Static Hedging and Model Risk for Barrier Options

Transcription

1 Static Hedging and Model Risk for Barrier Options Morten Nalholm Rolf Poulsen Abstract We investigate how sensitive different dynamic and static hedge strategies for barrier options are to model risk. We find that using plain vanilla options to hedge offers considerable improvements over usual -hedges. Further, we show that the hedge portfolios involving options are relatively more sensitive to model risk, but that the degree of misspecification sensitivity is robust across commonly used models. 1 Introduction Models may produce similar plain vanilla option prices, yet give markedly different prices of barrier options. This in documented for instance in Hirsa, Courtadon & Madan (2002). We investigate the natural next question: How does this affect hedge portfolios? The question is particularly interesting because there are several alternatives to traditional dynamic - hedging. We look at static hedges that are portfolios of plain vanilla options constructed such that no dynamic trading is necessary to achieve replication of the barrier option. It has been argued 1 that this use of more natural hedge instruments reduces sensitivity to misspecification of model dynamics; model risk. However, most constructions of the static hedge portfolios hinge strongly on either the Black/Scholes model or a zero drift assumption, so the net effect of model misspecification is not obvious. We carry out a simulation study of the performance of hedge strategies static and dynamic, Black/Scholes-based and market data adjusted against different classes of non- Black/Scholes models. As many others, we find that the use of plain vanilla options to hedge barrier options offers considerable improvements over traditional dynamic -hedging in a Black/Scholes framework. We show that this result also holds under model misspecification. Further we demonstrate that hedges involving options are relatively more model risk sensitive, and that adjustments to reflect either the theoretical model structure or market Both authors are from the Department of Applied Mathematics and Statistics, Institute for Mathematical Sciences, Universitetsparken 5, University of Copenhagen, DK-2100, Denmark. s: {nalholm, rolf}@math.ku.dk. We thank Søren Willemann, Søren Lolle, and Peter Ritchken for helpful comments 1 See for instance Peter Carr s remarks on page 41 of Risk Magazine s January 2003 issue. 1

2 F 0 B Alive region: F solves B/S-PDE K F (B, t) = 0 F (x, T ) = (x K) + B F (B, t) = 0 K F (x, T ) = (x K) + F 0 Alive region: F solves B/S-PDE T 0 T Figure 1: The PDEs for the down-and-out (left) and up-and-out call options. data can have sizeable effects. When adjustments are made, we find static hedge accuracy to be robust across models producing similar prices of plain vanilla options (but different barrier option prices). The rest of the paper is organized as follows. In Section 2 we briefly summarize the literature on pricing and statically hedging barrier options. Section 3 describes the experimental design and the Black/Scholes benchmark case. Section 4 is a simulation study of hedge performance against the skew-generating models constant elasticity of variance (CEV), Heston s stochastic volatility, Merton s jump diffusion, and the infinite jump intensity model Variance Gamma. Section 5 concludes. 2 Barrier Options and Static Hedges It is well-known (see Wilmott (1998) for instance) that in a model where stock-price dynamics under the risk-neutral measure Q are ds(t) = (r d)s(t)dt + σ(s(t), t)s(t)dw (t), (1) the (still alive) price of a knock-out option is of the form π(t) = F (S(t), t) where the function 2

3 F solves the partial differential equation (PDE) F t x2 σ 2 (x, t)f xx + (r d)xf x = rf on the alive region, F (B, t) = 0 for t < T (knock-out at the barrier), F (x, T ) = g(x) (pay-off at expiry). The option can be replicated by a -hedging strategy that holds (t) = F x (S(t), t) units of stock time t and is kept self-financing via the bank-account. In more complicated models (with stochastic volatility or jumps) arbitrage-free knock-out option prices are found from the martingale relation π(t) = E Q t (exp( r(τ t))pay-off(τ)), where τ is the minimum of expiry and the first hitting time to the barrier. Analytical representations of the price become more complicated such as PDEs with multi-dimensional space variables or non-local terms. 2.1 Static Hedging with Calendar-Spreads Derman, Ergener & Kani (1995) show how to create static hedge portfolios for barrier options using plain vanilla options with different expiry dates. With the down-and-out part of Figure 1 in mind, the technique is easily understood. Suppose that the call underlying the barrier option is bought at time 0, and that put options with strikes equal to the barrier and different (nearer) expiry-dates are added to the portfolio in such a way that the portfolio s value along the barrier is forced to 0. The put option weights are easily calculated recursively, and the end result is a portfolio (formed at time 0) whose value equals that of the barrier option along the barrier and at expiry. So buying this portfolio at time 0 and liquidating at the appropriate time replicates the barrier option. As long as a PDE-type representation of the barrier option price is possible, this approach can be used, but obviously matters get more complicated; see Andersen, Andreasen & Eliezer (2002) Fink (2003) for examples. 2.2 Static Hedging with Strike-Spreads Carr & Chou (1997) and Carr, Ellis & Gupta (1998) give a different way to hedge barrier options. The key is the existence of simple claim that is equivalent to the barrier option in the sense that their values coincide at expiry and along the barrier, and thus in the entire alive region. This simple claim s pay-off function can be written down directly in terms of 3

4 characteristics of the barrier option. In the Black/Scholes model the adjusted payoffs for the down-and-out and up-and-out calls are: { h DO g(x) if x > B (x) = ( x B )p g( B2 x ) if x B and h UO (x) = where g(x) = (x K) + and p = 1 2(r d)/σ 2. { ( x B )p g( B2 x ) if x > B g(x) if x B, A static hedge is the obtained by matching the adjusted pay-off with a portfolio of plain vanilla puts or calls with different strikes. Little is known about generalizations of this approach that go beyond the B/S model. Results exist for stochastic volatility models, but they appear practical only in the case 0-drift case, i.e. when r = d. 3 Experimental Design and the Benchmark Case We will look at two cases that capture the essence of pricing, hedging and model risk problems for barrier options: The down-and-out call and the up-and-out call. Closed-form expressions for B/S model prices of these barrier options were given already in Merton (1973), so they are as old as the B/S formula itself. Note that (see Figure 1) the PDE for the up-and-out call has a discontinuity at the point (B, T ), and that induces unpleasant behavior (discontinuous or unbounded derivatives) in the vicinity of the barrier. The rest of the paper compares variants of hedging strategies by simulation experiments, the overall design of which is like this: Specify the true model. Besides the benchmark B/S model, we look at CEV, Heston s stochastic volatility, Merton s jump diffusion and a Variance Gamma model. Table A at the end of the paper summarizes parameter symbols, values and interpretations. Suggest different hedging strategies. model. Which and how many varies from model to Simulate stock-price paths from the true model. These paths run from time 0 to time τ, which is either expiry or the first time the barrier is hit; whichever comes first. All simulations are done under the risk-neutral measure. The expiry-dates or strikes of auxiliary options used for the static hedges are chosen uniformly over the life of the barrier option or the adjustment area of the modified pay-off function. 4

5 The dynamic strategies are adjusted along the stock-price path. At each trading date we adjust our stock-position such that we hold exactly the number of stocks that the continuous-time -hedge strategy prescribes. The strategy is kept self-financing by trading in the bank-account (ie. borrowing or lending). At time τ all strategies are liquidated and we record the discounted hedge error or the Profit/Loss (P/L), by which we mean ɛ j i = exp( rτ i) ( Value of hedge portfolio j (τ i ) Value of barrier option(τ i ) ), for the j th hedge strategy and the i th stock-price path. Note that for many strategies we don t actually need to be able to price the barrier option in the true (and presumably complicated) model. Repeat this for many stock-price paths; say N of them. errors and estimate their distributional characteristics. Record discounted hedge 3.1 The Black/Scholes Model The rest of this section deals with the B/S model. We compare four hedging strategies for the two barrier options: Standard -hedging, the cal-hedge (calendar-spreads), the str-hedge (strike-spreads), and a fourth strategy called mixed -hedging where we buy the plain vanilla option underlying the barrier option (always a strike-110 call), and then -hedge the residual. Since a large part of the hedge error for discrete -hedging is due to the kink in the plain vanilla call s pay-off function, this should to be an improvement over standard -hedging. The down-and-out call The second to fourth columns in Table 1 show descriptive statistics for hedge errors for a down-and-out call. When we talk about accuracy of the hedge in the following we mean standard deviation of the discounted hedge errors. This choice is parsimonious, but not the only possible one; for the sake of brevity we have left out risk measures such as short-fall probability or semi-variance. We first notice a large accuracy improvement as we go from pure to mixed -hedging; the hedge error standard deviation drops from about 10% of the barrier option price to around 2%. This confirms that much of the inaccuracy in discrete hedging stems not from the barrier, but from the kink in the pay-off function at the strike. We also see that for either type of static hedge it takes only 3 options (in addition to the underlying plain-vanilla call) to achieve a hedge that is comparable (slightly better) than what is achieved by a mixed -hedge that is adjusted every second day. 5

6 BLACK/SCHOLES MODEL Hedge method Barrier option type down-and-out call up-and-out call price mean std. dev. price mean std. dev. Pure % 10.1% % 94.9% Mixed % 2.0% % 93.5% str-static % 1.9% % 52.3% cal-static % 1.4% % 62.8% Table 1: Hedging barrier options in the Black/Scholes model. Moments are relative to the true barrier option prices which are equal to the prices of the -hedging strategies. Both static hedges use (1+3) options in the down-and-out case and (1+11) options in the up-and-out case. The up-and-out call Table 1 also shows hedge performance for an up-and-out call in the B/S model. The most notable change from the down-and-out hedging is the much higher standard deviations; these now range from 50% to 100% of the true barrier price, even after raising the number of options in the static portfolios to 11. The option is thus hard to hedge. This phenomenon is well known and is caused by barrier discontinuity (or more graphically exploding greeks). We see that there is no improvement from mixed -hedging. Looking at pay-off functions, this is not surprising. Adding the regular call does not remove the discontinuity and the kink is just shifted from K to B. 4 Misspecification A direct way to see that option markets do not behave as the Black/Scholes analysis tells us is to look at the implied volatilities across strikes and see that this curve is not flat, but has a (typically) smile or skew shape. In this section we look at four Black/Scholes extensions that can generate such smiles/skews. The models we consider in the following are similar in the sense that their parameters have been calibrated to produce the same (skew in) implied volatilities of 1-year call options, but their mathematical structures are quite different, and therefore the barrier option prices in the various models are not the same. 6

7 4.1 The constant elasticity of variance model In the notation of Equation (1) the CEV model has σ(x, t) = σx α 1. When α is less than 1 volatility increases when the stock-price falls, which makes sense economically when thinking of stocks as options on firm value. The model generates a skew. There are closedform solutions for plain vanilla options in the CEV model, see Cox (1996), and barrier options are easily priced by solving the PDE (1) numerically. As for hedge strategies, more choices can be made. Dynamic hedge strategies The naive -hedger just uses the for the option from the B/S-model. To make this work a volatility has to be plugged in. We assume he uses the implied volatility of the (K, T )-call; 20% percent just as before in the pure B/S world. We also consider a -hedger who uses the correct barrier option in the CEV model calculated from the numerical PDE solution. Calendar-spreads One way to do it is exactly as in the true B/S model with the (K, T )-implied volatility as σ (this we refer to as hedge volatility in the following). But for the CEV model we can, almost as easily, do it correctly by applying the true CEV option pricing formula along the boundary when we construct the replication portfolio. Note that this explicitly uses the theoretical structure of the CEV model, but does not require knowledge of its barrier option prices. Strike-spreads Again, the first idea simply is to use the hedge portfolio from the B/S model with the (K, T )-implied volatility playing the role of σ. Corrections can be made following this line of reasoning: Constructing the str-replicating portfolio involves trading options with strikes in the adjustment area of the pay-off function. The naive approach assumes that these trade at B/S prices. But implied volatility skew in the market tells us that they don t. In other words, we see a (large) discrepancy between the value (of the components) of our hedge portfolio and the value of the barrier option in the B/S model. Two ways to adjust the portfolio of auxiliary options, say w, seem natural. First, w could be scaled by a factor such that the price of the str-portfolio matches the B/S model s barrier option price, i.e. uniform scaling by β = B/S model s barrier option price market price of underlying plain vanilla option i strikes w. i market price of auxiliary option i (2) 7

8 CEV MODEL Hedge method Barrier option type down-and-out call up-and-out call price mean std. dev. price mean std. dev. Pure ; B/S-based % 10.3% % 126% Pure ; CEV-based % 10.1% % 120% Mixed ; B/S-based % 4.3% % 129% Mixed ; CEV-based % 2.8% % 122% str-static; B/S-based % 14.0% % 36% Uniformly scaled str-static % 6.2% % 55% Smile-scaled str-static % 0.8% % 26% cal-static; B/S-based % 1.8% % 75% cal-static; CEV-based % 1.5% % 65% Table 2: Hedging the down-/up-and-out calls in the CEV model. Moments are given relative to the true barrier option prices. These are for the down-and-out call, and for the up-and-out call. The static hedges use the same (1+3) and (1+11), respectively, options as in Table 1. B/S-based strategies use the implied volatility of a the underlying plain vanilla call as hedge volatility. But not all options are equally mispriced relative to the Black/Scholes model. Adjustments should reflect that. So suppose that the traded amount of auxiliary option i is set such that the revenue we get from trading it is what we would get in the true B/S model, i.e. use the smile-scaled portfolio weights ŵ i = B/S model s price of auxiliary option i market price of auxiliary option i w i. (3) Both adjustments are general we just have to know market prices of the hedge instruments and smile-scaling allows a higher degree of individual adjustment of hedge portfolio components. In Table 2 we report the results of the nine strategies for hedging down-and-out and up-andout calls in the CEV model. Looking at the down-and-call first, we see that when -hedging with the stock, the gain from using the CEV model is barely visible; the accuracies are 10.3% and 10.1% of the true price. When options are incorporated in the hedge portfolios the effects of using a misspecified or naively applied model are evident. When going from B/S- to CEV-based mixed -hedging, the accuracy improves from 4.3% to 2.8%. For the strike-spread hedges the accuracy improvement is even larger, from 14% 8

9 (naive) to 6.0% (uniform scaling) to 0.8% (smile-scaled). The effect on calendar-spread hedges is lower (a drop from 1.8% to 1.5%), but still considerable in relative terms. The reason the strike-spread hedges benefit the most from adjustments is that while the calendarspread hedges use strikes along the barrier, the strike-spread hedges involve options with strikes well beyond the barrier for which B/S has a large pricing error. Turning to the up-and-out call, we first notice that it is still much harder to hedge, no matter if we use static or dynamic hedging. At hedge frequencies that it does not seem feasible to go beyond in practice, the accuracies are of the same order of magnitude as the barrier option s price itself. Again, the misspecification effects on dynamic hedges are low (accuracies range between 120% and 130%; as in the B/S-case mixed- does not help here). The static hedges clearly outperform the dynamic ones, and in this case the strike-spreads (25% - 50% accuracies) hold the advantage over calendar-spreads (65% - 75% accuracies). The effects from adjustments are clearly seen. Calendar-spreads with the true CEV model and smile-scaling (misspecification correction reflecting different degrees of B/S mispricing) causes accuracies to improve by about 10%-points, but the uniform scaling deteriorates the accuracy (from 36% to 55%). This is because there is a large difference between the price of the barrier option in the true CEV model (2.651) and the B/S-price (2.277), which is what the uniformly scaled portfolio s price matches, and that causes and over-adjustment to the naive B/S strike-hedge (whose price was 2.996). 4.2 Stochastic Volatility In virtually all financial markets, price change variances display considerable variation over time. A model that captures this is the Heston (1993) model, ds(t) = (r d)s(t)dt + v(t)s(t)dw 1 (t) (4) dv(t) = κ(θ SV v(t))dt + η v(t)(ρdw 1 (t) + 1 ρ 2 dw 2 (t)). Here volatility, or more precisely local variance, is stochastic and modeled by a Cox/Ingersoll/Rosstype process, that may be correlated with price movements themselves. Call prices can be expressed in closed form up to the integration of a known function. If the correlation is non-zero we know of no closed-form barrier option formula. In this model perfect dynamic hedging with the stock and bank-account is no longer possible, but there is no difficulty in creating operational -hedge strategies by plugging an implied volatility into the call price formula. The immediate hedge strategies to consider are the pure and mixed -hedging strategies, the pure, the uniform and the smile-scaled str-hedges and the cal-hedge all implemented using implied (K, T )-volatility where needed. 9

10 HESTON MODEL Hedge method Barrier option type down-and-out call up-and-out call price mean std. dev. price mean std. dev. Pure % 15.4% % 85.9% Mixed % 4.5% % 93.2% str-static % 13.3% % 38.0% Uniformly scaled str-static % 2.5% % 58.7% Smile-scaled str-static % 1.3% % 19.3% cal-static % 2.9% % 57.4% cal-static with conditional vol % 3.2% % 49.1% cal-static with Fink s extension % 3.1% % 31.6% Table 3: Hedging the down-/up-and-out calls in the Heston model. Moments are given relative to the true barrier option prices. These are for the down-and-out call, and for the up-and-out call. We consider two further versions of the cal-hedges that both use the theoretical structure of the Heston model. First, we use a hedge volatility equal to the square root of the expected (local) variance conditional on the underlying being at the barrier. This can be calculated in the spirit of Dupire (1994) or Derman & Kani (1998) from the formula E Q (v(t) S(t) = B) = 2 T Call + (r d)b K Call + rcall B 2 2 KK Call, (5) where Call denotes call-option prices, T and K partial derivatives wrt. expiry and strike, and all functions on the right hand side should be evaluated at (S(0), 0 B, t). Second, we employ the extension suggested in Fink (2003), where for each expiry-date, 5 different options with strikes beyond the barrier (we use strike-step sizes of 5) are used to create a portfolio whose value is 0 for 5 possible possible levels of v(t) (chosen symmetrically around the hedge volatility). In Table 3 we report the performance of the eight hedge strategies. We first notice the similarity to the results of the CEV misspecification analysis from Table 2. Discontinuous barrier options are hard to hedge (accuracies between 30% and 100% of the true price), static hedges with few options outperform dynamic -hedges, but there can be considerable variation depending on how exactly adjustments to reflect a non-b/s markets are made. For the down-and-out call str-hedge accuracies range from 13% to 1% depending on the scaling, for the up-and-out call accuracies can change by a factor of 2. The strike-hedges 10

11 display larger sensitivity than calender-hedges to such adjustments. Again, this is because the former use auxiliary options with strikes beyond the barrier, thus having larger B/S mispricing. We also see that simple adjustments may be too simple (uniform scaling of the strike-hedge halves the accuracy for the up-and-out call), but if the naive B/S-hedges are adjusted, then the more information of the model or the market that is used, the better the results are. For calender-hedges smile-scaling dominates uniform scaling, and for calender-hedges the use of expected volatility along the barrier or Fink s adjustment improves performance. In other words, theoretical work helps. 4.3 Jumps Merton s jump diffusion In the classical Merton (1976) model stock returns are hit by Poisson arrivals of (displaced) lognormal jumps, ie. ds(t) = (r d λk)s(t)dt + σ JD S(t)dW (t) + (J(t) 1)S(t)dq(t) where ln J(t) N(γ, δ), dq(t) Poisson(λ), and k = E Q [J(t) 1]. Call prices can be written as (infinite) sums of B/S prices. Prices for barrier options have to be found numerically. There are several ways to do this, but since we only need the true price as a benchmark, we just use simulation. Models with random jump sizes are (grossly) incomplete, so perfect -hedging with any finite number of assets is impossible. However, the same operational dynamic hedge strategies are considered, that is the pure and mixed strategies. For the static hedges we use the both the pure and the smile-scaled strike-spread hedges and the calendar-spread using implied (K, T )-volatilities in the Black-Scholes formula. Variance Gamma The Variance Gamma model from Madan, Carr & Chang (1998) is a pure jump process with many small jumps but infinite arrival rate. With γ( ; α, β) denoting a Gamma process, ie. the Levy process whose increments over unit intervals follow a Gamma(α, β)-distribution, the basic Variance Gamma model can be written like this: X (t; θ V G, σ V G, ν) = θ V G γ(t; 1, ν) + σ V G W (γ(t; 1, ν)), S(t) = S(0) exp ((r d + ω)t + X (t; θ V G, σ V G, ν)), ω = 1 ( ν ln 1 θ V G ν σ2 V G ν ). 2 The model can be thought of as a Black/Scholes model that runs after a different, stochastic clock that captures varying trading activity over time. This and related models have recently 11

12 MERTON JUMP DIFFUSION MODEL Hedge method Barrier option type down-and-out call up-and-out call price mean std. dev. price mean std. dev. Pure % 31.6% % 132.8% Mixed % 6.1% % 150.0% str-static % 11.2% % 29.7% Uniformly scaled str-static % 2.5% % 66.1% Smile-Scaled str-static % 2.3% % 31.1% cal-static % 2.9% % 54.1% VARIANCE GAMMA MODEL Hedge method Barrier option type down-and-out call up-and-out call price mean std. dev. price mean std. dev. Pure % 29.8% % 150.5% Mixed % 5.1% % 165.8% str-static % 11.2% % 32.2% Uniformly scaled str-static % 2.5% % 69.5% Smile-Scaled str-static % 2.3% % 41.6% cal-static % 2.8% % 56.5% Table 4: Hedging the down-/up-and-out calls in jump models. Moments are given relative to the true barrier option prices. In the Merton model they these are for the downand-out call, and for the up-and-out call. In the Variance Gamma model they are and

13 become popular because to their ability to match both the dynamics of the underlying and option market prices. Simulation is fairly straight-forward, see e.g. Glasserman (2003), and call prices can be expressed with special functions or found by integration of B/S prices. We use simulation to find the true prices of the barrier options. Table 4 shows the performance of the hedge strategies in the two jump models. We first notice the similarity of the two panels, ie. even though the two jump models are quite different in nature, hedge performances are alike. Comparing Table 4 to the other misspecification analyses, we also see a high likeness. We do, however, note that the -hedge performances are quite strongly adversely affected by jumps (from 15/85 % accuracies in Heston to about 30/140 % here). Jumps and barriers are a hard combination, and we have little theory to guide us in building static hedges. We see that for the up-and-out call none of the strhedge adjustments yield better results than the naive B/S implementation, but note that if adjustments are made, then higher sophistication improves performance (smile scaling dominates uniform scaling), which is encouraging for further research. 5 Conclusion In the Black/Scholes model it is well known that static hedging is likely to outperform dynamic -hedging and that discontinuous barrier options are difficult to hedge irrespective of the type of hedge. We showed that these two results carry over to commonly used Black/Scholes extensions. We further documented that because the static hedges use outof-the-money options for which Black/Scholes mispricing is large, they are relative more model risk sensitive meaning that performance is strongly affected by the extent to which theoretical model characteristics or market data (skew/smile) are taken into account in their implementation. Finally, we showed that for extended models that have the same implied volatilities but different barrier option prices static hedge performances are more robust than -hedges to the exact choice of Black/Scholes extension. References Andersen, L., Andreasen, J. & Eliezer, D. (2002), Static replication of barrier options: some general results, Journal of Computational Finance 5, Carr, P. & Chou, A. (1997), Breaking Barriers, Risk Magazine 10(September). Carr, P., Ellis, K. & Gupta, V. (1998), Static Hedging of Exotic Options, Journal of Finance 53,

14 Cox, J. C. (1996), The Constant Elasticity of Variance Option Pricing Model, Journal of Portfolio Management pp Derman, E., Ergener, D. & Kani, I. (1995), Static Options Replication, Journal of Derivatives 2, Derman, E. & Kani, I. (1998), Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility, International Journal of Theoretical and Applied Finance 1, Dupire, B. (1994), Pricing with a smile, Risk Magazine 7(January). Fink, J. (2003), An Examination of the Effectiveness of Static Hedging in the Presence of Stochastic Volatility, Journal of Futures Markets 23, Glasserman, P. (2003), Monte Carlo Methods in Financial Engineering, Springer. Heston, S. (1993), A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies 6, Hirsa, A., Courtadon, G. & Madan, D. B. (2002), The Effect of Model Risk on the Valuation of Barrier Options, Journal of Risk Finance 4(Winter), Madan, D. B., Carr, P. & Chang, E. C. (1998), The Variance Gamma Process and Option Pricing, European Finance Review 2(1), Merton, R. (1973), The Theory of Rational Option Pricing, Bell Journal of Economics and Management Science 4, Merton, R. (1976), Option pricing when the underlying stock returns are discontinuous, Journal of Financial Economics 5, Wilmott, P. (1998), Derivatives, Wiley. 14

15 A Parameters in the Simulation Experiments Quantity Symbol Numerical value Initial stock-price S(0) 100 Interest rate r 0.06 Dividend yield d 0.02 B/S-volatility σ 0.20 Carr/Chou-p p = 1 2(r d)/σ 2-1 Strike of underlying call K 110 Expiry of underlying call T 1 Down-and-out barrier B DO 90 Up-and-out barrier B UO 140 -hedging time step t 2/252 No. simulated paths N 10,000 CEV model Variance elasticity α 0.5 CEV-volatility σ CEV Heston model Mean reversion of variance κ Long term variance level θ SV (= ) Volatility of variance η Correlation (stock, variance) ρ Merton model Jump intensity λ Mean jump size γ Jump size variance δ Volatility of diffusion part σ JD Variance Gamma model Drift of diffusion part θ V G Volatility of diffusion part σ V G Variance rate of gamma part ν