Hedging under Model Mis-Speciﬂcation: Which Risk Factors ...

Hedging under Model Mis-Specification:

Which Risk Factors Should You Not Forget?

Nicole Branger§ Christian Schlag‡ Eva Schneider‡ Norman Seeger‡

This version: May 31, 2008

§Finance Center Munster, University of Munster, Universitatsstr. 14-16, D-48143 Munster, Germany.E-mail: [email protected]

‡Finance Department, Goethe University, Mertonstr. 17-21/Uni-Pf 77, D-60054 Frankfurtam Main, Germany. E-mail: [email protected], [email protected],[email protected] (corresponding author)

Earlier versions of this paper were presented at the 11th conference of the Swiss Society for FinancialMarket Research in Zurich 2008, and the Workshop on Finance, Stochastics and Insurance in Bonn 2008.The authors would like to thank the conference participants and discussants for useful comments andsuggestions.

Hedging under Model Mis-Specification:

Which Risk Factors Should You Not Forget?

Nicole Branger§ Christian Schlag‡ Eva Schneider‡ Norman Seeger‡

This version: May 31, 2008

Abstract

Option pricing models that include stochastic volatility and/or stochastic jumpsare often hard to distinguish from each other based on the prices of Europeanplain-vanilla options only, where one reason are rather high bid-ask spreads. Wefirst analyze the hedging error induced by this model mis-specification. This impactof model risk is economically significant. We find that it is largest for delta-vegaand smallest for minimum variance hedges. Furthermore, we show that a simpleBS ad-hoc hedge is biased towards a minimum-variance hedge, which explains itssurprisingly good performance. Second, we analyze whether hedging errors can helpin identifying model mis-specification and ideally also the correct model. We showthat there are substantial differences between realized hedging errors (if the incorrecthedge model is applied to the true data-generating process) and anticipated hedgingerrors (if the hedge model were applied to prices generated by the hedge model), inparticular depending on whether or not stochastic volatility is included in the hedgemodel. Hedging errors can thus provide useful support in model identification.

Keywords: Hedging, Model Risk, Model Identification, Delta-Hedge, Delta-VegaHedge, Minimum-Variance Hedge

JEL: G13

§Finance Center Munster, University of Munster, Universitatsstr. 14-16, D-48143 Munster, Germany.E-mail: [email protected]

‡Finance Department, Goethe University, Mertonstr. 17-21/Uni-Pf 77, D-60054 Frankfurtam Main, Germany. E-mail: [email protected], [email protected],[email protected] (corresponding author)

Earlier versions of this paper were presented at the 11th conference of the Swiss Society for FinancialMarket Research in Zurich 2008, and the Workshop on Finance, Stochastics and Insurance in Bonn 2008.The authors would like to thank the conference participants and discussants for useful comments andsuggestions.

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 sim-

ilar 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 (2004), 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 differ-

ences 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

1See e.g. Merton (1976), Heston (1993), Bates (1996), Bakshi, Cao, and Chen (1997), Bates (2000),

Duffie, Pan, and Singleton (2000), Carr and Wu (2004).

1

such that the maximal absolute pricing difference is minimized. Given the calibrated mod-

els, 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 pro-

vide 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 distri-

bution 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 (2003)). 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 (2005). The

ad-hoc Black-Scholes delta-hedge, which is based on individual option implied volatilities,

2

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 signif-

icantly 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 so-

phisticated 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 (2004), 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

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 (2003) calibrate stochastic volatility mod-

els, jump diffusion models, and Levy 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 vari-

ables are appropriate. Additionally, we are interested in the use of hedging errors for the

purpose of model identification. An and Suo (2003) 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 (2003) is subject to the simultaneous problems of mis-priced European options and

incorrect model assumptions. Finally, Poulson, Schenk-Hoppe, and Ewald (2007) 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

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:

dSt

St−=

(r + ηV Vt + λPEP [

eXt − 1]− λQEQ [

eXt − 1])

dt

+√

VtdW St +

(eXt − 1

)dNt − EP [

eXt − 1]λPdt

dVt = κv (v − Vt) dt + σv

√Vt

(ρdW S

t +√

1− ρ2dW Vt

)

The jump size distribution is log-normal, i.e. Xt ∼ N(ln(1 + µJ) − 0.5σ2J , σ2

J), and the

jump intensity under P is given by λP. Other models are nested as special cases in our

specification. Setting σv = 0, λP = λQ = 0, and V0 = v = σ2, 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 λP = λQ = 0 results in Heston (1993), and eliminating

the stochastic volatility part via setting σv = 0 and V0 = v = σ2 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

2.2 Hedging Strategies

We consider three different hedging strategies to hedge the ’target option’. In a delta-

hedge, 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.

CT = ws · S + wm ·MCT

S = ws · 1 + wm · 0

where CT and CTS denote the initial price and the delta of the target option, respectively.

ws denotes the number of stocks in the hedge portfolio, and the amount wm 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,

CT = ws · S + wm ·M + wi · CI

CTS = ws · 1 + wm · 0 + wi · CI

S

CTV = ws · 0 + wm · 0 + wi · CI

V

where wi denotes the number of units in the instrument option with price CI , delta CIS,

and vega CIV .

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 ws = CTS , 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

2Poulson, Schenk-Hoppe, and Ewald (2007) 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

the impact of stochastic volatility. One obtains

ws =Cov[dSt, dCt]

V ar(dSt)

=CT

S VtS2t dt + CT

V VtStρσV dt

VtS2t dt

= CTS +

CTV ρσV

St

. (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. 2018).

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

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 90, 95, 100, 105, and 110 for each maturity. The current stock

price is 100. 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:

minΘ

∑i

(PMarket

i − PModeli (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 short-

term 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 103 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., bid-

ask spreads of the options and of the underlying stock, differences between borrowing and

lending rates, non-synchronicity, or rounding (see Hentschel (2003)). The consequence is

8

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

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 100,000 runs, and the associated stochastic

differential equations are discretized using an Euler scheme with 100 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 100,000 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 0.5% 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 hedg-

ing 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.

10

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 con-

sider 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

4.2 Hedging

4.2.1 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 espe-

cially 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 2.

4.2.2 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

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 (2005), 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, σBS(M))

∂S(2)

− ∂CBS(S,K, σBS(M))

∂σ· ∂σBS(M)

∂M· M

S

where moneyness is defined as M = KS

and 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

thus biases the hedge in the ’right direction’.

When it comes to hedging, an investor is not only interested in the standard devi-

ation 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 0.5%

smallest as well as the 0.5% 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 40 cents in absolute value, but that the extreme errors can be up to

1.60 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.

4.2.3 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

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.

4.2.4 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

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 delta-

vega 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, σBS(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-

3In 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

ity is of first-order importance as compared to the inclusion of stochastic jumps. This

supplements the findings of Poulson, Schenk-Hoppe, and Ewald (2007) 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 model4 (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

4To 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

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, mar-

ket incompleteness, and the use of an incorrect hedge model. Consequently, the realized

hedging errors have a much higher standard deviation than anticipated. While the hedg-

ing errors are still bounded from below, the lower 1%- and 5%-quantiles are by a factor

of 5-20 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 signifi-

cantly 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

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 0.25% 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

high losses (which are also the largest ones in absolute terms) are reduced in the minimum-

variance 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

20

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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Quantitative Analysis 38, 779–810.

22

Heston, S.L., 1993, A Closed-Form Solution for Options with Stochastic Volatility with

Applications to Bond and Currency Options, Review of Financial Studies 6, 327–343.

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Working Paper.

23

V0 κ θ σV ρ λ µJ σJ

SVJ 0.040000 2.03000 0.040000 0.380000 -0.57000 0.590000 -0.050000 0.070000Heston 0.044336 2.66636 0.043832 0.376326 -0.64931Merton (σ2) 0.025590 0.483361 -0.157131 0.100047

Table 1: Calibrated Parameters

The table summarizes the results of the calibration. The true model is assumed to beBakshi, Cao, and Chen (1997) (SVJ). The calibration is done by minimizing the sum ofsquared absolute pricing errors of 25 options with maturities of 1, 3, 6, 12, and 18 monthsand strike prices equal to 90%, 95%, 100%, 105%, and 110% of the current stock price.

24

Strike 90 Strike 100 Strike 110BS SVJ BS SVJ BS SVJ

1 week 0.01 -0.97 -1.33 -12.47 -20.94 -0.09 -0.130.05 -0.96 -1.13 -12.42 -17.25 -0.09 -0.120.10 -0.95 -0.98 -12.27 -15.14 -0.09 -0.100.25 -0.83 -0.65 -11.13 -11.19 -0.07 -0.080.50 -0.43 -0.22 -6.54 -5.49 -0.03 -0.040.75 0.33 0.25 4.54 4.82 0.03 0.010.90 1.28 0.68 21.43 20.98 0.10 0.050.95 2.02 0.94 34.61 34.29 0.15 0.080.99 5.70 1.48 64.45 65.38 0.50 0.14

std. dev. 0.015 0.093 0.167 0.231 0.002 0.036skewness 7.58 73.31 2.40 11.97 20.52 136.88kurtosis 115.06 6157.01 7.83 305.28 860.42 19075.31

1 month 0.01 -1.90 -5.58 -5.77 -25.87 -1.23 -7.090.05 -1.89 -4.18 -5.74 -19.08 -1.22 -5.080.10 -1.87 -3.41 -5.68 -15.36 -1.21 -4.030.25 -1.70 -2.08 -5.17 -8.94 -1.10 -2.270.50 -1.01 -0.51 -3.13 -1.41 -0.65 -0.230.75 0.63 1.26 1.91 6.87 0.41 1.900.90 3.19 3.25 9.88 15.56 2.02 3.990.95 5.24 4.83 16.36 22.04 3.34 5.420.99 10.57 9.56 31.88 38.54 6.74 9.00


3 months 0.01 -2.10 -16.54 -3.27 -34.25 -2.08 -27.910.05 -2.09 -12.14 -3.26 -24.72 -2.07 -19.840.10 -2.07 -9.70 -3.22 -19.59 -2.05 -15.590.25 -1.89 -5.50 -2.94 -10.76 -1.87 -8.440.50 -1.14 -0.61 -1.79 -0.67 -1.13 -0.310.75 0.68 4.66 1.06 9.78 0.68 7.980.90 3.59 9.97 5.60 19.73 3.55 15.560.95 5.96 13.57 9.32 26.00 5.90 20.270.99 11.83 21.95 18.31 39.40 11.56 29.53


6 months 0.01 -1.73 -23.15 -2.27 -36.82 -1.92 -37.310.05 -1.72 -16.84 -2.26 -26.57 -1.92 -26.760.10 -1.70 -13.35 -2.24 -20.92 -1.89 -21.100.25 -1.55 -7.44 -2.04 -11.49 -1.73 -11.410.50 -0.94 -0.53 -1.24 -0.55 -1.05 -0.400.75 0.56 6.76 0.73 10.74 0.63 10.880.90 2.95 13.80 3.89 21.29 3.29 21.220.95 4.90 18.31 6.47 27.95 5.48 27.530.99 9.69 27.84 12.74 41.34 10.75 40.10


Table 2: Quantiles of hedging errors (BS delta hedge)

The table compares the quantiles of the hedging errors (multiplied with 100) of the BSdelta hedge obtained when the BS model is true (column BS) and when the SVJ modelis true (column SVJ) for different target options. The values are the result of a MonteCarlo simulation based on the parametrization of Table 1.

25

Strike 90 Strike 100 Strike 110Merton SVJ Merton SVJ Merton SVJ

1 week 0.01 -1.62 -1.15 -10.98 -21.18 -0.01 -0.080.05 -1.50 -0.99 -10.95 -17.48 -0.01 -0.070.10 -1.43 -0.87 -10.82 -15.40 -0.01 -0.070.25 -1.31 -0.59 -9.94 -11.47 -0.01 -0.060.50 -1.17 -0.22 -6.31 -5.76 -0.01 -0.040.75 -1.01 0.19 2.25 4.74 0.00 -0.020.90 -0.86 0.57 15.23 21.49 0.00 0.010.95 -0.77 0.80 25.54 34.96 0.00 0.030.99 -0.57 1.28 50.82 67.51 0.01 0.12


1 month 0.01 -1.67 -5.38 -5.48 -27.44 -0.94 -5.810.05 -1.67 -4.42 -5.47 -20.40 -0.93 -4.260.10 -1.65 -3.88 -5.42 -16.69 -0.92 -3.440.25 -1.56 -2.84 -5.04 -10.07 -0.84 -2.010.50 -1.27 -1.34 -3.50 -2.16 -0.54 -0.360.75 -0.78 1.16 0.20 6.97 0.18 1.460.90 -0.20 4.93 6.06 17.79 1.29 3.430.95 0.20 7.96 10.91 26.31 2.20 5.020.99 1.12 15.66 23.26 47.51 5.01 10.11


3 months 0.01 -1.98 -18.53 -3.33 -36.60 -2.20 -28.360.05 -1.98 -14.21 -3.32 -26.78 -2.19 -20.160.10 -1.96 -11.74 -3.29 -21.46 -2.17 -15.850.25 -1.83 -7.30 -3.09 -12.20 -2.02 -8.570.50 -1.35 -1.54 -2.25 -1.34 -1.38 -0.310.75 -0.32 5.50 -0.23 10.50 0.14 8.110.90 1.08 13.37 2.99 22.38 2.59 15.840.95 2.12 18.78 5.65 30.26 4.66 20.570.99 4.70 30.94 12.76 47.60 10.32 30.08


6 months 0.01 -2.23 -24.55 -2.68 -38.51 -2.26 -39.120.05 -2.22 -18.17 -2.67 -28.01 -2.25 -28.010.10 -2.19 -14.67 -2.64 -22.28 -2.23 -22.120.25 -1.98 -8.47 -2.47 -12.43 -2.09 -12.020.50 -1.25 -0.99 -1.76 -0.94 -1.49 -0.540.75 0.15 7.31 -0.07 11.30 -0.01 11.390.90 1.94 15.66 2.54 23.05 2.38 22.360.95 3.24 21.17 4.59 30.45 4.34 29.110.99 6.41 32.88 10.06 45.95 9.86 42.66


Table 3: Quantiles of hedging errors (Merton delta hedge)

The table compares the quantiles of the hedging errors (multiplied with 100) of the Mertondelta hedge obtained when the Merton model is true (column Merton) and when the SVJmodel is true (column SVJ) for different target options. The values are the result of aMonte Carlo simulation based on the parametrization of Table 1.

26

Strike 90 Strike 100 Strike 110Merton SVJ Merton SVJ Merton SVJ

1 week 0.01 -44.51 -57.03 -19.50 -27.27 -0.07 -0.050.05 -31.77 -40.04 -19.42 -23.50 -0.06 -0.050.10 -24.93 -31.15 -19.15 -21.33 -0.05 -0.050.25 -13.65 -16.20 -17.18 -16.99 -0.03 -0.040.50 -0.79 0.16 -9.52 -9.18 -0.01 -0.040.75 12.23 16.47 7.41 8.29 0.02 -0.030.90 23.93 30.75 30.49 33.59 0.04 -0.010.95 30.98 39.10 47.27 52.20 0.05 0.000.99 45.03 55.37 86.90 95.41 0.07 0.04


1 month 0.01 -40.07 -37.92 -23.14 -32.34 -1.09 -5.140.05 -29.12 -28.85 -22.30 -24.92 -1.09 -3.990.10 -23.04 -23.55 -20.60 -20.64 -1.08 -3.340.25 -12.83 -13.49 -15.15 -13.14 -1.01 -2.210.50 -0.94 -1.16 -5.09 -3.44 -0.70 -0.740.75 11.26 12.29 9.07 9.23 0.16 1.210.90 22.36 24.82 25.07 24.58 1.74 3.960.95 29.09 32.41 36.09 35.69 3.19 6.350.99 42.55 47.68 61.62 61.03 7.46 12.51


3 months 0.01 -29.41 -25.33 -25.66 -37.15 -7.24 -24.770.05 -22.10 -18.56 -21.63 -27.06 -7.09 -18.010.10 -17.83 -14.87 -18.55 -21.44 -6.73 -14.310.25 -10.35 -8.38 -12.16 -12.06 -5.24 -7.980.50 -1.21 -0.79 -2.90 -0.98 -2.07 -0.670.75 8.63 7.49 8.53 10.77 2.79 7.120.90 17.93 15.64 20.50 22.20 8.68 14.780.95 23.71 20.96 28.46 29.70 12.93 19.820.99 35.59 31.97 46.22 46.08 23.35 31.46


6 months 0.01 -21.60 -26.19 -22.52 -37.71 -13.30 -35.070.05 -16.51 -18.70 -18.21 -27.30 -11.65 -25.290.10 -13.45 -14.67 -15.30 -21.38 -10.20 -19.930.25 -7.98 -8.00 -9.65 -11.75 -6.96 -10.830.50 -1.11 -0.27 -1.96 -0.44 -1.94 -0.450.75 6.47 7.62 7.15 11.08 4.57 10.260.90 13.79 14.94 16.44 21.75 11.68 20.260.95 18.42 19.41 22.53 28.30 16.54 26.410.99 28.10 28.45 35.84 41.83 27.74 39.28


Table 4: Quantiles of hedging errors (Merton MV hedge)

The table compares the quantiles of the hedging errors (multiplied with 100) of the MertonMV hedge obtained when the Merton model is true (column Merton) and when the SVJmodel is true (column SVJ) for different target options. The values are the result of aMonte Carlo simulation based on the parametrization of Table 1.

27

Strike 90 Strike 100 Strike 110Heston SVJ Heston SVJ Heston SVJ

1 week 0.01 -0.09 -0.50 -20.47 -21.03 -0.02 -0.050.05 -0.07 -0.45 -16.99 -17.36 -0.02 -0.050.10 -0.07 -0.41 -14.98 -15.27 -0.01 -0.050.25 -0.05 -0.34 -11.30 -11.34 -0.01 -0.050.50 -0.03 -0.23 -5.70 -5.64 0.00 -0.040.75 0.01 -0.09 5.38 4.75 0.00 -0.030.90 0.06 0.10 22.73 21.22 0.01 0.000.95 0.11 0.26 36.62 34.64 0.02 0.020.99 0.49 0.74 68.12 66.65 0.07 0.09


1 month 0.01 -5.24 -5.10 -25.82 -27.01 -7.67 -8.040.05 -4.07 -3.97 -19.32 -20.03 -5.56 -5.800.10 -3.40 -3.34 -15.76 -16.38 -4.43 -4.600.25 -2.25 -2.21 -9.59 -9.75 -2.47 -2.580.50 -0.78 -0.81 -2.06 -1.96 -0.20 -0.260.75 1.14 0.95 6.86 6.90 2.19 2.200.90 4.08 3.48 17.87 17.16 4.64 4.630.95 6.95 5.87 27.03 25.15 6.30 6.250.99 14.30 12.55 48.08 45.55 9.90 9.84

std. dev. 0.038 0.092 0.146 0.192 0.036 0.057skewness 2.55 52.80 1.29 12.99 0.48 43l96kurtosis 12.45 3837.88 3.64 403.32 1.03 3950.59

3 months 0.01 -16.73 -17.36 -35.04 -37.24 -28.79 -31.520.05 -12.65 -13.14 -25.80 -27.33 -20.76 -22.520.10 -10.45 -10.75 -20.90 -21.95 -16.44 -17.690.25 -6.42 -6.49 -12.16 -12.58 -9.05 -9.670.50 -1.29 -1.22 -1.57 -1.50 -0.48 -0.510.75 4.93 5.00 10.30 10.69 8.41 9.000.90 12.01 11.89 22.69 23.12 16.95 17.920.95 17.15 16.70 31.43 31.34 22.49 23.380.99 28.56 27.59 49.82 49.38 33.65 34.71


6 months 0.01 -22.84 -25.36 -36.76 -41.18 -37.33 -42.460.05 -17.12 -18.89 -27.16 -30.12 -27.14 -30.560.10 -13.98 -15.26 -21.87 -24.05 -21.62 -24.140.25 -8.26 -8.87 -12.55 -13.60 -12.01 -13.290.50 -1.15 -1.14 -1.31 -1.29 -0.78 -0.780.75 6.95 7.59 11.06 12.14 11.09 12.360.90 15.40 16.48 23.44 25.29 22.53 24.730.95 21.14 22.27 31.69 33.71 29.93 32.450.99 33.29 34.69 48.81 51.24 44.91 48.00


Table 5: Quantiles of hedging errors (Heston delta hedge)

The table compares the quantiles of the hedging errors (multiplied with 100) of the Hestondelta hedge obtained when the Heston model is true (column Heston) and when the SVJmodel is true (column SVJ) for different target options. The values are the result of aMonte Carlo simulation based on the parametrization of Table 1.

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1 week 0.01 -0.11 -0.46 -20.30 -20.96 -0.02 -0.060.05 -0.09 -0.42 -16.73 -17.25 -0.01 -0.060.10 -0.08 -0.39 -14.75 -15.15 -0.01 -0.060.25 -0.06 -0.33 -11.06 -11.17 -0.01 -0.050.50 -0.03 -0.23 -5.51 -5.49 0.00 -0.040.75 0.02 -0.10 5.24 4.92 0.00 -0.020.90 0.08 0.07 22.13 21.06 0.01 0.000.95 0.13 0.23 35.62 34.17 0.01 0.020.99 0.45 0.70 65.92 65.80 0.08 0.04


1 month 0.010 -5.75 -5.73 -23.92 -25.83 -5.83 -6.060.050 -4.28 -4.28 -17.64 -19.01 -4.23 -4.400.100 -3.45 -3.49 -14.11 -15.24 -3.38 -3.520.250 -2.07 -2.12 -8.17 -8.77 -1.95 -2.030.500 -0.40 -0.48 -1.08 -1.25 -0.27 -0.320.750 1.49 1.35 6.64 6.91 1.55 1.560.900 3.72 3.33 15.17 15.44 3.52 3.500.950 5.60 4.89 21.46 21.52 5.01 4.930.990 10.71 9.27 36.74 37.34 9.45 9.65


3 months 0.010 -15.83 -17.03 -30.50 -33.76 -22.61 -25.330.050 -11.40 -12.30 -21.90 -24.34 -16.18 -18.230.100 -8.97 -9.73 -17.19 -19.17 -12.73 -14.370.250 -4.91 -5.37 -9.28 -10.38 -6.95 -7.860.500 -0.30 -0.35 -0.35 -0.46 -0.33 -0.420.750 4.58 4.84 8.92 9.70 6.56 7.240.900 9.23 9.75 17.54 19.13 13.06 14.470.950 12.30 12.96 23.05 24.99 17.24 19.030.990 18.93 19.61 33.54 36.96 25.78 28.93


6 months 0.010 -20.19 -23.10 -31.18 -36.11 -29.97 -35.320.050 -14.48 -16.70 -22.31 -25.97 -21.46 -25.390.100 -11.38 -13.16 -17.49 -20.43 -16.86 -19.930.250 -6.14 -7.14 -9.40 -10.99 -9.10 -10.800.500 -0.25 -0.31 -0.29 -0.38 -0.31 -0.390.750 5.91 6.75 9.15 10.53 8.81 10.270.900 11.65 13.36 17.85 20.58 17.25 20.080.950 15.22 17.49 23.18 26.79 22.41 26.180.990 22.36 25.69 33.39 38.75 32.40 38.18


Table 6: Quantiles of hedging errors (Heston MV hedge)

The table compares the quantiles of the hedging errors (multiplied with 100) of the HestonMV hedge obtained when the Heston model is true (column Heston) and when the SVJmodel is true (column SVJ) for different target options. The values are the result of aMonte Carlo simulation based on the parametrization of Table 1.

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1 week 0.01 -0.08 -0.53 -10.19 -10.18 -0.01 -0.050.05 -0.07 -0.45 -9.74 -9.70 -0.01 -0.050.10 -0.07 -0.40 -9.45 -9.38 -0.01 -0.050.25 -0.06 -0.32 -8.55 -8.49 -0.01 -0.050.50 -0.02 -0.22 -4.91 -5.02 0.00 -0.040.75 0.03 -0.09 3.71 3.28 0.00 -0.030.90 0.06 0.07 16.48 15.53 0.01 0.000.95 0.09 0.20 26.35 25.19 0.01 0.020.99 0.37 0.63 48.64 48.33 0.06 0.09

std. dev. 0.001 0.094 0.127 0.153 0.000 0.036skewness 20.07 73 08 2.30 8.52 43.91 136.82kurtosis 843.80 6105 93 7.00 165.36 4359.97 19074.20

1 month 0.01 -1.60 -1.85 -3.99 -4.490.05 -0.99 -1.09 -2.15 -2.370.10 -0.73 -0.82 -1.36 -1.540.25 -0.43 -0.50 -0.38 -0.480.50 -0.21 -0.28 0.20 0.170.75 0.27 0.21 0.52 0.640.90 0.93 0.84 1.09 1.380.95 1.47 1.32 1.55 1.920.99 3.30 2.55 2.79 3.15

std. dev. 0.009 0.060 0.012 0.038skewness 4.05 74.24 -0.956 63.50kurtosis 42.88 6626.06 9.52 7496.57

3 months 0.01 -12.81 -13.28 -26.18 -26.00 -22.82 -23.160.05 -6.61 -6.52 -13.44 -12.85 -11.77 -11.590.10 -4.03 -3.88 -8.17 -7.62 -7.14 -7.030.25 -0.82 -0.70 -1.66 -1.40 -1.46 -1.340.50 1.24 1.36 2.55 2.69 2.22 2.410.75 2.08 2.17 4.29 4.27 3.70 3.910.90 2.28 2.40 4.68 4.94 4.05 4.320.95 2.46 2.63 4.84 5.29 4.33 4.720.99 2.95 3.22 5.15 5.94 5.07 5.74

std. dev. 0.033 0.051 0.066 0.107 0.058 0.099skewness -2.60 -15.87 -2.64 -18.79 -2.65 -24.00kurtosis 9.78 430.29 10.00 574.18 10.34 975.49

6 months 0.01 -21.55 -21.97 -33.72 -33.90 -33.87 -34.450.05 -11.12 -10.72 -17.27 -16.74 -17.35 -17.060.10 -6.73 -6.36 -10.49 -9.94 -10.51 -10.250.25 -1.35 -1.18 -2.12 -1.91 -2.12 -1.960.50 2.11 2.17 3.28 3.33 3.30 3.420.75 3.52 3.46 5.50 5.47 5.52 5.510.90 3.82 4.35 5.99 6.83 6.00 6.830.95 3.99 4.92 6.21 7.61 6.23 7.660.99 4.44 6.11 6.69 9.07 6.82 9.26

std. dev. 0.055 0.095 0.085 0.144 0.086 0.144skewness -2.64 -19.95 -2.65 -20.16 -2.66 -21.48kurtosis 10.04 612.22 10.13 648.81 10.25 780.50

Table 7: Quantiles of hedging errors (Heston delta-vega hedge)

The table compares the quantiles of the hedging errors (multiplied with 100) of the Hestondelta-vega hedge obtained when the Heston model is true (column Heston) and when theSVJ model is true (column SVJ) for different target options. The values are the result ofa Monte Carlo simulation based on the parametrization of Table 1.

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0.9 1 1.10.15

0.2

0.25

0.3Maturity: 1 week

0.9 1 1.10.15

0.2

0.25

0.3Maturity: 1 month

0.9 1 1.10.15

0.2

0.25

0.3Maturity: 3 months

0.9 1 1.10.15

0.2

0.25


0.9 1 1.1

−0.10

0.10.20.3

Maturity: 1 week

0.9 1 1.1

−0.10

0.10.20.3

Maturity: 1 month

0.9 1 1.1

−0.10

0.10.20.3

Maturity: 3 months

0.9 1 1.1

−0.10

0.10.20.3

Maturity: 6 months

Figure 1: Implied volatilities and option prices

The figure compares the prices in the true model and the calibrated models as a functionof the strike price. The left panel shows the implied BS-volatilities of the true model(solid line), the Heston model (dashed line), and the Merton model (dotted line). Theright panel shows the differences between the true option prices and the prices in thecalibrated models. The parameters can be found in Table 1.

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0.9 1 1.10

0.1

0.2

Maturity: 1 week

0.9 1 1.10

0.1

0.2

Maturity: 1 month

0.9 1 1.10

0.1

0.2

Maturity: 3 months

0.9 1 1.10

0.1

0.2

Maturity: 6 months

Figure 2: Standard deviation of hedging errors without model mis-specification

The graphs show the standard deviation of the hedging errors as a function of the strikeprice of the target option for different times to maturity of the target option. The hedginginterval is one day, and the instruments used in the hedge portfolio are the stock, themoney market account, and – in case of the delta-vega-hedge – also the 1 month ATMcall. The solid line represents the delta hedge, the dashed line the MV hedge, and thedotted line the delta-vega hedge. The parameters of the true model are given in Table 1.

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standard deviation

0.9 1 1.10

0.1

0.2

Maturity: 1 week

0.9 1 1.10

0.1

0.2

Maturity: 1 month

0.9 1 1.10

0.1

0.2

Maturity: 3 months

0.9 1 1.10

0.1

0.2

Maturity: 6 months

expectation upper0.5%-quantile

0.9 1 1.10

1

2

Maturity: 1 week

0.9 1 1.10

1

2

Maturity: 1 month

0.9 1 1.10

1

2

Maturity: 3 months

0.9 1 1.10

1

2

Maturity: 6 months

expectation lower0.5%-quantile

0.9 1 1.1

−0.5

0Maturity: 1 week

0.9 1 1.1

−0.5

0Maturity: 1 month

0.9 1 1.1

−0.5

0Maturity: 3 months

0.9 1 1.1

−0.5

0Maturity: 6 months

Figure 3: Performance of delta hedges

The left graphs show the standard deviation of the hedging errors as a function of thestrike price of the target option for different times to maturity of the target option. Themiddle and the right graphs show the expectation of the upper/lower 2.5% quantile. Thehedging interval is one day, and the instruments used in the hedge portfolio are the stockand the money market account. The solid line represents the case of the true model, thedashed line the case of the SV model, the dotted line the SJ model and the dash-dottedline represents the case of the BS ad hoc model. All parameters are from Table 1.

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0.9 1 1.1−0.05

0

0.05

Maturity: 1 week

0.9 1 1.1−0.05

0

0.05

Maturity: 1 month

0.9 1 1.1−0.05

0

0.05

Maturity: 3 months

0.9 1 1.1−0.05

0

0.05

Maturity: 6 months

0.9 1 1.1

−20

−10

0

Maturity: 1 week

0.9 1 1.1

−20

−10

0

Maturity: 1 month

0.9 1 1.1

−20

−10

0

Maturity: 3 months

0.9 1 1.1

−20

−10

0

Maturity: 6 months

Figure 4: Delta and vega for true and calibrated models

The left (right) graph shows the difference between the option delta (vega) in the cal-ibrated and the true model as a function of the strike price. The solid line representsthe case of the true model, the dashed line the case of the SV model, the dotted linethe SJ model and the dash-dotted line represents the case of the BS ad hoc model. Allparameters are from Table 1.

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standard deviation

0.9 1 1.10

0.2

0.4Maturity: 1 week

0.9 1 1.10

0.2

0.4Maturity: 1 month

0.9 1 1.10

0.2


0.9 1 1.10

0.2



0.9 1 1.10

1

2Maturity: 1 week

0.9 1 1.10

1

2Maturity: 1 month

0.9 1 1.10

1

2Maturity: 3 months

0.9 1 1.10

1

2Maturity: 6 months


0.9 1 1.1

−0.4

−0.2

0Maturity: 1 week

0.9 1 1.1

−0.4

−0.2

0Maturity: 1 month

0.9 1 1.1

−0.4

−0.2

0Maturity: 3 months

0.9 1 1.1

−0.4

−0.2

0Maturity: 6 months

Figure 5: Performance of minimum-variance hedges


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standard deviation

0.9 1 1.10

0.1

0.2

Maturity: 1 week

0.9 1 1.10

0.1

0.2

Maturity: 1 month

0.9 1 1.10

0.1

0.2

Maturity: 3 months

0.9 1 1.10

0.1

0.2

Maturity: 6 months


0.9 1 1.10

1

Maturity: 1 week

0.9 1 1.10

1

Maturity: 1 month

0.9 1 1.10

1

Maturity: 3 months

0.9 1 1.10

1

Maturity: 6 months


0.9 1 1.1−3

−2

−1

0Maturity: 1 week

0.9 1 1.1−3

−2

−1

0Maturity: 1 month

0.9 1 1.1−3

−2

−1

0Maturity: 3 months

0.9 1 1.1−3

−2

−1

0Maturity: 6 months

Figure 6: Performance of delta-vega hedges


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Hedging under Model Mis-Speciﬂcation: Which Risk Factors ...

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