MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS DR TILL C. SCHRÖTER AND DR MICHAEL MONOYIOS

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1 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS DR TILL C. SCHRÖTER AND DR MICHAEL MONOYIOS University of Oxford Mathematical Institute St Giles Oxford OX1 3LB, United Kingdom Tel.: +44 (0) DR MARIO ROMETSCH AND PROF KARSTEN URBAN University of Ulm Institute of Numerical Mathematics Helmholtzstr Ulm Germany Tel.: +49 (0) 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 from hedging a payoff on the basis of an incorrect hedging model. The study is performed by hedging an Asian option in simulated real-world financial markets. These real-world markets are modelled by a stochastic volatility model with stochastic correlation, by a stochastic volatility model with jumps in the asset and the volatility, and by a jump-dominated Lévy model. The results of the study show that the use of more sophisticated hedging models is not in general warranted by a better hedging performance or by a higher robustness of these models to structural changes in the financial market. addresses: till.schroter@maths.oxon.org, monoyios@maths.ox.ac.uk, mario.rometsch@alumni.uni-ulm.de, karsten.urban@uni-ulm.de. 1

2 2 1. Introduction This paper is a contribution to the understanding of the robustness of classical option hedging schemes to model error. It is generally accepted that no parametric model can ever hope to capture perfectly the risk-neutral dynamics of underlying factors which generate the observed dynamic implied volatility surface in traded options markets. This means that an agent who dynamically hedges an option on the basis of some parametric model will almost inevitably face model risk: the risk 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 nonexchange-traded option, since an agent seeking to price such an option will choose parameters in a model (the so-called hedging model) which, at the time of trade, result in the model correctly pricing related exchange-traded options. Sometimes, for pricing purposes, no model at all is needed, as any sensible formula mapping parameters to prices can be used (see Figlewski [Fig02]). But for classical hedging, one is concerned with the full distribution of the underlying security until the derivative s expiry date and does indeed need a model in some shape or form. Hence, model risk becomes a much more serious issue. In this paper we are concerned with the model risk that arises from using a mis-specified hedging model, for an agent that is dynamically hedging a position in an exotic, non-exchange-traded option. One way to avoid model risk is to use completely model-free, static hedges, which occasionally can be found for contracts such as European vanilla options, barrier options or Asian options (see Carr and Wu [CW09], Cox and Ob lój [CO11] and Albrecher et al. [ADGS05], for example). These methods are promising but often give only wide bounds for super-hedges, or else (at present) are limited to a narrow class of claims. For these reasons, dynamic hedges based on a specific model formula are still very popular amongst financial practitioners and model risk is a serious issue for a wide class of claims. Naturally, market participants are interested in minimising their exposure to model risk and will therefore aim to use hedging models that are relatively robust with respect to different possible structures of the underlying market. To give a good hedging performance, a hedging model does not necessarily have to constitute a realistic model of the actual dynamics of the underlying asset as long as the model parameters sensibly capture parts of the market dynamics. For example, the Black-Scholes model, if used with a judiciously chosen volatility parameter, is known to be able to provide an effective dynamic hedge in a diffusion environment, even one involving stochastic volatility and other factors, as shown in El Karoui et al. [EKJPS98]. In general, it is not implausible that a relatively simple model, involving few free parameters, can perform well. The philosophy behind this idea is that since all models are wrong, and since all models need to be calibrated (and usually re-calibrated) to observed price data, a model in which only a few parameters need to be adjusted may work better than one in which 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 the hedging error distributions that result from dynamically hedging a path-dependent claim using the Black-Scholes [BS73], Heston [Hes93] and SABR [HKLW02] models. The hedges are performed in three simulated true market environments (which we refer to as market models) that are based on: (i) a three-dimensional Itô diffusion featuring stochastic volatility and correlation, (ii) a stochastic volatility model with jumps in both the stock price and the volatility and (iii) a jump-dominated Lévy process. In particular, we analyse the robustness of the hedging performance of the three hedging models to changes in the underlying market regime, and the effectiveness of the hedging models 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 models has been discussed by Hull and Suo [HS02] and An and Suo [AS09]. As few liquid markets for exotic options exist, model risk is a particular concern to the hedger of exotic, path-dependent instruments, 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

3 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 3 model s capability of approximating the full distribution of the underlying asset process. This motivates our choice of a path-dependent, arithmetic Asian option as the target option which we aim 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 a comprehensive 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 robustness of hedging models, but this subject is hard to tackle, in particular when the true or the mis-specified asset price process are not modelled by diffusions. When the true and the mis-specified asset price processes are diffusions, bounds on the hedging error arising from dynamic hedging schemes are computed by Corielli [Cor06] and [EKJPS98]. Boyle and Emanuel [BE80] study errors arising from discrete hedging in a Black-Scholes market, Anagnou-Basioudis and Hodges [ABH03] decompose the hedging error to investigate the effects of discretisation and volatility forecasting errors on the hedging performance, and Ahn et al. [AMS99] minimise hedging error in a model with misspecified volatility via a worst-case scenario approach. Gibson et al. [GLPT99] define model risk and give a more qualitatively focused account of its sources. Given the advantages of simulation studies, it is not surprising that these studies have been conducted frequently to study model risk in equity markets. Among the first, Figlewski [Fig89] investigates the effects that a misspecified volatility and market frictions have on hedges of European vanilla options in a market generated by the Black-Scholes model. Jiang and Oomen [JO01] are concerned with misspecified volatility parameters and study the impact of a wrongly selected hedging model on hedging performance. Specifically, they ask how well the Black-Scholes and Heston models hedge standard European options in stochastic volatility environments (with and without jumps). Hull and Suo [HS02] test different local volatility models by pricing and hedging exotic options in a stochastic volatility market. Coleman et al. [CKLV01] investigate how well local and implied volatility models hedge European vanilla options when the market data is generated by a non mean-reverting Ornstein-Uhlenbeck process. The question of whether volatility derivatives are suited to hedge European vanilla options in a stochastic volatility environment is examined in Psychoyios and Skiadopoulos [PS06]. Poulson et al. [PSHE09] simulate a stochastic volatility market to test the effectiveness of locally risk minimising strategies. The performance of semi-static hedges for 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 hedging strategies for European vanilla options in a stochastic volatility market that contains jumps in the underlying. These studies indicate that simple models such as the Black-Scholes model can sometimes be effective, and our study goes much further in confirming this broad conclusion. It would be of great interest to theoretically determine conditions under which a low-dimensional model can be an effective hedging tool in a higher-dimensional true market environment, and this would be an important future research challenge. The simulation study of this paper is based on a two-step procedure, outlined in [HS02]. In the first step, we generate 50, 000 realisations of the market data in each of the market models over a three month time-span. The market data comprises the stock price trajectories, the trajectories of all other stochastic factors that drive the market (e.g. stochastic volatility), and the option prices of 15 European vanilla options. These option prices are calculated along every trajectory and for every day. In a second step, we evaluate the hedging performance of the hedging models on the basis of the generated market data. This is accomplished by setting up a portfolio consisting of a stock and at most two European vanilla options to dynamically hedge the Asian claim. The hedging portfolio is rebalanced daily, using sensitivity parameters derived from the different hedging models. These hedging models are, at every rebalancing time, re-fitted (or re-calibrated) to the available market data. At the end of the three month period, the difference between the value of the hedging portfolio and the price of the Asian claim in the market model gives us the terminal hedging error along a

4 4 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS single trajectory. The hedging error distribution is obtained from the hedging error along each of the 50, 000 trajectories. Our simulation study is distinct from the related literature in several ways. (i) We model the market environments by a three-factor Markovian model, by a stochastic volatility model with jumps in the underlying and the volatility and by a Lévy process model. These models have not been used in previous studies of the same type and they span a wide range of potential market environments 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 opposed to 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, 000 market 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 is performed on the basis of how financial practitioners hedge, that is on the basis of frequently rebalanced portfolios and frequently recalibrated hedging models. The importance of this procedure for reasonable estimates of the hedging error has been pointed out in [HS02]. In particular, the frequent recalibration is computationally demanding. All computations have been performed on the UZWR cluster [UZW] consisting mostly of processors with a clock speed of 2.6 GHz. To complete the 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 seemingly simple and easy to implement Black-Scholes model achieves a hedging performance that is comparable to or better than any other hedging model in terms of minimal variance and near zero shortfall 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, in general, the best performing hedging models do not yield better results when they are used on the basis of globally calibrated parameters. This latter finding is, of course, a reflection of the fact that any model is bound to be mis-specified, and attempting to make it match all observed traded option prices will render it ineffective as a hedging tool. The best one can do is to make sure a model locally fits traded options of similar maturity and moneyness to the target option, and this maximises its hedging effectiveness. These findings illustrate that a parsimonious model, where the calibration issues are condensed into 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, given the sometimes more favourable expectation of the hedging errors in the Heston model, it seems that the Heston model should be used if many hedges of the same type are to be performed. In this case we can hope to realise the favourable expected error. However, when the goal is to conduct one hedge as accurately as possible, the Black-Scholes model is the preferred model. Finally, we study the robustness of the previous findings. In particular, we find that the Heston model performs more strongly than the Black-Scholes model in environments that are characterised by a high kurtosis of the market returns. From a numerical perspective, our contribution is to show how the efficient simulation of an equity market 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 option more than times to simulate just one single hedging error distribution. To achieve this with reasonable computational effort, we make use of a reduced model by employing a proper orthogonal decomposition 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 probabilistic setting. Section 3 contains a discussion of the market models that describe the underlying 1 This finding illustrates the importance of assessing the full distribution of the hedging error. If this study, like most previous studies, were based on the expected error or the variance only, the Heston model or the Black-Scholes model respectively would have delivered the best hedging performance. Both results would be slightly misleading.

5 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 5 true market environments. The different hedging models are reviewed in Section 4. The methodology 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 physical measure. 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 filtration F = (F t ) t T, such that F T F. F is assumed to satisfy the usual conditions of right-continuity and completeness. By W P = (W P,1,..., W P,n ), we denote an n-dimensional (n N) standard P - Brownian motion on the filtered probability space and N = (N t ) t T stands for a one-dimensional Poisson process on that stochastic basis. The risk-neutral pricing measure is denoted by Q. This measure is not necessarily unique and will be fixed in Section 3 for different market models. A Brownian motion under Q is written as W Q = (W Q,1,..., W Q,n ). For a local martingale M, M 0 = 0, with quadratic variation [M], the stochastic exponential is introduced via ) ([M] E(M) st = e Mt Ms 1 2 t [M] s, 0 s t T, and we write E(M) t = E(M) 0t. For any given process (U t ) t T we frequently write U = (U t ) t T to denote the entire process. The stock that underlies all derivatives in the market is modelled by the F-adapted process S. Finally, the interest rate is constant and set to r = Throughout this study, we encounter two sorts of European derivatives. First, a fixed strike Asian option that depends on the average price of the underlying stock during its lifespan and whose price at t T is given by (1) C A t = e r(t A t) E Q [ ( 1 T A Ỹ TA K A ) + F t ] where T A T denotes the maturity, K A R + the strike price and Ỹ t = t 0 S u du. Second, European vanilla calls that serve as hedging instruments and are needed for calibration purposes. Their prices at t T are given by C t = e r(t t) E Q [ (S T K) + F t ], 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 denote the former situation and the superscript H to identify the latter scenario, for example, we write C M for the price of a European vanilla call in the market model. The only exception to this is the price of an Asian option, where we write C A M and CA H. 3. The Market Models The market models have two distinct functions in this study: (i) they are used to simulate the trajectories of the factors that drive the market and (ii) they provide the prices of the European vanilla and Asian options which constitute the derivative data in the market. For this reason, the market models are specified under the physical and the risk-neutral measure and their parameters are chosen to reflect pertinent time series and option price data. The equity market and equity market models have been studied widely in the literature. Therefore, several approaches exist to analyse the structure of this market. One popular approach to obtain information on the factors that drive the equity market is to study the properties of European vanilla option prices via the related implied volatility surface. On the basis of a principal component analysis of the implied volatility surface, such studies have been performed, for example,,

6 6 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS by Skiadopoulos et al. [SHC99], Cont and da Fonseca [CdF02] and Fengler et al. [FHS02]. Depending on the focus and the methodology, these studies show that 70 90% of the surface movements can be explained by only two to three independent stochastic driving factors. These findings are somewhat reflected in our choice of a market model that is driven by three independent diffusion factors (Subsection 3.1). Alternatively, the fit of different models has been examined in terms of their pricing and hedging performance in the equity market and in terms of their potential to replicate the statistical properties of financial time series data. Several studies document the importance of jumps and stochastic volatility for a good pricing and hedging performance. By evaluating out-of-sample pricing and hedging errors of S&P 500 options, Bakshi et al. [BCC97] demonstrate that, within the framework of their study, stochastic volatility models with jumps in the underlying perform best when it comes to pricing options and that stochastic volatility models are of first-order importance for minimising the hedging error. Comparable results are also given in Pan [Pan02] and [Bat00], who additionally point 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 process but also as part of the stochastic volatility. Models with jumps in stochastic volatility seem to perform best in mimicking market dynamics. This has been attributed to the greater persistence of jumps in the volatility process which, contrary to transient jumps in the underlying, actively change the distribution of the underlying over periods of time, see Eraker et al. [EJP03], Chernov et al. [CGGT03] and Eraker [Era04]. However, [Era04] finds that the improved time series fit of models with jumps in the volatility does not carry on to a better out-of-sample pricing performance of these models for European vanilla options. The importance of jumps in the underlying and the volatility 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 trajectories must be regarded as an approximation to the true price process. To account for that behaviour, we have included the jump-based CGMYe model of Carr et al. [CGMY02] as third choice of a market model. Below we discuss the market models in greater detail The Three-Factor Model. The three-factor model (3F model, 3FM) is a stochastic volatility model whose correlation is driven by an additional stochastic variable. It therefore consists of three independent factors driving the market. Das and Sundaram [DS99] show that a negative correlation between an asset and its volatility leads to the volatility smile being tilted. From that point of view, a stochastic correlation effectively models a stochastic skew in the implied volatility surface. In the three-factor model, the asset S follows an Itô diffusion, the variance v is described by a Cox-Ingersoll-Ross process and the correlation ρ is modelled by the translation of a Jacobi 2 process. For M {P, Q}, the dynamics of the three-factor model are (2) ds t = µ M S t dt + v t S t dw M,1 t, dv t = α M (β M v t )dt + σ v vt dbt M, dρ t = κ M (λ M ρ t )dt + σ ρ 1 ρ 2 M,3 t dwt, where dbt M = ρ t dw M,1 t + 1 ρ 2 M,2 t dwt. 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 ρ t = 2J t 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 describe the SDEs under the physical and risk-neutral measures are given in Table 1. 2 The Jacobi process, J, is a mean-reverting process that solves the SDE djt = a(b J t)dt + c J t(1 J t)dw t, for a, b, c > 0 and Brownian motion W.

7 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 7 P - Physical Measure µ P α P β P σ v κ P λ P σ ρ Q - Risk Neutral Measure µ Q α Q β Q σ v κ Q λ Q σ ρ Table 1. The annualised values of the modelling parameters under P and Q in the three-factor model. The asset and volatility parameters are taken from Chernov and Ghysels [CG00]. The correlation parameters are chosen such that the correlation dynamics do not not change under the measures P and Q. This choice is somewhat arbitrary but feasible as we are only interested in a reasonable setting 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 parameter estimates from market prices do not always satisfy the theoretical conditions that guarantee the existence 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 the three-factor model in greater detail The Stochastic Volatility Model with Jumps in Stock and Volatility. Market models with jumps in volatility have been suggested in [Bat00], [Pan02] and others after it became clear that jumps in the asset price alone are not sufficient to model a market correctly. The rationale for 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, but the impact of a jump is transient: a jump in returns today has no impact on the future distribution of returns. On the other hand, diffusive volatility is highly persistent, but its dynamics are driven by a Brownian motion. For this reason, diffusive stochastic volatility can only increase gradually via a sequence of small normally distributed increments. Jumps in volatility fill the gap between jumps in returns and diffusive volatility by providing a rapidly moving but persistent factor that drives the conditional volatility of returns. In the stochastic volatility model with jumps in stock and volatility (SVJJ model), the market is driven by the log-stock price L, L = ln S, and the variance v. For M {P, Q}, the dynamics of the factors in the SVJJ model are (3) dl t = ( µ M 1 2 v t ) dt + v t dw M,1 t + ξ r,m dn t, dv t = α M (β v t )dt + σ v vt db M t + ξ v dn t, where dbt M = ρdw M,1 t + 1 ρ 2 dw M,2 t for some constant ρ [ 1, 1]. In (3), N is a Poisson process with constant jump arrival intensity λ N > 0, identical under any measure. The parameters that describe the dynamics of L and v under the physical and the risk-neutral measures are given in 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 µ v ) and by a normally distributed random variable, that is ξ v exp (µ v ), ξ r,m ξ v N(µ M r + ρ J ξ v, σ 2 r).

8 8 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS P - Diffusion Parameters µ P α P β σ v ρ P - Jump Parameters µ P r σ r ρ J µ v λ N Table 2. The annualised values of the modelling parameters under P in the SVJJ model. This jump specification accounts for two empirical observations. First, volatility does not become negative. This is accounted for by the exponential distribution. Second, jumps in volatility and the underlying often occur contemporaneously, as a crash in the underlying usually leads to a subsequent increase in volatility. This observation is reflected in the choice of a single jump process N that drives both factors. In the SVJJ model, a semi-closed form formula exists for the price of a European vanilla option. This attractive feature has been developed in Duffie et al. [DPS00]. Q - Diffusion Parameters µ Q α Q β σ v ρ Q - Jump Parameters µ Q r σ r ρ J µ v λ N Table 3. The annualised values of the modelling parameters under Q in the SVJJ model. Note that the risk-neutral drift parameter is given by µ Q = r λ N (θ(1, 0) 1) (for details and notation see (4.5) in [DPS00]) The CGMYe Model. To account for the diffusion-like behaviour of small, daily price variations, we previously introduced a diffusion and a jump-diffusion model. Trading, however, takes place 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, Drăgulescu and Yakovenko [DY02]) and may contain some jumps, other classes of processes offer greater flexibility in modelling the probability distributions and the path properties of stock prices. Popular, jump-dominated processes introduced to model the behaviour of stocks are the CGMY 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 Lévy processes. The Lévy density of the distribution that underlies the CGMY process is given by C M exp ( GM x ) for x 0, x k CGMY (x) = 1+Y M C M exp ( M M x ) for x 0, x 1+Y M for parameters C M > 0, G M 0, M M 0 and Y M < 2. The first three of these parameters influence the skewness and kurtosis of the distribution of the CGMY process and the parameter Y M controls the jump behaviour of that process. For a CGMY process X and corresponding

9 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 9 density k CGMY (x), the characteristic function is explicitly known and given by E [ e iuxt] ( { = exp t e iux 1 } ) k CGMY (x)dx ( [ (M = exp tc M Γ( Y M ) M iu ) ]) Y M (M M ) Y M + (G M + iu) Y M (G M ) Y M =: φ CGMY ( u, t; C M, G M, M M, Y M), 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 of the CGMY process X and an additional Brownian motion W M,1 independent of X, i.e. (4) S t = S 0 exp ((µ M + ω M 12 ) ) ηm t + η M W M,1 t + X t (C M, G M, M M, Y M ), with ω M defined by e ωm t = φ CGMY ( i, t; C M, G M, M M, Y M). The parameters that describe the dynamics of the asset price under the physical and risk-neutral measures are given in Table 4. These parameter choices have been made on the basis of [CGMY02]. P - CGMYe µ P C P G P M P Y P η P Q - CGMYe µ Q C Q G Q M Q Y Q η Q 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 the dynamics of CH A, 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 Itô diffusion X and that the market contains d traded assets. Then, the price of any asset in the hedging model at time t T can be written as A H lt = AH l (t, X t ), l {1,..., d}, where A H l (t, x), x R m, denotes the pricing function of the l-th asset in the hedging model. By Itô s formula, the asset dynamics in the hedging model are (5) da H lt = dt + AH lt d X t, while those of the Asian option can be described by (6) dc A Ht = dt + CA Ht d X t. Also, the market contains a bank account, B, that satisfies the differential equation db t = rb t dt.

10 10 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS In (5) and (6), we omit the locally risk-free components of the asset dynamics as they play no role in 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 A H lt = ( x1 A H l (t, X t ),..., xm A H l (t, X t ) ). The hedging portfolio Π consists of the assets A and the bank account B, d (7) Π t = Ψ B t B t + Ψ l ta lt, where Ψ B denotes the amount of money in the bank account. The number of shares held in asset l is recorded in the l-th entry of the vector Ψ R 1 d. As we are only concerned with self-financing strategies, the portfolio Π has the dynamics d dπ t = rψ B t dt + Ψ l tda lt, for Ψ B = Π d l=1 Ψl A l. By (5), the portfolio evolution can be written as (8) dπ t = dt + Ψ t l=1 l=1 A H 1t. A H dt d X t. To replicate the dynamics of CH A with the portfolio Π, a comparison of (8) with (6) shows that the vector of portfolio weights, Ψ, must solve the linear system (9) CHt A = Ψ t A H 1t. A H dt This system of equations is solvable, whenever the matrix in (9) is invertible. The general solvability of (9) is a non-trivial question. Romano and Touzi [RT97] discuss this question for a stochastic volatility model. Some results for more general m-factor Itô diffusions are obtained in [Dav04]. The hedging instruments in our simulation are the stock S and two European vanilla options C 1 and C 2. This corresponds to setting A 1 = S, A 2 = C 1 and A 3 = C 2. Then, the portfolio (7) becomes Π = Ψ 1 S + Ψ 2 C 1 + Ψ 3 C 2 + Ψ B B. The market models are based on a maximum of three stochastic driving factors. Therefore, in the hedging 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 X 1 = S, X2 = v and X 3 = ρ. With that choice, (9) becomes explicitly 3 S C A Ht v C A Ht ρ C A Ht =. 1 S C1t H S C2t H 0 v C1t H v C2t H 0 ρ C1t H ρ C2t H Ψ1 t Ψ 2 t Ψ 3 t Clearly, if this equation has a solution it is S C (10) Ψ1 Ht A Ψ2 t S C1t H Ψ3 t S C2t H t Ψ 2 vcht t = A vch 2t Ψ3 t Ψ 3 vc1t H t ρcht A vch 1t vca Ht ρch 1t vc1t H ρch 2t ρch 1t vch 2t 3 Sometimes, the first derivatives of a pricing formula with respect to the underlying asset and the volatility are called delta and vega...

11 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 11 If we hedge with less than three instruments, the corresponding portfolio is obtained by setting the redundant portfolio weights to zero. For instance, if we hedge with the stock only, we set Ψ 2 = Ψ 3 = 0 and therefore have Ψ 1 = S CH A. Although the Black-Scholes model is driven by only one stochastic factor and the Heston and SABR models by two stochastic factors, the portfolio (10) has been set up to always accommodate a 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 to eliminate all risk in the market. It is, however, difficult to exactly isolate the risk factors that drive a stock price or a derivative. As a result, no hedging model accurately reflects the dynamics of the real-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 static in 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 are calculated as outlined in (10). This hedge would not be necessary if the hedging model were to be a good 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 volatility parameter 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 the sources of risk it recognises. For example, even in the framework of the Black-Scholes out-of-model hedge, the hedger only takes into account price changes of the target option that can be explained by changes in the underlying stock and changes in the implied volatility surface. If the price of the target option is also affected by other sources of risk, say a change in the curvature of the implied 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 for calibration purposes and to set up the hedging portfolios. It is therefore sufficient to specify them under the risk-neutral measure Q The Black-Scholes Model. In the model of Black and Scholes [BS73], the risk-neutral dynamics of the stock price are given by (11) ds t = rs t dt + σs t dw Q,1 t, S 0 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 s strike 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 our study. To set up a hedging portfolio, as outlined in (10), we need the partial derivatives of the pricing formulae for Asian and European vanilla options with respect to the underlying factors. For a European vanilla option, [BS73] establish a closed-form pricing formula. Hence, the respective partial derivatives can be easily calculated. No closed-form formula exists for the price of an Asian option in the Black-Scholes model. Therefore, prices and partial derivatives are obtained as solutions to a suitable PDE. We briefly present this PDE in Subsection 5.3.

12 12 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 4.2. The Heston Model. The model of Heston [Hes93] is a two-factor model of the stock, S, and its variance, v. In this model, the risk-neutral dynamics of the underlying factors are given by (12) (13) ds t = rs t dt + v t S t dw Q,1 t, S 0 R +, dv t = ν(φ v t )dt + ϕ v t db Q t, v 0 R +, where db Q t = ρdw Q,1 t + 1 ρ 2 dw Q,2 t 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 ϕ 2. For European vanilla options in the Heston model, [Hes93] establishes the existence of a semi closed-form pricing formula. This allows for simple calculation of the European vanilla option prices and of the related partial derivatives. No closed-form pricing formula exists for an Asian option in the Heston model. In Subsection 5.3, we therefore present a PDE that allows us to obtain the prices of the Asian options in the Heston model numerically 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 F t = S t e r(t t). Under the risk-neutral measure, the factors in the SABR model solve the SDEs (15) (16) df t = v t F β t dw Q,1 t, F 0 R +, dv t = ϕv t db Q t, v 0 R +, where db Q t = ρdw Q,1 t + 1 ρ 2 dw Q,2 t 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 therefore become very high. This unrealistic property highlights in particular that the SABR model is not intended as a realistic model of the market dynamics but aims to provide a useful parametrisation that relates observed prices to hedging parameters. The SABR model has been designed to provide a good fit to market data and to allow for convenient pricing formulae on the basis of a functional relationship (see [HKLW02] or Ob lój [Ob l08]) between the SABR parameters and the implied volatility surface. In fact, in terms of the forward price F t, the volatility level v t and the time-to-maturity τ at t T, the implied volatility surface can be approximated through the SABR parameters via (17) σ IV (K, τ; F t, v t ) ln ϕx(k; F t ) ( 1 2ρz(K;Ft)+z 2 (K;F t)+z(k;f t) ρ 1 ρ ) ( { ρϕvt } ) 3ρ2 ϕ 2 τ, 24 where x = ln (F t /K) denotes the log-moneyness and z = xϕ/v t. On the basis of (17), the option prices and corresponding partial derivatives in the SABR model can be calculated via the pricing formulae 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 in hedging Asian options. The hedging performance is evaluated over the hedging interval [t 0, T H ], with t 0 T H T A, and measured in terms of the hedging error. At any given time t T, the hedging error, ε, is defined as the difference between the market price of the Asian option C A M, and the value of the hedging portfolio Π, that is (18) ε t = Π t C A Mt.

13 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 13 The terminal hedging error, ε TH, is made up of a sequence of one-step hedging errors, ε i, that measure the contribution of the period (t i 1, t i ] (a daily interval in our study) to the terminal error, i.e. ε i = ε ti ε ti 1 = ( Π ti C A Mt i ) ( Πti 1 C A Mt i 1 ). In terms of the ε i, the terminal hedging error that results from revising the portfolio N N times can be written as N (19) ε TH = Π TH CMT A H = ε i + Π t0 CMt A 0. By (19), the hedging error ε TH depends on the one-step errors ε i, i = 1,..., N, and on Π t0 CMt A 0, the difference between the market price of the Asian option and the initial value of the hedging portfolio. Since this difference is F t0 -measurable, its value only impacts the expected hedging error and leaves all higher central moments of the hedging error distribution unchanged. When the hedging portfolio is started at Π t0 = CMt A 0, it follows from (19) that the terminal hedging error can be written as the sum of the one-step hedging errors only. Instead, when the initial value of the portfolio is Π t0 = CHt A 0, the terminal hedging error is composed of the sum of the one-step hedging errors and the additional term CHt A 0 CMt A 0. This difference reflects the degree to which the Asian option is mispriced by the given hedging model. Therefore, when Π t0 = CMt A 0, the terminal hedging error can be seen as a measure of the hedging model s pure hedging performance. Otherwise, when Π t0 = CHt A 0, the terminal hedging error must be understood as a measure of the hedging model s joint pricing and hedging performance. For example, situations in which the hedging portfolio is started at Π t0 = CHt A 0 occur when a bank sells a non-traded Asian derivative. In this case, the bank typically prices the non-traded claim with some hedging model and subsequently uses the same hedging 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 hedging error. As the price of the Asian option is known in the market models (by means of Monte Carlo simulations), 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 expected errors of the hedges performed in the different hedging models. There, the expected errors of hedges started 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, average hedging errors of different hedging models are compared to rank these models. For instance, 1 [BCC97] evaluate the average of M N one-step hedging errors, M M i=1 ε i, and the same measure 1 in absolute terms, M M i=1 ε i. However, the average (or expected) hedging error as the sole measure of a model s hedging performance is somewhat misleading, as different hedging models can give rise to hedging errors with similar expectations but strongly different variations around the mean. Instead of the expectation, [BKSS12] suggest to evaluate the variance of the one-step hedging error. 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 models give rise to hedging errors with similar variances we lack the information to compare these models further. Therefore, the hedging error should be evaluated on the basis of several moments of the hedging 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 expectation of the hedging error conditional on a negative hedging performance, i.e. E [ε TH ε TH < 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 t 0 = 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 t 0. We also assume that a year consists of 252 working days. Some hedging models are better suited than others to price and hedge options of longer or shorter maturities. For example, it has been observed that the Heston model tends to underprice i=1

14 14 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS Option Characteristics Asian Calls T A /K A 21/ / /50.0 European Vanilla Calls T/K 21/ / /47.0 T/K 21/ / /48.5 T/K 21/ / /50.0 T/K 21/ / /51.5 T/K 21/ / /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 these models by hedging Asian options of different maturities. For this reason, we hedge Asian calls who fit in one of the three categories: short term (days to expiration < 60), medium term (days to expiration [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 strike level of K A = 50. The European vanilla calls that constitute the option market data also mature after 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 the stochastic volatility. Third, we hedge with three hedging instruments, the stock and two European vanilla options, against changes in the values of the underlying, the stochastic volatility and the correlation. The two-instrument hedge is performed in all hedging models and, when performed in the Black-Scholes model, is an out-of-model hedge. The three-instrument hedge is only performed in the Heston and SABR models and always an out-of-model hedge. Whenever we hedge with one European vanilla option, the 126/50 4 -European vanilla call is used as the hedging instrument. Whenever we hedge with two European vanilla options, the 126/50-European vanilla call is used as the first and the 189/50-European vanilla call is used as the second hedging instrument. The hedging period consists of three months, i.e. T H = 63 days, for the medium and long term Asian option. The short term Asian call is hedged until its expiry, hence T H = 21 days. We do not hedge the medium term and the long term Asian option until their expiry dates, as this allows us to perform the simulations considerably faster. Also, since we are comparing hedging models in this study, it is mainly the difference between the hedging performances of the hedging models that interests us. By the previous choice of the hedging periods, we evaluate this difference after 21 days for hedges of the short term Asian option and after 63 days for hedges of the medium and long term Asian options. The basis study evaluates the performance of the hedging models when calibration is conducted locally. Local calibration means that the parameters of the hedging models are calibrated against the prices of the five European vanilla options that have the same expiry date as the Asian call we are hedging. The calibration is performed on every day of the hedge. 4 This 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.

15 MODEL UNCERTAINTY AND THE ROBUSTNESS OF HEDGING MODELS 15 To examine whether the calibration method has an impact on the performance of the hedging models, we also study the hedging error on the basis of globally calibrated parameters. Global calibration means, that the parameters of the hedging models are calibrated against the prices of all 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 the distribution 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 the Tables 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 the standard behaviour but instead exhibit frequent and strong up or downward movements which are mainly reflected in a considerably higher kurtosis of the market returns. To test whether the results of our hedging study are robust with respect to different market parameters, we reperform parts of our analysis with different parameters. In the 3FM market, a higher kurtosis can be obtained by increasing the volatility constants σ v and σ ρ while keeping α M, κ M, β M and λ M relatively low. Specifically, we posit σ v = 3, λ P = λ Q = 0.3 and leave all other parameters in Table 1 unchanged. By this choice, we increase the kurtosis and also ensure a somewhat stronger impact of the correlation on the dynamics of the 3FM market. In the SVJJ market, we tweak the kurtosis by increasing the frequency and magnitude of the jumps. This leads to the following new parameter values: λ N = 3, µ v = 0.12, σ r = 0.494, µ P r = 0.15 and µ Q r = All other entries in the Tables 2 and 3 remain unchanged. In the CGMYe market, the hedging errors are of high magnitude and all hedging models are more or less equally suited (or unsuited) for hedges in this market. Therefore, subtle differences in the performance of hedging models under different market parameters are comparatively meaningless given the overall magnitude of the hedging error. For this 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. In principle, the implementation is divided into three steps. In a first step, we generate the market data. In a second step, we calibrate the hedging models and in a third step, we calculate the hedge ratios and the hedging errors Market Data Generation. For each market model, we first simulate the daily values of the physical trajectories 5 of the underlying risk factors over the hedging horizon and generate the European vanilla and Asian call prices. In total, we simulate trajectories. The trajectories of the underlying risk factors in the 3FM model are started at the values S t0 = 50, v t0 = and ρ t0 = The starting values in the SVJJ model are given by S t0 = 50 and v t0 = and in the CGMYe model we use S t0 = 50. The prices of the European vanilla options are calculated along every physical trajectory for every day. The prices of the Asian calls are also calculated along every physical trajectory, but only on the initial and final day of the hedging horizon. The Asian call 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 SVJJ model, 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 trajectories on the basis of an algorithm by Tankov [Tan10]. One of the difficulties in simulating the variance SDEs is to ensure that they do not become negative. In theory, this can be achieved by imposing the 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 we observe a negative variance value, this value is replaced by zero. How discretised variance SDEs should be treated at their boundaries without distorting the distributional properties too much has been studied in Lord et al. [LKvD09]. 5 By this, we mean the trajectories under the physical measure P.