is a non-linear optimisation problemwith a non-linear constraint. The non-linear constraint arises in the Heston model from imposing(14) onto the parameters of the variance SDE (13). This constraint, however, is frequently ignoredwhen hedging models are calibrated by practitioners. We follow this practice. The variance itselfis not a market observable and therefore, in the Heston and SABR models, also obtained fromthe calibration procedure. From a technical point-of-view, the calibration is performed with thenon-linear least squares minimiser provided by the NAG C Library.

5.3. Hedge Ratios. The structure of the hedge ratios has been derived in (10). In this equation,the hedge ratios are given in terms of partial derivatives of the pricing formulae for European vanillaand Asian options. For European vanilla options, the required partial derivatives are easily calcu-lated. In the Black-Scholes and SABR models, closed-form expressions for these derivatives existand in the Heston model a corresponding semi closed-form expression can be efficiently evaluatedby quadrature methods.

For the Asian option, analytical pricing formulae do not exist in any of the hedging models andthe Black-Scholes price can be determined in terms of a solution to the PDE

∂tg +σ2

2(γt − y)2 ∂yyg = 0,

g(TA, y) = y+,

(20)

on [0, TA]× R that is subject to the boundary conditions

limy→∞

[g(t, y)− y] = 0,

limy→−∞

g(t, y) = 0.

In the Heston model, the corresponding PDE satisfies the equation

∂tg + ν(φ− v

)∂vg +

v

2(γt − y)2 ∂yyg +

1

2ϕ2v∂vvg + ϕρv (γt − y) ∂yvg = 0,

g(TA, y, v) = y+,(21)


on [0, TA]× R× R+ and is subject to the boundary conditions

limy→∞

[g(t, y, v)− y] = 0,

limy→−∞

g(t, y, v) = 0,

limv→∞

∂vg(t, y, v) = 0,

limv→0

∂vg(t, y, v) = 0.

The price of an Asian option in the Black-Scholes and Heston models, CAH , is related to solutionsof (20) and (21) via

CAHt = St g(t,Xt

St) and CAHt = St g(t,

Xt

St, vt)

respectively, where Xt, t ∈ [0, TA], is given as

(22) Xt =1

rTA

(1− e−r(TA−t)

)St + e−r(TA−t)

1

TA

∫ t

0Su du− e−r(TA−t)KA.

The relationship between solutions to (20) and (21) and the price of the Asian option has beenestablished by Vecer [Vec02] for the Black Scholes model. For the Heston model, we refer to Schroterand Monoyios [SM11], where we establish the relation between (21) and the Asian option price inthe Heston model and where we develop the relevant boundary conditions.

Since analytical formulae for the Asian option price do not exist, the partial derivatives, whichare required to calculate the hedge ratios, must be obtained numerically. This is a challengingtask, as the frequent recalibration of the hedging models necessitates to solve either one of thePDEs (20) and (21) about 3× 106 times for the simulation of just one single hedging error. In thisstudy, we solve the PDEs via finite element methods (FEM). Moreover, an application of the properorthogonal decomposition (POD) (see, for example, Sachs an Schu [SS08]) allows us to stronglyreduce the computation time required to solve the PDEs.

In general, setting up a finite element approximation begins with the choice of a set of N ∈ Ngeneric basis functions. Such basis functions are, for example, given by the hat functions of Figure 1.For good approximation results, a large numberN of these generic functions is required. This resultsin a high degree of computational complexity. To decrease the computational complexity, the FEMcalculations can be performed on a reduced basis. One way to reduce the basis is to replace thegeneric basis functions by new functions that are, in some sense, close to the solutions of the PDEs(20) and (21). In theory, functions that are sufficiently close can be found in the eigenfunctions ofthe PDE-operators. These eigenfunctions, however, are not explicitly known and moreover dependon the (due to recalibration) changing parameter vector θ that underlies the hedging models6. Byapplying a proper orthogonal decomposition, we are able to construct functions (the ‘POD basisfunctions’) that are stable with respect to changes in θ and that resemble the eigenfunctions of thePDE-operators. These functions allow us to set up a reduced basis for the FEM calculations.

We illustrate this procedure for the Black-Scholes price of the Asian option. In a first step, for afixed parameter θ, we approximate the solution gθ of the PDE (20), at time points t0 < · · · < tM ,M ∈ N, via

(23) gθ(tk, y) =N∑i=1

giθ(tk)bi(y), k ∈ 1, . . . ,M ,

on an equidistant partition y0 < · · · < yN of the domain of interest. The finite element approxima-tion in (23) is obtained by use of the hat basis,

bi(y) = max

1− |y − yi|

h, 0

, h =

yN − y0

N,

6In the Black-Scholes model we have θ = σ and in the Heston model we have θ = (ν, φ, ϕ, ρ).


0

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 1

(a) The hat basis.

0

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 1

y

g(TA, y)g(0, y)

(b) Solution to theBlack-Scholes PDE.

Figure 1. Hat basis on Ω = (−1, 1) with N = 8 and one solution of the Black-Scholes Asian PDE.

1e-20

1e-15

1e-10

1e-05

1

100000

1 10 100 1000

(a) The singular values of Y.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-1 -0.5 0 0.5 1

ψ1ψ2ψ3ψ4ψ5

(b) The reduced basis.

Figure 2. The decline of the singular values and the reduced basis in the Black-Scholes model.

for i = 1, . . . , N . In the sequel, we denote by gkθ =(g1θ(tk), . . . , g

Nθ (tk)

)Trthe N -dimensional vector

that contains the coefficients of the approximation (23).In a second step, we construct the functions of the reduced basis. For this, we fix p ∈ N

parameters θi, i ∈ 1, . . . , p, that cover the Black-Scholes parameter space sufficiently well andsolve the PDE (20) for each of these parameters. We also introduce the matrix Y that containssolutions of the PDE (20) at different time-points and for different parameter values,

Y =[g0θ1 , . . . ,g

Mθ1 ,g

0θ2 , . . . ,g

Mθ2 , . . . ,g

0θp , . . . ,g

Mθp

].

To compute the new POD basis functions, we perform a singular value decomposition (SVD) of Y.By this decomposition we get Y = UΣVT , for two unitary matrices U ∈ RN×N and V ∈ RpM×pMand a diagonal matrix Σ ∈ RN×pM . With the help of the matrices U and V, we are then able toconstruct the new (empirical) POD basis functions. The diagonal entries of Σ contain the so-called‘singular values’ of Y in decreasing order. Each singular value is associated to one POD basisfunction. It can be shown that POD basis functions that correspond to larger singular values aremore important for the approximation of a solution to the PDE (20). Ultimately, the goal is tomaintain the accuracy of the FEM computation with N basis functions by using L ∈ N, L < N ,of the most relevant POD basis functions. For good approximation results, it turns out that werequire 15 and 42 POD basis functions in the Black-Scholes and in the Heston models, respectively,and that 15 values of θ are sufficient to set up the new bases.

For the Black-Scholes model, the singular values and the first five POD basis functions aredepicted in Figure 2. In this figure, we see that the first POD basis function, ψ1, provides a coarse


σ CA0 Delta Vega Price error Delta error Vega error0.1 10.5968 0.989825 0.345335 8.3006e-05 8.38586e-05 0.0202638

0.188889 10.8198 0.926762 5.12669 0.000685423 0.000326805 0.02383530.277778 11.4769 0.845723 9.21924 0.000444494 0.000169424 0.01559030.366667 12.4005 0.787226 11.3620 0.000149486 0.000249543 0.01145990.455556 13.4670 0.748043 12.5556 0.000365294 0.000371058 0.0001644720.544444 14.6142 0.720466 13.2412 0.000161619 0.000211238 0.003581650.633333 15.8085 0.700741 13.6519 0.000218763 0.000282165 0.004550010.722222 17.0313 0.686913 13.8967 0.000465986 0.000449245 7.45315e-050.811111 18.2704 0.677614 14.0334 4.05614e-05 2.066e-05 0.00913167

0.9 19.5182 0.671855 14.0983 0.00100108 0.00104796 0.0125393

Table 6. Conventional FEM and POD calculations in the Black-Scholes model att = 0, with S0 = 100, TA = 0.5, K = 90 and r = 0.04. Price, delta and vega arebased on conventional FEM calculations. The errors relate to the difference betweenFEM and POD calculations.

g(TA, y, v)

-1-0.5

00.5

1

y0.5

11.5

2

v

-0.20

0.20.40.60.8

1

(a) Solution of the Heston PDE.

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

10000

1 10 100 1000

(b) The singular values of Y.

Figure 3. A solution of the Heston Asian PDE and the decline of the singularvalues in the Heston model.

approximation to the solution of the Asian PDE and that the other basis functions represent localdetails of the solution. By Figure 2, the decay in the singular values is exponential, hence relativelyfew POD basis functions are required to obtain good approximation results. Table 6 contains theapproximation results for the price of an Asian call in the Black-Scholes model for N = 801 hatbasis functions and L = 15 POD basis functions. In both approximations, we achieve the sameoverall accuracy (measured in terms of the price error), while the approximation on the reducedbasis can be performed 16 times faster than the corresponding approximation on the hat basis.

Similar observations can be made for the Heston model. The singular values in Figure 3 declineexponentially and thus a low number of POD basis functions is sufficient to obtain satisfactoryapproximation results.

In Figure 4, we see again that the first POD basis function, ψ1, constitutes a rough approxima-tion to the PDE surface and that the functions ψ2 and ψ3 model local details of the surface. Acomparison of the overall accuracy of an approximation with Ny = 81, Nv = 61 hat basis functionsand L = 42 POD basis functions is given Table 7. Again, we obtain the same level of accuracy (interms of the price error), while achieving 90 times faster calculations on the reduced basis.

6. Results and Conclusion

In this section, we discuss the results of the hedging simulation. We first discuss the performanceof the hedging models when their parameters are calibrated locally. Then, we review the hedging


ψ1

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-0.06-0.04-0.02

00.020.040.060.080.1

(c) ψ3

Figure 4. The first three functions of the reduced basis in the Heston model.

θi CA0 Delta Vega Price error Delta error Vega error1 11.402936 0.885104 4.485977 0.018236 0.000305 0.1981802 11.842398 0.851246 5.364510 0.010105 0.001115 0.1733573 12.022748 0.853884 4.436634 0.007271 0.002136 0.5265574 11.715899 0.853218 5.118843 0.000761 0.000500 0.0588515 10.910492 0.947674 3.534090 0.003323 0.001153 0.0143456 11.539249 0.862584 4.491026 0.001432 0.000637 0.0573777 11.457013 0.884826 6.363838 0.008435 0.001728 0.0242968 11.312174 0.887855 8.136989 0.004937 0.000604 0.0367059 11.150456 0.916820 5.748191 0.003046 0.001870 0.21372910 11.090177 0.903810 3.242317 0.008416 0.003473 0.39792811 10.986913 0.936491 6.183714 0.001551 0.000229 0.19522712 11.296781 0.907058 6.747697 0.010326 0.002407 0.42995013 10.849729 0.959915 1.469688 0.008794 0.001804 0.35385014 10.949302 0.939686 4.321778 0.003785 0.001334 0.13897215 11.547142 0.879573 5.907704 0.012027 0.001704 0.066837

Table 7. Conventional FEM and POD calculations in the Heston model at t = 0,with S0 = 100, v0 = 0.022 TA = 0.5, K = 90 and r = 0.04. Price, delta and vega arebased on conventional FEM calculations. The errors relate to the difference betweenFEM and POD calculations.

results of hedging models with globally calibrated parameters. Finally, we consider the hedgingmodel prices of the Asian options in the different market environments and examine how theseprices affect the hedging error.

In total, we evaluate five statistics of the hedging error: expectation, variance, skewness, (excess)kurtosis and the expected shortfall. Throughout the analysis, we assume that a hedger hopes fora hedging error with zero or positive expectation, a low variance and a near-zero shortfall —hedges that are favourable in that sense are referred to as good hedges. We sometimes refer toan ‘increasing’ (‘decreasing’) hedging error. By this, we mean a situation where some of the threestatistics expectation, variance and shortfall become less (more) favourable and all others (outof the three) remain roughly unchanged. If we say that the hedging error/performance in somecase A is better than (is more favourable than, has improved with respect to, etc.) the hedgingerror/performance in a situation B, we mean that the hedging error increases going from A to B.When the hedging error decreases going from A to B, we use a suitable opposite terminology. Also,for simplicity, we speak of a short term (medium term, long term) hedge when we mean the hedge


of the short term7 (medium term, long term) Asian call. All data is contained in the Tables 8 to17 of the Appendix B.

6.1. Local Calibration. Table 8 contains the hedging error statistics for hedges in the 3FMmarket when the stock is the only hedging instrument. There, we see that hedges based on theBlack-Scholes and the SABR models outperform similar hedges in the Heston model in terms ofvariances and shortfalls. In absolute terms, for all hedging models the expected hedging errors areof magnitude 0.002–0.3, the variances range between 0.009–0.035 and the shortfalls have a size of0.08–0.16. In the Black-Scholes and SABR models, the hedging performances slightly deterioratefor options with longer maturities. The same can be observed in the Heston model, there, however,the medium term and the corresponding long term hedge perform roughly the same. All of theseobservations are in line with a behaviour of the hedging errors we would expect, namely that theybecome less favourable when we hedge Asian options with longer maturities. The reasons for thisare (i) a longer hedging horizon for medium and long term options and (ii) that the impact of thestochastic volatility (which is not hedged when the stock is the only hedging instrument) gets morepronounced for options with longer maturities.

The statistics of the hedging errors for two-instrument hedges in the 3FM market can be foundin Table 9. In this table, we see that the inclusion of a second hedging instrument has littleimpact on the quality of the short term hedges but improves the quality of the medium term andlong term hedges. For the two-instrument hedges, the hedging error expectations centre aroundzero (in absolute terms they are 0.01 or less), the variances are of magnitude 0.03 or smallerand the shortfalls (in absolute terms) cover the range 0.04–0.13. While the additional hedginginstrument greatly improves the performances of hedges in the Black-Scholes and SABR models,the improvements are of smaller scale for hedges in the Heston model. In terms of the shortfalls andvariances of the hedging error distributions, the Black-Scholes and SABR hedges perform betterthan corresponding Heston hedges. The Heston model, however, achieves slightly positive averagehedging errors and therefore money is gained on average. In all hedging models, most statistics ofthe hedging error distributions improve when we hedge Asian options with longer maturities. Infact, for Asian options with longer maturities, the related hedging errors all have smaller variancesand smaller absolute shortfalls. These effects are more distinct for hedges in the Black-Scholes andSABR models. The reason that medium and long term hedges perform better than their short termcounterparts in the 3FM market, which is conflictive with the intuition of increasing hedging errorsover longer hedging horizons, lies in the effect that the stochastic volatility has on the hedgingperformances. The volatility process has only little impact on the prices of Asian options overshorter time frames. Therefore, short term hedges are only minimally affected by the inclusion ofa second hedging instrument. As a result, the one- and two-instrument short term hedges are verysimilar. For Asian options with longer maturities, the stochastic volatility becomes an importantcontributor to the Asian call prices and the call price with the longest maturity is most stronglyinfluenced by it. Therefore, the medium term and long term hedging errors greatly improve fromthe inclusion of a second hedging instrument.

The inclusion of a third hedging instrument (see Table 10) has little impact on the perfor-mance of the hedges. In fact, although the 3FM market is driven by three stochastic factors, thetwo-instrument hedges perform overall better than their three-instrument counterparts (only theexpectation of the Heston hedges is constantly higher for the three-instrument hedge). There aretwo possible reasons for this observation. Either the impact of the correlation on the Asian optionsin the 3FM market is so small that additional numerical noise from the three-instrument hedgeoutweighs a slightly better hedging performance, or the impact of the correlation on the Asianoptions in the 3FM market can not be modelled successfully by out-of-model hedges in the Hes-ton and SABR model. Finally, among the different three-instrument hedges, we observe that the

7Clearly, by short term (medium term, long term) Asian call we refer to the Asian option that matures on day 21(126, 189).


Heston model always achieves more favourable expectations and that the SABR model producesbetter variances and shortfalls.

In the SVJJ model, the picture is in parts similar. For hedges with the stock as the onlyhedging instrument, see Table 12, the Black-Scholes and SABR models produce equally good orbetter hedging results than the Heston model. For all hedging models and in absolute terms, theexpected hedging errors are in the range of 0.002–0.014, the variances vary between 0.06–0.26 andthe shortfalls have a magnitude of 0.18–0.44. For one-instrument hedges in the SVJJ market, thehedging errors increase when we hedge Asian options with longer maturities. Again, the reasons forthis observation are the longer hedging horizon of medium and long term hedges and the strongerimpact of the stochastic volatility on the prices of medium and long term Asian calls. One notableexception to this observation is the medium term Heston hedge which, in spite of a longer hedginghorizon, performs roughly as well as the short term Heston hedge.

The addition of a second hedging instrument to hedges in the SVJJ market, see Table 13, hasa positive influence on the hedging results. In fact, the hedging error distributions become moreright-skewed, the expected hedging errors are shifted closer to zero and their variances decrease.In numbers, the hedging error expectations are in the range of 0.001–0.006, the variances cover0.01–0.13 and the shortfalls have an absolute magnitude of 0.10–0.30. The impact of jumps onhedges in SVJJ market is observable in the, compared to the 3FM market, less favourable variancesand shortfalls of the hedging errors. Also, we observe relatively high kurtoses of the short termhedging errors, in particular, in the Black-Scholes and SABR models. This observation reflects thegreater sensitivity of the prices of short terms options to the jumps in the market. A comparison ofthe hedging results in the Black-Scholes and SABR models shows some degree of similarity betweenthese models. The Heston model is distinct. In the Heston model, short and medium term hedgesreact less strongly than their Black-Scholes and SABR counterparts to the presence of jumps. Thisis reflected in a 5–6 times smaller kurtosis of the corresponding hedging errors in the Heston model.Also, the hedging error distributions of hedges in the Heston model are more left-skewed. Thisleads to more negative shortfalls. For all hedging models, we observe that the hedging resultsimprove in hedges of Asian calls with longer maturities. This behaviour, as explained previously,is caused by the stronger impact that the stochastic volatility has on the prices of options withlonger maturities. The medium term Heston hedge is outstanding in that it performs better thanthe corresponding long term hedge. In fact, this hedge is better than all other Heston hedges in theSVJJ market. This reaffirms an observation we already made for one-instrument hedges, namelythat the medium term Heston hedge performs relatively strongly in the SVJJ market. However,despite the strong hedging performance of the Heston model for medium term Asian calls, overallthe Black-Scholes and SABR hedges are still superior in the SVJJ market.

The hedging error statistics for three-instrument hedges in the SVJJ market can be found inTable 14. There, we observe that the medium and long term, two-instrument hedges in the Hestonmodel perform better than their three-instrument counterparts. The opposite is true for the shortterm Heston hedge. With the SABR model, short and medium term hedges perform roughly thesame in the framework of two and three hedge instruments and the long term hedge is better whenconducted with two instruments. Since the SVJJ model is driven by only two diffusion processes,this mixed picture is not surprising. Among the various three-instrument hedges, SABR hedgesachieve better variances and shortfalls than their Heston counterparts.

The CGMYe model is a primarily jump-based model and therefore conceptionally different fromthe hedging models which are all based on SDEs with continuous trajectories. Hence, it is notsurprising that for all three hedging models the magnitudes of the hedging errors in the CGMYemarket are clearly higher than the magnitudes of these errors in the 3FM and SVJJ markets. In fact,Table 16 shows that for one-instrument hedges and in absolute terms, the hedging error expectationsare of size 0.27–1.06, the variances range between 0.2–1.5 and the shortfalls have a magnitude of0.4–1.57. Given the high magnitudes of the hedging errors, the performance differences of the threehedging models are small. No hedging model clearly outperforms all other models. In all hedging


models, the hedging performances get decidedly worse for Asian options with longer maturities.This reflects (i) the shorter hedging horizon of the short term Asian calls and (ii) the widening gapbetween the assumed distribution of the Asian payoff in the hedging models and the correspondingactual distribution in the CGMYe model.

By Table 17, the addition of a second hedging instrument to hedges in the CGMYe market doesnot lead to better hedging results. This observation is not unexpected given that the diffusionin the CGMYe market is small compared to its jump component. For hedges of the short termAsian option, the hedging results remain largely independent of the number of hedging instruments.For the hedges of the medium term and the long term Asian option, the one-instrument hedgesperform even slightly better than their two-instrument counterparts in terms of expectations andshortfalls. For two-instrument hedges in the CGMYe market, in absolute numbers the hedging errorexpectations range between 0.27–1.3, the variances cover 0.18–1.47, and the shortfalls have an order-of-magnitude of 0.39–1.71. Again, the differences between the hedging models are small given theoverall magnitudes of the hedging errors. Also, the hedging performances of two-instrument hedgesdeteriorate in all hedging models when we hedge Asian calls of longer maturities. This observationhas already been discussed in the context of one-instrument hedges in the CGMYe model.

The impact of a third hedging instrument in the CGMYe market can be studied in Table 18. Asin the two-instrument scenario, for the reasons discussed above, the addition of a third instrumentdoes in general not lead to improved results. With the Heston model, the two-instrument hedgesseem to realise better variances and shortfalls and the three-instrument hedges achieve betterexpectations. In the SABR model, the long term, three-instrument hedge performs better than thecorresponding two-instrument hedge. For medium and short term hedges with the SABR model,two- and three-instrument hedges perform roughly identically. Overall, the hedging errors are ofhigh magnitude in the CGMYe market and three-instrument hedges do not yield improvementsover one- and two-instrument hedges.

6.2. Global Calibration. When we follow a global calibration approach, we calibrate the hedgingmodels against the prices of all European vanilla options of Table 5. The idea behind a globalcalibration approach is that the structure of an option’s payoff should be reflected in the calibrationmethod. By the structure of the Asian payoff, Asian options depend on the distribution of theunderlying asset at several points in time. Therefore, the parameters of the hedging models shouldalso be calibrated on the basis of the risk-neutral stock price distribution at several time points.This is not the case when a hedging model is calibrated locally8. Since locally calibrated hedgingmodels are fitted to the prices of options with identical maturities, their calibrated parameters areonly based on the risk-neutral stock price distribution at one point in time. By fitting the hedgingmodels globally, we ensure that the globally calibrated parameters are not based on the stock pricedistribution at one single point in time but instead cover a range of time points. Whether thisyields better hedging results is the content of this subsection.

If global calibration were to influence the hedging results, we would expect the performance ofglobally calibrated short term hedges to deteriorate and the performance of globally calibratedlong term hedges to improve in comparison with their locally calibrated counterparts. The reasonto expect this pattern is that, when we calibrate hedging models globally, we calibrate againstadditional vanilla options that have no further relevance to the payoff of the short term Asianoption but yield a better picture of the risk-neutral stock price distribution over the life span ofthe long term Asian option. The previously described pattern, however, does not exist in the dataand therefore globally calibrated parameters hardly seem to impact the performance of the hedgingmodels.

In fact, when the parameters of the hedging models are calibrated globally most parts of theprevious analysis still hold. For all Asian options, the hedging results of the Black-Scholes and

8When we use a hedging model on the basis of globally (locally) calibrated parameters, we sometimes refer to themodel as globally (locally) calibrated. Also, a hedge on the basis of a globally (locally) calibrated hedging model issometimes called a globally (locally) calibrated hedge.


SABR models are largely stable with respect to the global calibration procedure in the 3FM andSVJJ markets (for one, two and three hedging instruments). In the CGMYe market, the globalcalibration procedure has little impact on the three-instrument SABR hedges and a slightly strongerimpact on the performance of the Black-Scholes and SABR hedges with one and two hedginginstruments. However, the impact has no clear direction and, given the higher magnitudes of thehedging errors in the CGMYe market, the differences in the hedging performances of locally andglobally calibrated hedging models are still negligible.

The situation is different in the Heston model. While the locally calibrated Heston model doesnot perform well in the 3FM market, the performance of the globally calibrated Heston modelbecomes even weaker for one- and two-instrument hedges in the 3FM market. In fact, the qualityof these hedges in the 3FM market decreases considerably and all expectations, variances andshortfalls are by a factor 1.5 (or higher) worse than the same figures in the locally calibratedmodel. For the globally calibrated three-instrument hedges in the 3FM market, the picture isdifferent. While the variances and shortfalls of the globally calibrated hedges improve for Asianoptions of all maturities, the hedging error expectations become slightly worse in comparison totheir locally calibrated counterparts. Within the SVJJ market, the Heston model’s decline inperformance for one- and two-instrument hedges is less extensive than in the 3FM market. Whilethe globally calibrated, medium term Heston hedges perform slightly worse than their locallycalibrated counterparts (one and two hedging instruments), the globally calibrated short and longterm Heston hedges perform roughly identical (one hedging instrument) or even slightly better(two hedging instruments) than their locally calibrated counterparts. For the three-instrumenthedges, the globally calibrated Heston model always achieves better results for variances, shortfallsand the expectation of the long term hedging error. Medium and short term hedging errors aremore favourable when the Heston model is applied on the basis of locally calibrated parameters.However, compared to any alternative hedging model, the globally calibrated Heston model doesstill not perform well in the SVJJ market. Finally, in the CGMYe market, the globally calibratedone-instrument Heston hedges perform slightly worse than their locally calibrated counterparts.Locally and globally calibrated Heston hedges perform more or less identically for two-instrumenthedges in the CGMYe market. For three-instrument hedges, the globally calibrated Heston modelperform better in terms of variances and shortfalls and the locally calibrated Heston model achievesbetter expectations.

6.3. Test of Robustness. In the Subsections 6.1 and 6.2, we saw that locally calibrated, two-instrument hedges yield on average the best hedging performances among all different scenarios andthat the Black-Scholes and SABR models in general yield a better performance than the Hestonmodel. To test the robustness of these results, we reperform parts of the previous analysis forlocally calibrated, two- and three-instrument hedges in the 3FM and SVJJ markets. This is doneon the basis of the parameter values outlined in Section 5 which introduce fatter tails, that is ahigher kurtosis, to the return distributions in the 3FM and SVJJ markets.

Table 11 contains the hedging error statistics for locally calibrated, two- and three-instrumenthedges in the 3FM market. There, we see that of all two- and three-instrument hedges, locallycalibrated, two-instrument Heston hedges lead to hedging errors with the lowest variances andshortfalls. Moreover, the expected hedging errors of these hedges are the closest to zero, makingthem the precisest of all hedges. Overall, among the two-instrument hedges, the most positivehedging error expectations are achieved by the SABR model. Among the three-instrument hedges,the Heston hedges lead to hedging errors with the most positive expectations. These findingsare for two reasons interesting. First, they confirm an observation of Subsection 6.1, namelythat in our 3FM framework three-instrument hedges do not perform better than two-instrumenthedges 9. Second, they are diametrically opposed to the findings of Subsection 6.1 where locally

9Based on the current choice of parameters this is somewhat surprising as this parameter choice leads to a strongerimpact of the correlation on the dynamics of the 3FM market.


calibrated two-instrument hedges with the Black-Scholes and SABR models yield hedging errorswith minimal variances and shortfalls and where the Heston model merely achieves good hedgingerror expectations.

For locally calibrated, two- and three-instrument hedges in the SVJJ market, Table 15 shows asimilar picture. Again, the locally calibrated, two-instrument hedges on the basis of the Hestonmodel achieve hedging errors with variances and shortfalls equal or better than those of any othertwo- or three-instrument hedge. For short and medium term Asian options, the best hedging errorexpectations are obtained in the SABR model, for long term options the best average hedging errorsare achieved by the Heston model. Thus, we find once more that the inclusion of a third hedginginstrument does not lead to a better hedging performance. In the SVJJ market, this finding islittle surprising since the market is driven by only two diffusion processes. Also, we observe thatthe hedging errors of locally calibrated, two-instrument hedges with the Heston model have lowervariances and shortfalls than the hedging errors obtained by any other model. Again, this is inopposition to our previous findings in Subsection 6.1.

6.4. The Prices of Asian Options in the Hedging Models. Next, we consider the pricingproperties of the hedging models in the different market environments. Information about thepricing properties of the hedging models can be obtained by comparing the prices of the Asianoptions in the hedging models with the prices of the Asian options in the market models. Thesecomparisons can simply be done by assessing the expectations of the hedging errors in Tables 8-17. In these tables, two sets of hedging error expectations are recorded. The first set is based onhedging portfolios whose initial values are the prices of Asian options in the market models. Thesecond set (in brackets) is based on hedging portfolios whose initial values are the prices of Asianoptions in the hedging models. From these hedging error expectations, the pricing tendencies ofany hedging model can be easily inferred with the help of (19). In fact, by (19), a hedging modelunderprices (overprices) an Asian option whenever the difference between the expected hedgingerror based on the market model price of the Asian option and the expected hedging error based onthe hedging model price of the Asian option is positive (negative). The magnitude of this differencereflects the degree of mispricing. In this study, we only observe exiguous differences between Asianoption prices in locally and globally calibrated hedging models. Therefore, in what follows, we onlydiscuss the option prices obtained on the basis of locally calibrated parameters.

In the 3FM market, the Black-Scholes and SABR models tend to underprice the Asian calls byan order of 0.007–0.014, whereas the Heston model overprices the same calls by a magnitude of0.2–0.4. In the Heston model, the degree of overpricing increases with the maturities of the Asianoptions. Black-Scholes and SABR models underprice the medium term Asian option stronger thanthe short term and the long term Asian options.

This picture changes in the SVJJ environment. There, the Heston model underprices the Asiancalls by a magnitude of 0.04–0.14, whereas the Black-Scholes and SABR models overprice thesecalls in the range of 0.01–0.03. For prices in the Black-Scholes model, the amount of overpricingdecreases for Asian options of longer maturities. The SABR model displays the opposite behaviour,there the amount of overpricing growths as the maturities of the Asian options get longer. Finally,for prices in the Heston model we find that longer maturities of Asian calls lead to a greater degreeof underpricing.

In the CGMYe market, all hedging models overprice the Asian options. The Black-Scholes andSABR models overprice the Asian calls in the range of 0.4–2.55 and the Heston model overpricesthese calls in the range of 0.1–1.5. Also, for all hedging models, the amount of overpricing increaseswhen the Asian options mature later.

Overall, the Heston model has the best pricing performance in the jump-dominated CGMYemarket. In the diffusion (3FM) and jump-diffusion (SVJJ) markets, the Black-Scholes and theSABR models perform better than the Heston model in pricing Asian options. In the 3FM market,the differences between the Black-Scholes and SABR models are negligible. In the SVJJ market,the Black-Scholes model tends to price Asians calls with longer maturities more accurately while the


SABR model performs better in pricing Asian calls with shorter maturities. Regarding the pricingperformance of the hedging models, we mainly observe that the amount of mispricing increases forAsian options with longer maturities. One reason for this observation are the diverging terminaldistributions of the Asian payoff in the hedging models and the market models. A notable exceptionto the previous observation is the Black-Scholes model in the SVJJ market. There the amount ofmispricing decreases for Asian calls that mature later. This behaviour can be partly attributed tothe presence of jumps in the SVJJ market.

6.5. Conclusion. In this paper, we study the performance of one-, two- and three-instrumenthedges in the Black-Scholes, Heston and SABR models by hedging Asian options in different mar-ket environments. These market environments are characterised by continuous trajectories (3FM),by a mixture of continuous trajectories and jumps (SVJJ) and by jump-dominated trajectories(CGMYe). Also, we examine the impact of different calibration methods on the hedging perfor-mance. Among all possible combinations of the calibration methods and the number of hedginginstruments, the locally calibrated two-instrument hedges achieve in general the best hedging per-formance in this study10. For this reason, we briefly review the performance of locally calibratedtwo-instrument hedges in the sequel.

In the 3FM market, the hedging errors in the Black-Scholes model achieve the lowest variancesand shortfalls throughout all maturity classes and hedging models. In the SVJJ market, thehedging performances of the Black-Scholes and SABR models are roughly identical with a (very)small advantage for the SABR model. Both models clearly outperform the Heston model in termsof variances and shortfalls. In the CGMYe market, all hedging models produce comparable hedgingresults and no model dominates the others. In terms of pricing, the Black-Scholes and SABR modelsperform better than the Heston model in the 3FM and SVJJ markets. However, the Heston modelis better suited for pricing Asian options in the CGMYe market.

Overall, in terms of variances and shortfalls, the hedging performances of the Black-Scholesand SABR models are better (3FM, SVJJ) or as good (CGMYe) as the hedging performanceof the Heston model. In terms of the expected hedging errors, the Heston hedges sometimesperform slightly better than corresponding hedges in the alternative hedging models. However,these differences are mostly negligible. Only when it comes to pricing Asian calls in the CGMYemarket, the Heston model clearly dominates all other models. Also, in all hedging models, thepresence of unhedgeable jumps (in the SVJJ and CGMYe markets) has a stronger impact on thehedging errors than the presence of an unhedged diffusion (the correlation in the 3FM market)and, as discussed in Subsection 6.1, the hedging errors of the short term options are more stronglyinfluenced by jumps than those of the medium and long term options.

The hedging performance of the Black-Scholes model is remarkable, given that it is the simplestof all tested hedging models. This simpler structure allows for a more efficient handling of the Black-Scholes model, for example, in terms of a faster calibration and faster PDE solvers. Based on thehedging performance and the simpler handling, the Black-Scholes model seems to be the preferredhedging model in all three market environments. On the basis of that simplicity, it is unreasonableto assume that the Black-Scholes model achieves its hedging performance because it constitutesa superior structural approximation to the distributions of the market models. Instead, it seemsmore sensible that the simplicity of the model allows for a very good calibration to market datawhich, in turn, leads to its good hedging performance. This explanation is also hinted at in [JO01],who show that, for a lower than daily rate of recalibration, two-instrument hedges in the Hestonmodel perform slightly better than the same hedges in the Black-Scholes model11. We thereforeconclude that a parsimonious model, where the calibration issues are condensed into a small number

10Sometimes the Heston model achieves better expected hedging errors in the context of three-instrument orglobally calibrated hedges. In those instances, however, all other hedging error statistics are on average inferior tocomparable two-instrument scenarios.

11However, the market model in [JO01] is a stochastic volatility model with jumps in the underlying, the resultsare obtained from hedging European standard options, and the hedging performance is measured in terms of the


of parameters, can be successful in hedging. More complex models, however, do not necessarilylead to greater robustness to model error. However, we want to stress that a more sophisticatedand problem-adapted calibration approach based on some metaheuristic algorithm like simulatedannealing (see Kirkpatrick et al. [KGV83] and Cerny [Cer85]) could improve the performanceof the non-linear least squares minimiser we used for calibration. It is subject to further studieswhether a more involved calibration procedure is able to leverage the better theoretical propertiesof the more complex hedging models.

One word of caution is needed in this context. In Subsection 6.3, we saw that in marketscharacterised by high kurtoses, the Heston model achieves on average a better hedging performancethan the Black-Scholes model. This potentially shows, that the Black-Scholes model mainly worksin relatively standard market regimes as modelled by the parameters in the Tables 1− 4 and thatdifferent hedging models are required in times where the market returns exhibit a higher kurtosisor similar characteristics.

In this paper, we have studied the performance of hedging models in the context of Asian options.An interesting question in this framework is whether our findings also hold for other path-dependentoptions. Previously, we concluded that the good hedging performance of the Black-Scholes modelseems to be partly due to its good calibration properties. If this is in fact the case, we can alsoexpect our findings to hold for other path-dependent options as long as these options do not haveterminal conditions which are particularly unsuited for the Black-Scholes framework.

Finally, the performance of the hedging models in the CGMYe market highlights a potentialneed for different hedging approaches in this market, for example, exponential or mean-variancehedging approaches. In this paper, we have investigated the performance of the hedging modelsin the CGMYe market on the basis of delta hedges. Whether different hedging approaches yieldbetter hedging results is largely an open question. In fact, relatively few studies in the pertinentliterature are concerned with the performance of misspecified hedging models in the framework ofalternative hedging approaches. While such a study is beyond the scope of this work, it remains atopic of future research. From a practical point of view, the good performance of the Heston modelin markets that are characterised by a high kurtosis shows that questions related to the choice ofappropriate hedging models for market regimes based on different parameter values (which thuslead to different statistical properties) warrant further theoretical and applied research. Also, littletheoretical work has been done to study model misspecification when either the misspecified or thetrue asset process are not modelled by an Ito diffusion. In this case, the discrepancy between thetrue and misspecified asset distributions could be gauged in terms of the entropy (or a differentmetric for the distance between two probability distributions). Questions regarding the feasibilityof an entropy-based approach in the context of model risk and the implementation of such anapproach in practice also remain subject to future research.

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29, 2010.

Appendix A. Change of Measure in the Three-Factor Model

To investigate the existence of an equivalent martingale measure in the three-factor model, werewrite the model of Section 3.1, for M ∈ P,Q, as

dSt = µMStdt+√vtStρtdWM,1

t +√

1− ρ2tdW

M,2t ,

dvt = αM (βM − vt)dt+ σv√vtdW

M,1t ,

dρt = κM (λM − ρt)dt+ σρ

√1− ρ2

tdWM,3t .

The market prices-of-risk that correspond to this structure preserving change of measure are, fort ∈ T , given by

λ1t =

αPβP − αQβQ + (αQ − αP )vtσv√vt

,

λ2t =

(µP − r)σv − (αPβP − αQβQ + (αQ − αP )vt)ρt√1− ρ2

tσv√vt

,

λ3t =

κPλP − κQλQ + (κQ − κP )ρt

σρ√

1− ρ2t

.

To ensure that the λi, i ∈ 1, 2, 3, are well defined, we must have vt ∈ (0,∞) and ρt ∈ (−1, 1) fort ∈ T . From the Feller test for explosions (see Karlin and Taylor [KT81]), this is the case whenthe parameters under the real-world measure satisfy

(24) κP ≥σ2ρ

1± λPand 2αPβP ≥ σ2

v .

For t ∈ T , we introduce the process Z3F by

Z3Ft = E(−λ1 ·WP,1)tE(−λ2 ·WP,2)tE(−λ3 ·WP,3)t.

By the Girsanov theorem, a martingale measure Q, with dQdP = Z3F

T, exists if E[Z3F

T] = 1. To verify

this condition, we first point out that, conditional on the path of v and ρ, the price-of-risk λ2 isdeterministic and well-defined upon imposing (24). Hence, E(−λ2 ·WP,2) is a martingale when


we have knowledge of v and ρ and we get E[E(−λ2 ·WP,2)T |(vt)t∈T , (ρt)t∈T ] = 1. This and theindependence of WP,1 and WP,3 allow us to write

E[Z3FT ] = E[E(−λ1 ·WP,1)T ]E[E(−λ3 ·WP,3)T ].

Thus, the existence of an equivalent martingale measure in the three-factor model follows when

E[E(−λ1 ·WP,1)T ] = E[E(−λ3 ·WP,3)T ] = 1.

By our choice of the parameters in Table 1, we have λ3 = 0. Therefore, we only have to establishwhether E[E(−λ1 ·WP,1)T ] = 1. This question has been treated in Cheridito et al. [CFK07] whoshow that, for M ∈ P,Q, the condition

2αMβM ≥ σ2v

is sufficient for E[E(−λ1 ·WP,1)T ] = 1.

Appendix B. The Data

This appendix contains the complete data generated in the simulation. The data is groupedwith respect to the underlying market models. For simpler comparison, the results of locally andglobally calibrated hedges are placed side by side. We summarise the results in terms of fivestatistical figures of the terminal hedging error: expectation, variance, skewness, (excess) kurtosisand expected shortfall. The expectation of the hedging error is computed twice: once based on ahedging portfolio whose initial value is the price of the Asian call in the market model, and oncebased on a hedging portfolio whose initial value is the price of the Asian call in the hedging model(the latter expectation is given in brackets, see also (19) and the following discussion). The kurtosisis computed as excess kurtosis with respect to the normal distribution. The expected shortfall isthe expectation of the hedging error conditional on a negative hedging outcome.


Th

ree-

Fac

tor

Mod

el(3

FM

)–

On

e-In

stru

men

tH

edge

s

Asi

anO

ptio

n–

21da

ys

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

0.00

2016

0.00

3306

0.00

2145

0.00

1939

0.03

9109

0.00

2055

(-0.

0050

53)

(0.2

1170

7)(-

0.00

5609

)(-

0.00

513)

(0.2

4751

)(-

0.00

5699

)

Var

ianc

e0.

0096

170.

0245

850.

0096

840.

0098

810.

0715

90.

0099

22

Skew

ness

-0.5

8054

5-0

.593

305

-0.5

8858

-0.6

2442

-0.2

8648

5-0

.626

61

Kur

tosi

s0.

7565

40.

3153

960.

7598

430.

8481

940.

1567

160.

8393

79

E.

Shor

tf.

-0.0

8283

6-0

.134

367

-0.0

8324

6-0

.084

337

-0.2

0083

8-0

.084

526

Asi

anO

ptio

n–

126

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-0.0

1173

1-0

.024

333

-0.0

1046

8-0

.011

853

-0.1

3222

5-0

.010

78(-

0.02

5918

)(0

.355

604)

(-0.

0239

71)

(-0.

0260

4)(0

.247

712)

(-0.

0242

83)

Var

ianc

e0.

0116

620.

0409

630.

0119

720.

0120

340.

0715

20.

0123

23

Skew

ness

-0.3

9868

60.

1306

16-0

.420

868

-0.4

1363

1-0

.281

006

-0.4

2409

8

Kur

tosi

s0.

5868

23-0

.338

340.

5623

910.

5789

430.

1310

160.

5464

47

E.

Shor

tf.

-0.0

9229

8-0

.170

584

-0.0

9382

8-0

.093

897

-0.2

7041

1-0

.095

269

Asi

anO

ptio

n–

189

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-0.0

1919

6-0

.030

237

-0.0

1726

2-0

.019

652

-0.1

9892

-0.0

1810

2(-

0.03

034)

(0.4

2363

1)(-

0.02

5893

)(-

0.03

0796

)(0

.254

948)

(-0.

0267

33)

Var

ianc

e0.

0134

80.

0356

970.

0140

10.

0138

0.06

221

0.01

4349

Skew

ness

-0.3

5614

60.

0031

81-0

.388

893

-0.3

2376

8-0

.193

913

-0.3

4176

3

Kur

tosi

s0.

466

-0.3

3155

10.

4383

390.

4177

670.

0927

080.

3873

62

E.

Shor

tf.

-0.1

0187

4-0

.165

832

-0.1

0373

1-0

.102

461

-0.2

9248

3-0

.104

552

Table

8.

Per

form

an

ceof

on

e-in

stru

men

th

edges

inth

eth

ree-

fact

or

mod

el.

Th

ree-

Fac

tor

Mod

el(3

FM

)–

Tw

o-In

stru

men

tH

edge

s

Asi

anO

ptio

n–

21da

ys

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

0.00

3041

0.00

2739

0.00

309

0.00

2756

-0.0

7528

0.00

2784

(-0.

0040

28)

(0.2

1114

)(-

0.00

4664

)(-

0.00

4313

)(0

.133

121)

(-0.

0049

7)

Var

ianc

e0.

0087

30.

0273

50.

0088

410.

0088

820.

0503

660.

0088

9

Skew

ness

-0.5

6430

6-0

.262

524

-0.4

8541

3-0

.619

932

-0.8

2589

4-0

.622

429

Kur

tosi

s0.

7583

141.

6901

461.

8676

770.

8635

850.

5406

090.

8681

55

E.

Shor

tf.

-0.0

7849

9-0

.134

146

-0.0

7848

1-0

.079

731

-0.2

2957

2-0

.079

702

Asi

anO

ptio

n–

126

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-0.0

034

0.00

1601

-0.0

1115

3-0

.002

881

-0.1

2486

-0.0

0454

2(-

0.01

7587

)(0

.381

538)

(-0.

0246

56)

(-0.

0170

68)

(0.2

5507

7)(-

0.01

8045

)

Var

ianc

e0.

0051

70.

0289

180.

0113

540.

0048

460.

0777

260.

0058

37

Skew

ness

-2.4

9779

50.

2242

26-4

.878

366

-1.4

9092

5-0

.422

139

-2.1

4140

3

Kur

tosi

s40

.784

51.

5280

161

.757

9128

.221

371.

0418

9349

.354

60

E.

Shor

tf.

-0.0

5584

-0.1

2937

4-0

.067

724

-0.0

5472

6-0

.273

829

-0.0

5662

9

Asi

anO

ptio

n–

189

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-0.0

0362

10.

0011

93-0

.008

906

-0.0

0438

9-0

.169

821

-0.0

088

(-0.

0147

65)

(0.4

5506

1)(-

0.01

7537

)(-

0.01

5533

)(0

.284

047)

(-0.

0174

31)

Var

ianc

e0.

0027

050.

0187

080.

0039

240.

0025

370.

0611

980.

0036

51

Skew

ness

-2.7

7730

10.

0231

63-4

.779

637

-3.1

4816

8-0

.231

85-4

.173

343

Kur

tosi

s75

.926

171.

8127

6312

1.72

9315

6.05

952.

1331

2413

5.59

69

E.

Shor

tf.

-0.0

3947

1-0

.105

982

-0.0

4312

8-0

.036

579

-0.2

6769

1-0

.040

375

Table

9.

Per

form

an

ceof

two-i

nst

rum

ent

hed

ges

inth

eth

ree-

fact

or

mod

el.


Th

ree-F

actor

Mod

el

(3F

M)

–T

hree-I

nstru

ment

Hed

ges

Asi

an

Op

tion

–21

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

08683

0.0

0266

0.0

05759

0.0

03012

(0.2

17084)

(-0.0

05094)

(0.2

1416)

(-0.0

04742)

Vari

an

ce

0.0

36335

0.0

09681

0.0

29622

0.0

08581

Skew

ness

-0.2

88268

-0.4

11171

-0.7

01362

-0.6

04615

Ku

rtosi

s1.7

3009

8.2

90155

0.5

0326

0.8

50605

E.

Sh

ort

f.-0

.15382

-0.0

78817

-0.1

51755

-0.0

77722

Asi

an

Op

tion

–126

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

05779

-0.0

1072

-0.0

01297

-0.0

04214

(0.3

85716)

(-0.0

24223)

(0.3

7864)

(-0.0

17717)

Vari

an

ce

0.2

58447

0.0

10617

0.0

60899

0.0

05825

Skew

ness

-0.0

51227

-5.0

06466

-0.1

36615

-2.0

88002

Ku

rtosi

s1.2

14618

63.8

9645

8.5

63608

42.0

6555

E.

Sh

ort

f.-0

.367588

-0.0

66484

-0.1

62124

-0.0

57499

Asi

an

Op

tion

–189

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

26359

-0.0

11513

-0.0

07841

-0.0

10723

(0.4

80227)

(-0.0

20144)

(0.4

46027)

(-0.0

19354)

Vari

an

ce

0.2

49556

0.0

06717

0.0

43693

0.0

05567

Skew

ness

0.0

04578

-5.0

45821

-0.1

0595

-4.1

55677

Ku

rtosi

s1.5

62022

70.9

6101

9.9

09783

67.4

4129

E.

Sh

ort

f.-0

.344624

-0.0

58507

-0.1

44991

-0.0

56154

Table

10.

Per

form

an

ceof

thre

e-in

stru

men

th

edges

inth

eth

ree-

fact

or

mod

el.

Th

ree-F

acto

rM

od

el

(3F

M)

–S

tress

Para

mete

rs

Asi

an

Op

tion

–21

day

s

Tw

oIn

st.

Hed

ge

Th

ree

Inst

.H

edge

BS

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

00416

0.0

05283

0.0

01241

0.0

04979

0.0

01301

Vari

an

ce0.0

86453

0.0

99498

0.1

00638

0.2

09106

0.0

95051

Skew

nes

s-3

.037214

-1.8

33721

-3.3

96103

-0.5

44402

-3.1

70578

Ku

rtosi

s17.8

7098

11.4

5616

20.5

8142

19.0

1015

24.6

0428

E.

Sh

ort

f.-0

.257014

-0.2

63724

-0.2

8616

-0.3

28999

-0.2

58564

Asi

an

Op

tion

–126

day

s

Tw

oIn

st.

Hed

ge

Th

ree

Inst

.H

edge

BS

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

31542

0.0

14989

0.0

53711

0.0

81174

0.0

53711

Vari

an

ce0.7

47138

0.7

41934

0.9

64686

2.5

27892

0.9

64686

Skew

nes

s-1

.98455

-1.2

36579

-1.8

54601

-0.3

15383

-1.8

54601

Ku

rtosi

s17.5

3094

14.6

9856

13.7

449

4.8

73458

13.7

449

E.

Sh

ort

f.-0

.831867

-0.6

15328

-0.9

62119

-1.0

73406

-0.9

62119

Asi

an

Op

tion

–189

day

s

Tw

oIn

st.

Hed

ge

Th

ree

Inst

.H

edge

BS

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

08372

0.0

01522

0.0

24952

0.0

49213

0.0

60619

Vari

an

ce0.6

33159

0.5

40602

0.9

10849

2.8

09624

1.1

80005

Skew

nes

s-1

.939938

-1.5

19626

-1.6

70322

-0.0

53937

-2.2

14406

Ku

rtosi

s22.0

7533

19.6

0088

16.1

9049

4.5

55125

11.2

5241

E.

Sh

ort

f.-0

.654462

-0.4

58535

-0.7

83627

-0.9

823

-1.1

05724

Table

11.

Per

form

an

cein

the

thre

e-fa

ctor

model

,st

ress

scen

ari

o.


SV

JJ

Mod

el–

On

e-In

stru

men

tH

edges

Asi

anO

ptio

n–

21da

ys

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

0.00

3049

0.00

2488

0.00

2704

0.00

3396

0.07

3949

0.00

3121

(0.0

3603

9)(-

0.04

2597

)(0

.021

246)

(0.0

3638

6)(0

.028

864)

(0.0

2166

3)

Var

ianc

e0.

0617

390.

1572

810.

0667

670.

0605

720.

1615

050.

0632

15

Skew

ness

-4.8

8780

3-2

.495

504

-4.6

8010

4-5

.016

496

-2.4

7932

6-4

.787

338

Kur

tosi

s46

.676

0115

.274

7743

.000

0351

.165

9415

.329

3944

.753

5

E.

Shor

tf.

-0.1

7769

8-0

.338

445

-0.1

8972

9-0

.174

367

-0.3

2829

7-0

.181

639

Asi

anO

ptio

n–

126

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

0.01

2436

0.01

3085

0.00

9869

0.01

2913

0.37

5631

0.01

0436

(0.0

4264

8)(-

0.06

5604

)(0

.035

399)

(0.0

4312

5)(0

.296

942)

(0.0

3596

6)

Var

ianc

e0.

1522

710.

1717

970.

1867

90.

1511

990.

2615

690.

1818

48

Skew

ness

-2.1

1965

9-1

.724

217

-1.9

6946

5-2

.101

858

-1.4

5819

2-1

.999

692

Kur

tosi

s9.

4527

857.

4068

7.78

7545

9.37

5901

4.79

8658

8.04

018

E.

Shor

tf.

-0.3

2352

5-0

.339

844

-0.3

7747

3-0

.321

317

-0.4

1927

1-0

.371

211

Asi

anO

ptio

n–

189

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

0.01

3871

0.01

0511

0.01

0529

0.01

3942

0.54

3473

0.01

0378

(0.0

3333

2)(-

0.13

4883

)(0

.046

804)

(0.0

3340

3)(0

.398

079)

(0.0

4665

3)

Var

ianc

e0.

1694

270.

2621

770.

2106

80.

1702

820.

2813

330.

2191

44

Skew

ness

-1.7

7858

1-1

.419

504

-1.7

033

-1.7

8894

5-1

.461

93-1

.691

239

Kur

tosi

s6.

9722

264.

3974

675.

7707

087.

0181

794.

3070

825.

7114

66

E.

Shor

tf.

-0.3

4593

6-0

.447

527

-0.4

0926

9-0

.347

899

-0.4

5279

5-0

.417

704

Table

12.

Per

form

an

ceof

on

e-in

stru

men

th

edges

inth

eS

VJJ

mod

el.

SV

JJ

Mod

el–

Tw

o-I

nst

rum

ent

Hed

ges

Asi

anO

ptio

n–

21da

ys

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

0.00

2347

0.00

1021

0.00

2487

0.00

2913

0.07

4402

0.00

2679

(0.0

3533

7)(-

0.04

4061

)(0

.021

029)

(0.0

3590

3)(0

.029

317)

(0.0

2122

1)

Var

ianc

e0.

0511

380.

1302

730.

0498

620.

0492

850.

1219

250.

0499

07

Skew

ness

-5.3

2690

1-1

.721

413

-5.3

1564

9-5

.207

173

-1.9

3464

5-5

.293

554

Kur

tosi

s57

.424

067.

9953

5556

.824

6655

.019

079.

3306

0956

.521

38

E.

Shor

tf.

-0.1

6033

2-0

.312

178

-0.1

5786

5-0

.156

596

-0.2

9342

5-0

.157

576

Asi

anO

ptio

n–

126

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

0.00

5566

0.00

617

0.00

6067

0.00

6591

0.38

1304

0.00

6506

(0.0

3577

8)(-

0.07

2519

)(0

.031

597)

(0.0

3680

3)(0

.302

615)

(0.0

3203

6)

Var

ianc

e0.

0413

990.

0417

540.

0402

760.

0408

70.

0897

510.

0406

05

Skew

ness

-3.2

1927

4-0

.944

847

-3.2

1230

3-3

.171

955

-0.3

4480

99-3

.210

829

Kur

tosi

s18

.334

64.

6474

9718

.334

617

.950

761.

5471

9418

.230

45

E.

Shor

tf.

-0.1

6753

-0.1

5395

3-0

.165

354

-0.1

6627

-0.1

9745

2-0

.166

158

Asi

anO

ptio

n–

189

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

0.00

4034

0.00

4606

0.00

4379

0.00

3878

0.54

3179

0.00

5933

(0.0

2349

5)(-

0.14

0788

)(0

.040

654)

(0.0

2333

9)(0

.397

785)

(0.0

4220

8)

Var

ianc

e0.

0168

80.

0792

060.

0166

540.

0170

850.

0629

530.

0161

97

Skew

ness

-3.3

5835

20.

4387

47-3

.325

878

-3.3

8428

6-0

.322

627

-3.2

7066

3

Kur

tosi

s20

.220

2327

.832

6619

.858

2720

.503

581.

0732

1219

.252

82

E.

Shor

tf.

-0.1

0821

8-0

.207

082

-0.1

0724

4-0

.108

618

-0.1

6547

1-0

.106

103

Table

13.

Per

form

an

ceof

two-i

nst

rum

ent

hed

ges

inth

eS

VJJ

mod

el.


SV

JJ

Mod

el

–T

hree-I

nstru

ment

Hed

ges

Asi

an

Op

tion

–21

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

04667

0.0

02446

0.0

01964

0.0

02254

(-0.0

40418)

(0.0

20988)

(-0.0

43121)

(0.0

20796)

Vari

an

ce

0.1

46819

0.0

46026

0.0

89434

0.0

46218

Skew

ness

-0.4

06926

-5.2

7492

-2.5

87778

-5.3

39088

Ku

rtosi

s26.7

1848

58.3

4686

18.3

0299

57.9

9038

E.

Sh

ort

f.-0

.257176

-0.1

50208

-0.2

49588

-0.1

50764

Asi

an

Op

tion

–126

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

1857

0.0

05813

0.0

11454

0.0

05905

(-0.0

60119)

(0.0

31343)

(-0.0

67235)

(0.0

31435)

Vari

an

ce

1.0

85315

0.0

39861

0.3

07905

0.0

40249

Skew

ness

-0.0

30786

-3.2

38427

-0.6

53791

-3.2

29856

Ku

rtosi

s7.0

94854

18.5

5831

25.0

8522

18.4

7475

E.

Sh

ort

f.-0

.61133

-0.1

63628

-0.3

37963

-0.1

64439

Asi

an

Op

tion

–189

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

-0.0

13424

0.0

05706

0.0

16531

0.0

05134

(-0.1

58818)

(0.0

41981)

(-0.1

28863)

(0.0

41409)

Vari

an

ce

1.9

47977

0.0

44646

0.3

84936

0.0

4458

Skew

ness

-0.0

29953

-2.5

15602

-0.6

55351

-2.4

358

Ku

rtosi

s3.8

44548

12.9

9791

25.2

715

12.0

3183

E.

Sh

ort

f.-0

.903786

-0.1

7447

-0.3

56818

-0.1

74256

Table

14.

Per

form

an

ceof

thre

e-in

stru

men

th

edges

inth

eS

VJJ

model

.

SV

JJ

Mod

el

–S

tress

Para

mete

rs

Asi

an

Op

tion

–21

day

s

Tw

oIn

st.

Hed

ge

Th

ree

Inst

.H

edge

BS

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.0

87213

0.0

59234

0.1

02233

0.0

54031

0.0

91457

Vari

an

ce8.3

97447

7.9

38554

8.7

19403

7.6

72912

8.2

52107

Skew

nes

s-3

.936084

-4.1

40713

-4.0

47581

-4.1

6202

-4.0

399

Ku

rtosi

s18.0

3627

18.8

8984

18.4

3452

19.4

7471

19.1

264

E.

Sh

ort

f.-5

.710182

-5.4

98943

-5.9

41856

-5.3

50161

-5.6

9875

Asi

an

Op

tion

–126

day

s

Tw

oIn

st.

Hed

ge

Th

ree

Inst

.H

edge

BS

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.5

05547

0.1

16637

0.7

0241

0.4

84463

0.6

92179

Vari

an

ce21.0

6128

16.7

545

25.2

6212

22.5

3917

25.1

2514

Skew

nes

s-1

.415336

-2.0

21561

-1.0

85284

-1.0

89071

-1.0

90466

Ku

rtosi

s6.2

81846

7.9

00555

5.3

34872

6.1

15268

5.3

88825

E.

Sh

ort

f.-5

.364221

-4.7

75659

-5.8

97217

-4.7

43466

-5.8

69128

Asi

an

Op

tion

–189

day

s

Tw

oIn

st.

Hed

ge

Th

ree

Inst

.H

edge

BS

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

0.1

75359

0.3

10624

0.1

95356

0.7

79702

0.3

22837

Vari

an

ce17.5

6728

13.6

0815

22.7

4609

24.1

6794

26.6

2431

Skew

nes

s-0

.9512

-1.8

57531

-0.9

64712

-0.5

95742

-1.3

23267

Ku

rtosi

s9.3

41644

11.6

6122

6.7

53523

6.4

82284

4.8

17019

E.

Sh

ort

f.-4

.546458

-3.9

26153

-5.3

82438

-4.5

76368

-6.2

1656

Table

15.

Per

form

an

cein

the

SV

JJ

mod

el,

stre

sssc

enari

o.


CG

MY

eM

od

el–

On

e-In

stru

men

tH

edges

Asi

anO

ptio

n–

21da

ys

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-0.2

7044

1-0

.271

121

-0.2

7261

7-0

.265

679

-0.2

7287

7-0

.266

467

(0.2

0143

3)(-

0.16

0453

)(0

.144

501)

(0.2

0619

5)(-

0.16

2209

)(0

.150

651)

Var

ianc

e0.

2021

110.

2310

910.

2181

610.

1985

490.

3130

020.

2008

06

Skew

ness

-2.6

4740

7-2

.412

686

-2.6

9523

3-2

.541

804

-2.4

7434

2-2

.596

792

Kur

tosi

s13

.249

8411

.165

1513

.362

1112

.482

8710

.989

3512

.864

20

E.

Shor

tf.

-0.4

0968

7-0

.443

848

-0.4

2519

4-0

.411

286

-0.5

2159

3-0

.411

001

Asi

anO

ptio

n–

126

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-0.9

7061

4-0

.970

228

-0.9

7651

1-0

.953

074

-0.9

7356

8-0

.957

64(0

.820

326)

(0.0

8371

2)(0

.778

039)

(0.8

3786

6)(0

.080

372)

(0.7

9691

)

Var

ianc

e0.

5243

310.

5353

560.

6110

480.

5271

550.

6601

770.

5896

59

Skew

ness

-0.6

7112

-0.7

7715

-0.5

6584

5-0

.680

457

-0.7

6374

5-0

.602

288

Kur

tosi

s1.

3963

091.

2212

020.

9874

691.

4583

50.

8658

451.

1633

72

E.

Shor

tf.

-1.0

6320

7-1

.038

815

-1.1

1556

7-1

.053

378

-1.0

8992

4-1

.091

651

Asi

anO

ptio

n–

189

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-1.0

6599

3-1

.065

719

-1.0

7187

1-1

.040

648

-1.0

5458

4-1

.047

927

(1.4

6094

7)(0

.490

391)

(1.4

8305

9)(1

.486

292)

(0.5

0152

6)(1

.507

003)

Var

ianc

e1.

3993

961.

3749

271.

5574

61.

3979

823.

1723

21.

6005

81

Skew

ness

0.25

7265

-0.0

4935

10.

2687

390.

2540

12-0

.367

999

0.26

2476

Kur

tosi

s-0

.280

224

-0.4

8467

2-0

.357

337

-0.2

5976

923

.214

87-0

.350

145

E.

Shor

tf.

-1.5

1396

5-1

.492

499

-1.5

7162

6-1

.499

23-1

.661

645

-1.5

6987

1

Table

16.

Per

form

an

ceof

on

e-in

stru

men

th

edges

inth

eC

GM

Ye

mod

el.

CG

MY

eM

od

el–

Tw

o-I

nst

rum

ent

Hed

ges

Asi

anO

ptio

n–

21da

ys

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-0.2

7925

1-0

.288

776

-0.2

7449

5-0

.274

954

-0.2

8949

1-0

.274

938

(0.1

9262

3)(-

0.17

8108

)(0

.142

623)

(0.1

9692

)(-

0.17

8823

)(0

.142

18)

Var

ianc

e0.

1822

880.

1946

270.

1894

880.

1783

870.

1810

310.

1797

64

Skew

ness

-2.6

0064

9-2

.243

752

-2.6

0374

8-2

.485

291

-1.8

0842

9-2

.548

048

Kur

tosi

s12

.873

859.

5826

2714

.855

1912

.010

366.

4100

7212

.494

52

E.

Shor

tf.

-0.3

9633

6-0

.423

361

-0.3

9408

-0.3

9764

1-0

.435

625

-0.3

9556

3

Asi

anO

ptio

n–

126

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-1.0

9622

7-1

.147

918

-1.0

9623

8-1

.078

458

-1.1

4949

4-1

.078

506

(0.6

9471

3)(-

0.09

3978

)(0

.658

312)

(0.7

1248

2)(-

0.09

5554

)(0

.676

044)

Var

ianc

e0.

4537

390.

4363

950.

4520

30.

4552

490.

4356

70.

4528

04

Skew

ness

-0.0

2586

-0.0

9490

7-0

.025

832

-0.0

2728

1-0

.164

211

-0.0

2499

9

Kur

tosi

s-0

.333

5-0

.883

619

-0.3

2695

5-0

.341

379

-0.9

1214

6-0

.327

1

E.

Shor

tf.

-1.1

5718

2-1

.165

1-1

.156

345

-1.1

4624

5-1

.161

551

-1.1

4560

4

Asi

anO

ptio

n–

189

days

Loc

alC

alib

rati

onG

loba

lC

alib

rati

on

BS

HE

STSA

BR

BS

HE

STSA

BR

Exp

ect.

-1.2

6523

6-1

.301

552

-1.2

6925

7-1

.239

81-1

.298

382

-1.2

4050

3(1

.261

704)

(0.2

5455

8)(1

.285

673)

(1.2

8713

)(0

.257

728)

(1.3

1442

7)

Var

ianc

e1.

4416

991.

4269

361.

4692

331.

4405

171.

4120

821.

4621

24

Skew

ness

0.72

5445

0.57

7335

0.71

1782

0.72

591

0.62

8717

0.70

5167

Kur

tosi

s-0

.376

629

-0.6

1553

-0.4

0889

3-0

.378

698

-0.5

4568

-0.4

1501

5

E.

Shor

tf.

-1.7

0327

3-1

.714

818

-1.7

1311

2-1

.685

586

-1.7

1213

5-1

.690

83

Table

17.

Per

form

an

ceof

two-i

nst

rum

ent

hed

ges

inth

eC

GM

Ye

mod

el.


CG

MY

eM

od

el

–T

hree-I

nstru

ment

Hed

ges

Asi

an

Op

tion

–21

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

-0.2

41067

-0.2

56775

-0.2

59111

-0.2

64353

(-0.1

30399)

(0.1

60343)

(-0.1

48443)

(0.1

52765)

Vari

an

ce

1.0

87376

0.2

86053

0.2

78655

0.1

66926

Skew

ness

0.1

67799

-1.7

7225

-2.1

93163

-2.5

20188

Ku

rtosi

s10.2

5072

26.0

5432

9.3

74397

12.2

459

E.

Sh

ort

f.-0

.683668

-0.4

07926

-0.4

91184

-0.3

80927

Asi

an

Op

tion

–126

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

-1.0

85919

-1.0

96238

-1.0

24105

-1.0

96138

(-0.0

31979)

(0.6

58312)

(0.0

29835)

(0.6

58412)

Vari

an

ce

7.6

40608

0.4

5203

1.9

64392

0.4

52804

Skew

ness

0.5

03264

-0.0

25832

-0.5

3057

-0.0

24999

Ku

rtosi

s0.3

80529

-0.3

26955

24.6

6231

-0.3

271

E.

Sh

ort

f.-2

.396855

-1.1

56345

-1.3

95072

-1.1

56956

Asi

an

Op

tion

–189

days

Local

Cali

bra

tion

Glo

bal

Cali

bra

tion

HE

ST

SA

BR

HE

ST

SA

BR

Exp

ect.

-0.1

37116

-1.2

2485

-1.1

80707

-1.2

23684

(1.4

18994)

(1.3

3008)

(0.3

75403)

(1.3

31246)

Vari

an

ce

12.6

6709

1.4

24093

3.4

94497

1.4

19084

Skew

ness

0.0

44998

0.5

44462

-0.0

21402

0.5

41099

Ku

rtosi

s-0

.874073

-0.4

34139

11.9

5814

-0.4

39252

E.

Sh

ort

f.-2

.998364

-1.6

527

-1.8

57647

-1.6

50126

Table

18.

Per

form

an

ceof

thre

e-in

stru

men

th

edges

inth

eC

GM

Ye

mod

el.

model uncertainty and the robustness of hedging … · model uncertainty and the robustness of...

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