Hedging under Model Mis-Specification: Which Risk Factors Should You Not Forget?

Transcription

1 Hedging under Model Mis-Specification: Which Risk Factors Should You Not Forget? Nicole Branger Christian Schlag Eva Schneider Norman Seeger This version: May 31, 28 Finance Center Münster, University of Münster, Universitätsstr , D Münster, Germany. Finance Department, Goethe University, Mertonstr /Uni-Pf 77, D-654 Frankfurt am Main, Germany. (corresponding author) Earlier versions of this paper were presented at the 11th conference of the Swiss Society for Financial Market Research in Zurich 28, and the Workshop on Finance, Stochastics and Insurance in Bonn 28. The authors would like to thank the conference participants and discussants for useful comments and suggestions.

2 Hedging under Model Mis-Specification: Which Risk Factors Should You Not Forget? Nicole Branger Christian Schlag Eva Schneider Norman Seeger This version: May 31, 28 Abstract Option pricing models that include stochastic volatility and/or stochastic jumps are often hard to distinguish from each other based on the prices of European plain-vanilla options only, where one reason are rather high bid-ask spreads. We first analyze the hedging error induced by this model mis-specification. This impact of model risk is economically significant. We find that it is largest for delta-vega and smallest for minimum variance hedges. Furthermore, we show that a simple BS ad-hoc hedge is biased towards a minimum-variance hedge, which explains its surprisingly good performance. Second, we analyze whether hedging errors can help in identifying model mis-specification and ideally also the correct model. We show that there are substantial differences between realized hedging errors (if the incorrect hedge model is applied to the true data-generating process) and anticipated hedging errors (if the hedge model were applied to prices generated by the hedge model), in particular depending on whether or not stochastic volatility is included in the hedge model. Hedging errors can thus provide useful support in model identification. Keywords: Hedging, Model Risk, Model Identification, Delta-Hedge, Delta-Vega Hedge, Minimum-Variance Hedge JEL: G13 Finance Center Münster, University of Münster, Universitätsstr , D Münster, Germany. nicole.branger@wiwi.uni-muenster.de Finance Department, Goethe University, Mertonstr /Uni-Pf 77, D-654 Frankfurt am Main, Germany. schlag@finance.uni-frankfurt.de, schneider@finance.uni-frankfurt.de, seeger@finance.uni-frankfurt.de (corresponding author) Earlier versions of this paper were presented at the 11th conference of the Swiss Society for Financial Market Research in Zurich 28, and the Workshop on Finance, Stochastics and Insurance in Bonn 28. The authors would like to thank the conference participants and discussants for useful comments and suggestions.

3 1 Introduction and Motivation State-of-the-art option pricing models include one or several stochastic volatility factors, stochastic jumps in returns, and sometimes also stochastic jumps in variance. 1 Although the dynamics of stock prices can be rather different, the models can produce quite similar prices for European plain-vanilla options and be calibrated to the volatility smile. Consequently, they are often hard to identify empirically from a cross section of those prices only, which leads to model risk. In this paper, we first analyze the impact of this model mis-specification on the hedging performance of the models when European options are hedged. Second, we study whether hedging errors provide additional information for model identification. Exact model identification is hindered by noisy prices, e.g. due to bid-ask spreads. As argued by Dennis and Mayhew (24), we cannot distinguish between two models if the maximal pricing difference is smaller than the noise in the data. Using empirically observed prices and bid-ask spreads, they demonstrate that this problem may already arise for the models of Black-Scholes and Merton. This would not be a problem if differences between models never mattered. However, these differences could (and usually will) become important in the pricing of non-redundant derivatives like exotic options, in the process of hedging derivative positions, or in portfolio planning. The first contribution of our paper is to analyze the distribution of hedging errors in case of a mis-specified model. We focus on the (in our opinion most realistic) case of omitted risk factors, i.e. we assume that the hedge model incorrectly used by the investor contains less risk factors than the true data-generating process. In our simulation study, the true model is given by Bakshi, Cao, and Chen (1997) with stochastic volatility and jumps. The hedge model, however, does not contain a stochastic volatility component, or a stochastic jump component, or both. The resulting hedge models (Black and Scholes (1973), Merton (1976), and Heston (1993)) are calibrated to a cross section of option prices 1 See e.g. Merton (1976), Heston (1993), Bates (1996), Bakshi, Cao, and Chen (1997), Bates (2), Duffie, Pan, and Singleton (2), Carr and Wu (24). 1

4 such that the maximal absolute pricing difference is minimized. Given the calibrated models, the investor then implements a delta-hedge, a delta-vega hedge, and a local minimum variance hedge. The resulting hedging errors over the next time interval are obtained via Monte Carlo simulation. We also simulate the hedging errors when the correct model is used for hedging, which arise due to discrete trading and market incompleteness. They serve as the benchmark to which the hedging errors in case of model mis-specification are compared to. This setup allows us to assess whether the inclusion of stochastic volatility or of a jump component is more important when it comes to hedging. It also allows us to answer the question which hedging strategy is most sensitive to model mis-specification. The second contribution of our paper is to analyze whether hedging errors provide additional information for model identification beyond that contained in the cross section of option prices. For each hedge model and each hedging strategy, we compare the realized hedging errors already calculated above to the hedging errors under the null hypothesis that the hedge model describes the true data-generating process. An investor who believes in the calibrated hedge model anticipates to see exactly these hedging errors. Any statistically significant difference between these anticipated hedging errors and the realized hedging errors then indicates that the hedge model is mis-specified. Following this line of argument, hedging errors for plain-vanilla options provide additional information beyond that contained in the cross section of prices. To get the intuition, note that the prices of plain-vanilla options only depend on the terminal distribution of the price of the underlying. Hedging errors, on the other hand, depend on the joint dynamics of the underlying, the state variables and the price of the derivative over the next time interval (see Bates (23)). Thus, hedging errors indeed contain information that is not reflected in the cross-section of prices. We now give a brief overview of the main findings, where we start with the hedging performance in case of model mis-specification. For the delta-hedge, the hedge based on the hedge model is the closer to the true hedge based on the data-generating model the better the hedge model fits the true smile, which confirms the result of Bates (25). The ad-hoc Black-Scholes delta-hedge, which is based on individual option implied volatilities, 2

5 performs surprisingly good and may even lead to a lower standard deviation of the hedging errors than the true model. We show that this puzzling result can be explained by the fact that the ad hoc extension of the Black-Scholes model actually results in delta hedges which are biased towards minimum variance hedges. Naturally, this improves the quality of the hedge when hedging performance is measured via the standard deviation of hedging errors. For the minimum-variance hedge and in particular for the delta-vega hedge, model mis-specification matters more than for the delta-hedge. The ad-hoc Black-Scholes model, where both stochastic volatility and stochastic jumps are ignored, now performs significantly worse than the other hedge models for most combinations of time to maturity and moneyness, and it may even be worse than the (simpler) delta-hedge. The more sophisticated the hedging strategy is, the more the performance of the hedge thus depends on the hedge model really capturing the main properties of the data-generating process. Furthermore, the performance of the hedge is improved more when we include the state variable stochastic volatility in the hedge model than when we allow for jumps in prices. Furthermore, we find that hedging errors indeed contain useful information for model identification. For all hedge models and all hedging strategies we consider, the distribution of the realized hedging errors differs significantly from the distribution as anticipated by an investor who believes in his hedge model. The omission of stochastic volatility leads to realized hedging errors that are significantly more extreme than those anticipated by the investor. In particular, hedging errors are no longer bounded from below (or above) if volatility can change over time. Stochastic jumps, on the other hand, are harder to identify. The differences between the distributions of hedging errors are moderate during normal times, but extreme as soon as a jump takes place. We are not the first to analyze issues in model identification. The paper closest to ours is Dennis and Mayhew (24), who study the impact of noise in option prices on parameter estimation. They explicitly ask whether, given noisy prices, one can distinguish between a pure jump model and Black-Scholes. Different from their paper, we also include models with stochastic volatility. Furthermore, we focus on the economic significance of 3

6 model mis-specification for hedging to see whether an investor should care about the problems in model identification and show that hedging errors might indeed help to solve this problem. Schoutens, Simons, and Tistaert (23) calibrate stochastic volatility models, jump diffusion models, and Lévy models to a cross-section of plain-vanilla European options and then use these models to price exotic options. Despite the fact that basically all models fit the given prices equally well, they still obtain substantial differences in the prices of exotic derivatives. In contrast to these authors, our focus is on hedging, which represents a different way to judge if the assumed time series dynamics for the state variables are appropriate. Additionally, we are interested in the use of hedging errors for the purpose of model identification. An and Suo (23) test several option pricing models by analyzing the performance of hedging strategies for exotic derivatives. They recalibrate their hedge models daily, and find that the Black-Scholes model performs worst when measured by its in-sample fit. However, it still represents a very good and sometimes even the best choice for hedging for a wide range of hedging strategies. Compared to them, we work in a controlled simulation environment, whereas the analysis in An and Suo (23) is subject to the simultaneous problems of mis-priced European options and incorrect model assumptions. Finally, Poulson, Schenk-Hoppé, and Ewald (27) study the performance of delta-hedges and local minimum-variance hedges both in a simulation study and based on empirical data. They show that it is of first-order importance to use a model that allows for stochastic volatility, while the exact process assumed for this stochastic volatility matters much less. Different from their analysis, our focus is on the relative importance of stochastic volatility and stochastic jumps and on the information content of hedging errors. The reminder of the paper is organized as follows. The option pricing model and the hedging strategies are introduced in Section 2, while Section 3 describes the calibration of the hedging models and the methodology of the simulation study. Section 4 gives the numerical results for the performance of hedges in case of model mis-specification. The use of hedging errors to identify model mis-specification is discussed in Section 5. Section 6 concludes. 4

7 2 Option Pricing Models and Hedging Strategies 2.1 Model Setup The true option pricing model in our paper is given by a restricted specification of the model in Bakshi, Cao, and Chen (1997). We assume a constant interest rate, and the resulting model can also be seen as a special case of Bates (1996) with only one volatility factor. The dynamics of the stock price S and its local variance V are given by the following system of stochastic differential equations: ds t S t = ( r + η V V t + λ P E P [ e Xt 1 ] λ Q E Q [ e Xt 1 ]) dt + V t dwt S + ( e Xt 1 ) dn t E [ P e Xt 1 ] λ P dt ( dv t = κ v ( v V t ) dt + σ v Vt ρdwt S + ) 1 ρ 2 dwt V The jump size distribution is log-normal, i.e. X t N(ln(1 + µ J ).5σJ 2, σ2 J ), and the jump intensity under P is given by λ P. Other models are nested as special cases in our specification. Setting σ v =, λ P = λ Q =, and V = v = σ 2, where σ is the volatility of the stock price, yields the Black and Scholes (1973) model. Excluding the jump component by restricting the jump intensities to λ P = λ Q = results in Heston (1993), and eliminating the stochastic volatility part via setting σ v = and V = v = σ 2 yields the Merton (1976) model, where σ stands for the diffusion-based volatility of the stock price. With respect to model identification, it is important to note that both stochastic volatility (SV) and stochastic jumps (SJ) are able to generate excess kurtosis and skewness in the distribution of stock returns. Both risk factors are thus able to generate a smile or skew. The main difference between the two risk factors SV and SJ is the induced rate of time decay in the smile, in that the smile flattens much faster with SJ than with SV (see also Das and Sundaram (1999)). 5

8 2.2 Hedging Strategies We consider three different hedging strategies to hedge the target option. In a deltahedge, only the stock S and the money market account M are used. The hedge portfolio is chosen such that it has the same price and the same delta as the target option, i.e. where C T and C T S C T = w s S + w m M C T S = w s 1 + w m denote the initial price and the delta of the target option, respectively. w s denotes the number of stocks in the hedge portfolio, and the amount w m is invested in the money market account. In a delta-vega hedge, an additional instrument option is used for hedging. Now, the option and the hedge portfolio have the same price, the same delta and the same vega. Formally, C T = w s S + w m M + w i C I C T S = w s 1 + w m + w i C I S C T V = w s + w m + w i C I V where w i denotes the number of units in the instrument option with price C I, delta C I S, and vega C I V. The local minimum-variance hedge (MV hedge) again uses the stock and the money market account only. The hedge portfolio is chosen such that the local variance of the hedging error is minimized. The stock position in the MV hedge can be derived as shown in Bakshi, Cao, and Chen (1997). 2 In the BS model, one obtains the same stock position as in the delta hedge w s = CS T, which is not surprising, since the delta hedge is theoretically perfect with zero hedging error, and therefore also generates the smallest possible variance. In the Heston (1993) model the stock position is different from the option delta due to 2 Poulson, Schenk-Hoppé, and Ewald (27) determine the minimum-variance hedge in an incomplete market where the pricing function for the call is not given. In contrast to their paper, we take the risk-neutral measure as given. 6

9 the impact of stochastic volatility. One obtains w s = Cov[dS t, dc t ] V ar(ds t ) = CT S V ts 2 t dt + C T V V ts t ρσ V dt V t S 2 t dt = C T S + CT V ρσ V S t. (1) For the Bakshi, Cao, and Chen (1997) model, we also have to take jump risk into account, and the resulting stock position is given in Equation (21) in their paper. In case of the Merton (1976) model, the stock position in the MV hedge results as a special case of the Bakshi, Cao, and Chen (1997) model. 3 Hedging under Model Risk 3.1 Calibration of option pricing models The first step in an analysis of hedging under model risk is to estimate the parameters of the hedge model, usually from a cross-section of option prices. The true model in our paper is a slightly restricted version of the model suggested by Bakshi, Cao, and Chen (1997) where we ignore stochastic interest rates. The true parameters are taken from their empirical calibration All Options in Table III (Bakshi, Cao, and Chen, 1997, p. 218). In case of model mis-specification, the hedge model used by the investor differs from the true model. Here, we focus on the problem of omitted risk factors. For example, we assume that the hedge model does not contain an SV component, although the true model does. We do not consider hedge models that are larger than the true one. In our opinion, the case of omitted risk factors is empirically much more relevant, since the true data-generating will usually be quite involved and contain a large number of risk factors and parameters. The model used by the investor is usually much simpler, since the investor will stop to include any more risk factors into his model once he has achieved a sufficiently good description of the true stock price process and the option prices seen at the market. 7

10 At this point in time, there is also no more information left he can use to calibrate the parameters of further risk factors. Furthermore, the properties of more simple models are better known, and they are easier to implement when it comes to hedging or pricing of exotic options. Our setup captures this situation in a simplified way, in that we assume SVJ to be the true model, while the investor uses BS, SV, or SJ. Our calibration data set consists of 25 options with maturities of 1, 3, 6, 12, and 18 months and strike prices of 9, 95, 1, 15, and 11 for each maturity. The current stock price is 1. The option prices are calculated in the SVJ model. The calibration criterion we chose is to minimize the sum of squared differences between these given ( market ) prices and the prices generated by the hedge model: min Θ i ( P Market i P Model i (Θ) ) 2, where Θ denotes the parameter vector. We calibrate the Heston (1993) and the Merton (1976) model. We also consider an ad-hoc version of the Black and Scholes (1973) model, in which the volatility is set equal to the implied volatility of the claim to be hedged. This approach is widely used by practitioners. The results of the calibration are discussed in Section 4.1. We do two robustness checks. First, we calibrate the models to the prices of shortterm options only, which are usually the most liquid ones. We use 25 options with a maturity equal to three months and strike prices ranging from 97 to 13 in steps of 25 cents. Second, we minimize the squared sum of relative pricing errors instead of the squared sum of absolute pricing errors. The results remain in both cases basically the same. Note that perfect model identification would in principle be possible in our setup, since any model which does not fit the given option prices exactly can immediately be rejected. In reality, however, even the true model will not lead to a perfect fit, since market prices are noisy and thus deviate from the true model prices. The reasons are, e.g., bidask spreads of the options and of the underlying stock, differences between borrowing and lending rates, non-synchronicity, or rounding (see Hentschel (23)). The consequence is 8

11 that all models for which the pricing error is smaller than the amount of noise in the data cannot be rejected. 3.2 Comparison of hedging strategies We consider several target options to be hedged with a time to maturity equal to 1, 3 or 6 months and a strike price between 85 and 115. This allows us to analyze the impact of the time to maturity and the moneyness on the hedging error. We consider delta-hedges, delta-vega hedges and local minimum-variance hedges. Note that we do not take model risk into account when the hedge is set up. This is in a way a naive but certainly a pragmatic approach. In case of large hedging errors this indicates that it could be worthwhile to think about robust strategies. This, however, is beyond the scope of our paper and left for future research. The hedge portfolio is set up at time t. We consider the hedging error at the next rebalancing date t + t, i.e. the local hedging error over the next time interval. The length of this time interval is set equal to one day. The global hedging error until maturity is simply the sum of local hedging errors. The distribution of this global hedging error therefore follows from the distributions of the local hedging errors for all smaller times to maturity, different moneyness levels and different volatilities. The use of local hedging errors allows us study the impact of time to maturity, moneyness and local volatility without any averaging. We define the hedging error as the value of the option minus the value of the hedge portfolio, which consists of the stock, the money market account, and eventually some instrument options. If the hedging error is positive, the value of the option is larger than the value of the hedge portfolio, and an investor who wants to replicate a long position in the call (since he has sold this call, e.g.) makes a loss, while an investor who wants to replicate a short position makes a profit. In the following, we will analyze the distribution of the hedging errors. As a measure for the hedge performance, we only use the standard deviation of these hedging errors. 9

12 However, we are not interested in the mean hedging error. In particular, we do not want to favor hedging strategies for which the investor earns a high premium on the remaining risk exposure (which he would rather prefer to be zero). We thus analyze the distribution of the hedging error under the risk-neutral measure where the mean hedging error is always equal to zero. There is no closed-form solution for the distribution of the hedging error. We thus rely on a Monte Carlo simulation. We perform 1, runs, and the associated stochastic differential equations are discretized using an Euler scheme with 1 time steps per day. The variance process is monitored to stay strictly positive in all simulations. Given the true model and the parametrization from Table 1, the expected number of jumps over the next day would be 236 for 1, runs. We control the simulation such that the realized number of jumps equals the expected one. To eliminate the impact of simulation errors, we de-mean the hedging errors and then compute the standard deviation as well as the expectation of the highest and lowest.5% of hedging errors. The first issue we tackle is the impact of model risk on the performance of hedging strategies. We compare the hedging errors if an incorrect model is used to the benchmark hedging error when the true model is hedged as hedge model. This benchmark hedging error cannot be avoided due to discrete trading and also due to market incompleteness, since a continuous jump size cannot be hedged with finitely many options only. Note that this analysis can only be done in a simulation environment where we know the correct model, but cannot be done by the investor in real time (who believes in his hedge model). The second question is whether hedging errors can help to identify the true model. Note that a hedging error which is not identically equal to zero does not signal model mis-specification, since even the benchmark hedging error (which is based on the correct model) will exhibit some variation. The question here is rather whether the realized hedging error the investor observes is equal to the hedging error he anticipates, based on his assumption that the prices behave according to his hedge model. 1

13 4 Numerical Results: Hedging Performance 4.1 Calibration Table 1 gives the parameters for the true SVJ model and the calibrated SV and SJ models. For the Heston (1993) model the initial local volatility and its mean reversion level are higher than with SVJ, since the diffusive volatility now has to explain also the jump-based part of variance. Furthermore, the correlation is more negative than in the true model, since this parameter has to explain the additional negative skewness caused by on average negative jumps in the true model. The parameter values obtained for the calibration of the Merton model can be explained analogously. The mean jump size and the jump volatility are higher than in the SVJ model to generate the skewness which is no longer explained by stochastic volatility, while the jump intensity is lower. This is partly needed to offset the potentially larger negative jumps and helps to create additional skewness and kurtosis (due to rarer but larger negative jumps). The left panel of Figure 1 shows the fit of the calibrated models in terms of implied BS-volatilities. Of course, the BS ad hoc model matches all implied volatilities and prices perfectly. In the Merton model the smile is decreasing, but too steep at short maturities, and it flattens out too quickly for longer maturities. The Heston model produces the best fit, although the smile is a bit too flat for the shorter maturities. To assess the economic significance of model mis-specification for pricing, we consider price deviations, shown in the right panel of Figure 1. The absolute pricing errors are below 25 cents for all strikes and maturities. As expected the largest pricing errors are obtained for the Merton model with only few degrees of freedom, and the best fit is obtained for the Heston model with pricing errors far below 5 cents for all options. The relative pricing errors (not shown here) are well below 3% and thus below the usual level of microstructure noise. 11

14 4.2 Hedging Hedging performance without model mis-specification The hedging error if the true model is used as hedge model gives the benchmark hedging error which cannot be avoided due to jumps and discrete trading. Figure 2 shows the standard deviations of the hedging error for the delta, the MV and the delta-vega hedge for different strikes and maturities. They tend to be largest for ATM options, which are very sensitive to changes in stock price and volatility. The unavoidable hedging error over one day has a standard deviation which is usually below 25 cents. By definition, the MV hedge produces lower standard deviations than the delta hedge. The delta-vega hedge uses an additional hedge instrument, and we thus expect it to perform best. Indeed, we observe a significant improvement in hedging quality especially for options with short times to maturity. For longer maturities the delta-vega hedge behaves similar to the MV hedge. Of course, since the instrument option used here is a 1- month short-term ATM call, this contract will have a zero hedging error by construction, as can be seen from the second row in Figure Delta-Hedge under model mis-specification We first consider the performance of delta-hedges when an incorrect hedge model is used. The left column in Figure 3 shows the standard deviation of the hedging errors as a function of the moneyness of the target option. Similar to the benchmark case (where the correct hedge model is used), these standard deviations are in general largest for ATM options. The performance of the hedge based on the Heston model can barely be distinguished from the performance of the benchmark hedge. The Merton model performs slightly worse for short-term options and slightly better for long-term options. Somewhat surprisingly, the ad hoc version of the BS model performs best. The standard deviation of the hedging error can thus decrease if an incorrect hedge model is used instead of the correct one. To explain this result, note that the objective 12

15 of a delta-hedge is to eliminate the exposure to stock price risk, but not to minimize the variance of the hedging error. The standard deviation of the benchmark hedging error is thus not necessarily the lowest one, which is confirmed in Figure 2. If an incorrect hedge model biases the hedge towards the (true) minimum variance hedge, it will lead to a lower standard deviation. The results show that this is the case in particular for the ad-hoc Black-Scholes model. To explain the results for the delta hedge in more detail, we rely on some theoretical properties of delta. All the models we consider are homogeneous of degree one in the stock price and the strike price. As shown by Bates (25), the true delta is then equal to the delta in the Black-Scholes model plus a correction for the slope of the volatility smile: C(S, K) S = CBS (S, K, σ BS (M)) S CBS (S, K, σ BS (M)) σ σbs (M) M M S (2) where moneyness is defined as M = K and S σbs (M) denotes the implied BS-volatility of the option. If two models give exactly the same smile, they will thus also give exactly the same deltas. In case the calibrated volatility smile is too steep (flat), delta will be too large (small). Since the Heston model fits the true smile very well, the delta in the Heston model is nearly equal to the true delta, as also seen in Figure 2. For the Merton model, where deviations between the true and the calibrated smile are much larger, the differences in the hedge performance are larger, too. Finally, the ad-hoc Black-Scholes model ignores the slope of the volatility smile and thus the second term in Equation (2) when delta is calculated. For a downward sloping smile, delta is thus too low, as can also be seen in the left panel of Figure 4. In the SVJ model, a downward sloping smile is generated by a negative correlation between the stock price and the local variance and/or by jumps that are on average downward jumps. Equation (1) shows that a negative correlation leads to a hedge ratio for the minimum-variance hedge that is smaller than delta in the Heston model. This property of the minimum-variance hedge ratio carries over to most realistic parametrizations of the more general SVJ model. The underestimation of delta in the ad-hoc Black-Scholes model 13

16 thus biases the hedge in the right direction. When it comes to hedging, an investor is not only interested in the standard deviation of the hedging errors, but also in the probability and absolute size of large hedging errors. This is in particular true in our model setup, where the data-generating process includes jumps in the stock price. To check for the robustness of the results based on the standard deviation of hedging errors, we thus also analyze the average of the.5% smallest as well as the.5% largest hedging errors. The middle and right graphs in Figure 3 present these lower and upper partial moments. The ranking of the hedging models based on these partial moments is in line with that based on the standard deviation. Again we find that the ad hoc BS model performs better than the delta hedge based on the correct model. A look at the densities of hedging errors (not shown) reveals that most errors do not exceed 4 cents in absolute value, but that the extreme errors can be up to 1.6 dollars for long-term ATM options. These extreme hedging errors mainly occur when there are large changes in the underlying, mostly due to jumps. They do not arise due to model mis-specification, but could only be reduced if the investor used further hedge instruments Minimum-variance hedge under model mis-specification The results for the MV hedges are shown in Figure 5. The standard deviations are now smallest for the true model, which also has to be the case, since the objective of the minimum variance hedge is exactly to minimize this standard deviation. The Heston model comes very close, and the ad hoc BS model also performs very well. The bias of the BS hedge towards the minimum variance hedge thus has just the optimal size. The results for the Merton model, however, are considerably worse than for the other models in particular for short times to maturity. The ranking of the model is confirmed by the lower and upper partial moments shown in the middle and right column of Figure 5, respectively. For short times to maturity and OTM puts (or ITM calls), the Merton model leads to very large negative hedging 14

17 errors. A more detailed analysis shows that these losses are caused by large downward jumps in the stock price. In the hedge, the resulting large losses in the call price should be compensated by large gains from the short position in the underlying stock. However, the minimum variance hedge ratio in the Merton model is significantly lower than the minimum variance hedge ratio in the true model. The gain from the short position in the stock is thus not large enough, and the investor ends up with a significant loss. Apart from the bad performance of the Merton model for short term ITM calls, the hedging performance of the hedge models is very similar to each other and also very similar to the performance of the true model. Model mis-specifiation thus seems to matter less in an MV hedge than for a delta hedge Delta-vega hedge under model mis-specification Finally, we analyze the performance of delta-vega hedges. The crucial factors for the performance are how well the hedge model matches the delta and the vega in the true model. While a delta-vega hedge should produce smaller hedging error than a delta-hedge due to the use of an additional hedge instrument, its exposure to model mis-specification will also be larger. The performance of a delta-vega hedge relative to a delta hedge then depends on the trade-off. Figure 6 shows the standard deviations of the hedging errors and the upper and lower partial moments. Again, the hedge in the Heston model is very similar to the hedge in the true model. In particular for longer times to maturity, it may even lead to a lower standard deviation and is thus biased towards a minimum-variance hedge with two hedge instruments. The ad hoc BS model also performs slightly better than the true delta-vega hedge for short term options. On the other hand, the standard deviation can even double for longer times to maturity. Surprisingly, it can even become larger than the standard deviation of the hedging errors when the BS model is used to implement a simple delta hedge. The hedging errors which result from the use of the Merton model are rather high for nearly all times to maturity and all moneyness levels, so that the Merton model again 15

18 is the worst choice. The ranking of the models based on the standard deviations is confirmed by a comparison of the upper and lower partial moments. However, for medium to long times to maturity, the lower partial moments are much larger in absolute terms for the deltavega hedge than for the hedges which use the stock and the money market account only. The upper partial moments, on the other side, are lower, so that the delta-vega seems to trade-off extreme positive and negative errors differently from the other two hedges. To get the intuition for the results, the right column of Figure 4 compares the vegas in the hedge models and the vegas in the true model. 3 For the interpretation, it is important to keep in mind that the number of instrument options in the hedge portfolio depends on the ratio of the two vegas of the target and instrument option. Thus, the form of the vega function matters, while the absolute level does not. Again, the model of Heston is closest to the true model. For the ad-hoc Black-Scholes model, the functional form of the vega is still rather similar to that of the true model, while the differences are largest if the Merton model is used. To get the intuition, we analyze the relation between the true vega and the vega in the Black-Scholes model: C(S, K, V ) V = CBS (S, K, σ BS (M, V )) σ σbs (M, V ), V where V is equal to the local variance in the SV and SVJ model and equal to the diffusion volatility in case of SJ and BS. The formula shows that while a perfect fit to the current smile implies equality of the deltas, it does not imply equality of the vegas too. The reason is that vega also depends on the relation between the implied BS-volatility and the level of V, which is simply not captured by the current smile which only relates the implied volatility to moneyness. Overall, our results show that the inclusion of the state variable stochastic volatil- 3 In the BS and the Merton model, vega is computed as the partial derivative of the option price with respect to the parameter σ, while it is the partial derivative with respect to the local variance V in the SV and SVJ model. 16

19 ity is of first-order importance as compared to the inclusion of stochastic jumps. This supplements the findings of Poulson, Schenk-Hoppé, and Ewald (27) who show that it is more important to include stochastic volatility at all than to exactly identify the true process for stochastic volatility. 5 Numerical Results: Identification In the last section, we have analyzed the impact of model mis-specification on the hedging performance. This analysis was based on a comparison of the realized hedging errors in the true model when the hedge is based on the hedge model to the ideal hedging errors in the true model when the hedge is based on the true model. Note that the latter can only be calculated by the researcher in a simulation study, but not by the investor who is faced with model risk. The investor would rather compare the realized hedging errors he gets to the anticipated hedging errors which he expects to get if the hedge model he uses indeed described the true data-generating process. If the distribution of the realized hedging errors deviates significantly from the distribution of the anticipated hedging errors, the investor will conclude that the hedge model is indeed mis-specified and does not describe the true data-generating process. In this case, hedging errors provide additional information for model identification which is not included in the cross section of prices (since prices are explained sufficiently well by the calibrated hedge model). The main question is now whether the differences between the distributions are indeed large enough to allow for this model identification. Table 2 gives the quantiles of the distribution of the hedging errors when BS is used as the hedge model and when the true model is either the BS model 4 (the column labeled BS ) or the SVJ model (the column labeled SVJ ). In the BS world, the delta hedge would be perfect except for a (presumably small) discretization error. For small or 4 To be consistent with our previous analysis, we assume the ad hoc BS model to be the data-generating process, i.e. to compute the hedging error, we use the respective option s implied volatility in the BS dynamics. 17

20 no changes in the stock price, this discretization error leads to a negative, but bounded hedging error. For large changes of the stock price, on the other hand, the discretization error is positive, and there is no upper bound on this hedging error. The investor thus anticipates a highly asymmetric distribution of hedging errors. When the true model is SVJ, hedging errors arise due to discrete trading, market incompleteness, and the use of an incorrect hedge model. Consequently, the realized hedging errors have a much higher standard deviation than anticipated. While the hedging errors are still bounded from below, the lower 1%- and 5%-quantiles are by a factor of 5-2 larger than those anticipated by the investor. Our numerical analysis shows that these large losses happen for a moderate decrease in the stock price, combined with a drop in local variance. For longer times to maturity, the upper quantiles are also significantly larger than anticipated. The kurtosis of the hedging errors is also much larger in particular for short term options, while the skewness is larger in most cases, too, with the exception of long term options. Taken together, there is thus clear evidence for model mis-specification. Table 3 analyzes the hedging errors when Merton is used as the hedge model. The investor knows that the hedge is not perfect due to discrete trading and due to market incompleteness, caused by stochastic jumps. If Merton describes the true data-generating process, the hedging errors are still bounded from below, while jumps in either direction cause extremely large positive hedging errors. The quantiles of the hedging errors are for most options smaller than they have been with BS, which can be attributed to the lower standard deviation in the Merton model and thus to smaller stock price changes when there are no jumps. In case of a jump, the hedging errors are very large, which is also reflected in the higher moments, and the shorter the time to maturity, the larger these extreme hedging errors are. When the true data-generating process is SVJ, hedging errors due to diffusive stock price and volatility changes are larger than expected, which can be attributed to the larger (diffusive) volatility and also to the presence of stochastic volatility. Extreme hedging errors in case of a jump, however, are much smaller in the SVJ model (where the average jump size is -5%) than in the Merton model (where the average 18

21 jump size is -15%). The fact that hedging errors are less extreme than expected based on the Merton model, however, does not help the investor to identify the true model, since jumps are very rare events and thus subject to a peso problem. Model identification should rather rely on the hedging errors for normal stock price movements, and in case of model mis-specification, realized hedging errors are significantly larger than expected. Table 4 does the same analysis for the minimum-variance hedge. Now, the quantiles in the SJ and in the SVJ model are rather similar, so that model identification becomes even more difficult in this case. Finally, we consider the case where the investor uses the Heston model as a hedge model. The results for the delta-hedge are given in Table 5. The quantiles of the hedging errors in the SVJ model and in the Heston model are very similar for all strikes and all times to maturity. In particular, the lower quantiles differ much less than in the cases where we considered Merton or BS as a hedge model. To get the intuition, note that negative hedging errors can now not only be explained by small changes in the stock price, but also by a drop in volatility in both models (and not just in the SVJ model as before). Given the similarity of the quantiles, it is nearly impossible to tell from a comparison of the quantiles that the model of Heston is mis-specified. The picture changes when we look at the moments of the distributions. While the standard deviation is still rather similar, the skewness is larger if SVJ is the true model, and the kurtosis is much larger. Both these differences can be attributed to jumps in the SVJ model which the investor does not allow for in the Heston model. If the stock price drops significantly, there is a large positive hedging error, and the investor will conclude that Heston is most probably mis-specified. Note, however, that the probability of a jump over the next day is below.25% in our setup, so that jumps do not have an impact on the upper or lower 1%-quantiles. The results for the minimum variance hedge are given in Table 6. Again, the quantiles of the distributions are rather similar for all times to maturity and all moneyness levels, while skewness and kurtosis are much larger when SVJ is the true data-generating process instead of SV. The main difference as compared to the delta-hedge is that very 19

22 high losses (which are also the largest ones in absolute terms) are reduced in the minimumvariance hedge. Intuitively, the reason is that the reduction of exactly those losses that are largest in absolute terms helps most in reducing the variance of the hedging errors. Table 7 gives the results for the delta-vega hedge. A comparison of the quantiles in the SVJ model and the Heston model shows that the quantiles do not provide significant additional information when it comes to model identification. As for the delta hedge and the minimum variance hedge, however, skewness and kurtosis are again much more extreme in the SVJ model due to jumps than in the SV model. An interesting finding is that the negative 1%-quantile is now in absolute terms much larger than the positive 99%-quantile, in particular for long times to maturity. A further analysis of the hedging errors for longer term options shows that, different from all other hedges where only the stock and the money market account are used, the hedging error is now positive for small changes of the stock price, and negative for large jumps in either direction. To get the intuition, we take a closer look at how the hedge is set up. In the first step, the vega of the target option is hedged by an appropriate position in the instrument option, and in the second step, the resulting position is delta-hedged by the stock. If the maturity of the instrument option (one month in our case) is smaller than the maturity of the target option, this resulting position is concave in the stock price for a realistic range of stock price changes. The dependence of the hedging error on the size of stock price changes then follows from this concavity. 6 Conclusion The paper deals with hedging under model mis-specification. Given that the incorrect hedge model is calibrated to the cross section of prices and fits them sufficiently well, the question is how much model mis-specification matters when it comes to hedging. We show that the performance of the hedges is indeed subject to model risk. For a delta-hedge, the use of an SV model changes the optimal hedge least, since the SV model gives the best fit to the smile, and the SJ model performs worst. Surprisingly, an ad-hoc BS hedge, which 2

23 is the worst choice from a theoretical point of view, may even reduce the variance of the hedging error further, since it biases the delta-hedge towards a minimum variance hedge. This may partly explain the popularity of this approach. For a delta-vega hedge, model risk matters much more, and SV is now significantly better than both SJ and the ad-hoc BS model. The inclusion of stochastic volatility in the hedge model is thus of first-order importance. We then discuss whether a comparison of realized hedging errors and anticipated hedging errors can help to identify model mis-specification, i.e. whether hedging errors contain additional information beyond that in the cross section of option prices. We find that both the SJ model and the ad-hoc BS model lead to realized hedging errors that are significantly more extreme than those anticipated by the investor. For SV, on other hand, the distributions are rather similar and differ mainly with respect to extreme hedging errors due to jumps in the stock price. The question whether to include stochastic volatility or not can thus be answered based on normal hedging errors, while the presence of jumps can basically only be inferred from the observation of large jumps in the data, but not from the stock prices and hedging errors in calm times. Further research could focus on exotic options like Barrier options or options on several underlyings. The first question is certainly whether the calibrated models give approximately the same prices for these options. Given that these contracts are often not liquidly traded and have to be replicated or hedged by the issuer, the next question is about the performance of the corresponding hedging strategies under model mis-specification. 21

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