Hedging Exotic Options

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1 Kai Detlefsen Wolfgang Härdle Center for Applied Statistics and Economics Humboldt-Universität zu Berlin Germany

2 introduction 1-1 Models The Black Scholes model has some shortcomings: - volatility is not constant - returns are not normally distributed Hence, alternative models have been considered. Strengths and weaknesses of different models? 0 0.4

3 introduction 1-2 Bakshi (1997) Bakshi et al. compared stochastic volatility models with jumps and stochastic interest rates by - in-sample fit - stability of parameters - hedging performance of European options 0 0.4

4 introduction 1-3 Aims - We repeat Bakshi s analysis for a European data set from 01/2000 to 06/ We consider exotic options for hedging. Thus, we extend the analysis of Bakshi to exotic options and repeat it with recent DAX data

5 introduction 1-4 Schoutens (2004) Schoutens et al. studied modern option pricing models that - all led to good in-sample fits - but had different prices for exotic options Additional aim: Does our study also lead to different prices for exotic options? 0 0.4

6 introduction 1-5 Overview 1. introduction 2. data 3. calibration 4. Monte Carlo simulation 5. hedging 6. conclusion 0 0.4

7 data 2-1 Components of data set Our data is a time series from January 2000 to June 2004 that contains for each trading day - an implied volatility surface of settlement prices - the value of the DAX - the interest rate curve (EURIBOR)

8 data 2-2 Components of data set Our data is a time series from January 2000 to June 2004 that contains for each trading day - an implied volatility surface of settlement prices - the value of the DAX - the interest rate curve (EURIBOR)

9 data 2-3 Number of observations moneyness sum maturity sum

10 data iv*E years Figure 1: Time series of mean implied volatilities for long maturities. (blue: in the money, green: at the money, red: out of the money) 0 0.4

11 data 2-5 iv years Figure 2: Time series of mean implied volatilities for mean maturities. (blue: in the money, green: at the money, red: out of the money) 0 0.4

12 data 2-6 iv years Figure 3: Time series of mean implied volatilities for short maturities. (blue: in the money, green: at the money, red: out of the money) 0 0.4

13 data DAX*E years Figure 4: DAX

14 data ir*E years Figure 5: Interest rates for maturity 1 year

15 data 2-9 Preprocessing In order to delete arbitrage opportunities in the data we have used - a method by Hafner, Wallmeier to correct tax effects - a method by Fengler to smooth the whole ivs

16 models 3-1 The option pricing models We consider - the Merton model (jump diffusion/exponential Lévy model) - the Heston model (stochastic volatility model) - the Bates model (stochastic volatility model with jumps) The Bates model is the combination of the Merton and the Heston model

17 models 3-2 The Merton model In this model, the price process is given by N t S t = s 0 exp(µt + σw t + Y i ). W is a Wiener process, N a Poisson process with intensity λ and the jumps Y i are N(m, δ 2 ) distributed. µ is the drift and σ the volatility. i=

18 models Figure 6: Implied volatility surface of the Merton model for µ M = 0.046, σ = 0.15, λ = 0.5, δ = 0.2 and m = (Left axis: time to maturity, right axis: moneyness) 0 0.4

19 models 3-4 The Heston model In this model, the price process is given by ds t = µdt + V t dw (1) t S t and the volatility process is modelled by a square-root process: dv t = ξ(η V t )dt + θ V t dw (2) t, where the Wiener processes W (1) and W (2) have correlation ρ

20 models 3-5 The Heston model II The other parameters in the Heston model have the following meaning: - µ drift of stock price - ξ mean reversion speed of volatility - η average volatility - θ volatility of volatility The volatility process stays positive if ξη > θ

21 models Figure 7: Implied volatility surface of the Heston model for ξ = 1.0, η = 0.15, ρ = 0.5, θ = 0.5 and v 0 = 0.1. (Left axis: time to maturity, right axis: moneyness) 0 0.4

22 models 3-7 The Bates model In this model, the price process is given by ds t = µdt + V t dw (1) t + dz t S t dv t = ξ(η V t )dt + θ V t dw (2) t where W (1) and W (2) are Wiener processes with correlation ρ and Z is a compound Poisson process with intensity λ and independent jumps J with ln(1 + J) N{ln(1 + k) 1 2 δ2, δ 2 }. The meaning of the parameters is similar to the interpretations in the Merton and the Heston model

23 models Figure 8: Implied volatility surface of the Bates model for λ = 0.5, δ = 0.2, k = 0.1, ξ = 1.0, η = 0.15, ρ = 0.5, θ = 0.5 and v 0 = 0.1. (Left axis: time to maturity, right axis: moneyness) 0 0.4

24 calibration 4-1 FFT For the calibration it is essential to have a fast algorithm for calculating the prices/implied volatilities of plain vanilla options. We have used the FFT based method by Carr and Madan which uses the characteristic functions of the log price processes

25 calibration 4-2 Error functional As measure for the errors we have used the squared distance between the observed iv σ obs and the model iv σ mod. In order to give the at the money observations with long maturities more importance we used vega weights V. In order to make the errors on different days comparable we included additional weights

26 calibration 4-3 Error functional II error(p) def = τ K 1 n τ n S (τ) V (K, τ){σmod (K, τ, p) σ obs (K, τ)} 2 where p is a vector of model parameters, n τ is the number of times to maturity of the observed surface and n S (τ) is the number of strikes with time to maturity τ

27 calibration 4-4 Minimization algorithms The calibration problem can be stated as min error(p) p where the minimum is taken over all possible parameter vectors p. For this optimization, we have considered - Broyden-Flechter-Goldfarb-Shanno algorithm - simulated annealing algorithm

28 calibration 4-5 Minimization algorithms II These algorithms have been tested with fixed starting values, moving starting values and the problem has been regularized. Simulated annealing with moving starting values without regularization seems to give the best results with respect to computation time and fit

29 calibration 4-6 Results: errors The Bates model gives the smallest errors (median 0.7). The errors in the Heston model are similar (median 1.0). The Merton model performs worse than the other two models (median 3.9)

30 calibration 4-7 squared error years Figure 9: Error functional after calibration in the Bates model (blue), the Heston model (green) and the Merton model (red)

31 calibration 4-8 squared error years Figure 10: Error functional after calibration in the Bates model (blue) and the Heston model (green)

32 calibration iv moneyness Figure 11: Original iv (black) and calibrated iv in the Bates model (blue), the Heston model (green) and the Merton model (red)

33 calibration iv*E moneyness*E-2 Figure 12: Original iv (black) and calibrated iv in the Bates model (blue), the Heston model (green) and the Merton model (red)

34 calibration iv*E moneyness Figure 13: Original iv (black) and calibrated iv in the Bates model (blue), the Heston model (green) and the Merton model (red)

35 calibration iv*E moneyness Figure 14: Original iv (black) and calibrated iv in the Bates model (blue), the Heston model (green) and the Merton model (red)

36 calibration price*e moneyness*E3 Figure 15: Original prices (black) and prices from iv calibration in the Merton model (red)

37 calibration price*e moneyness*E3 Figure 16: Original prices (black) and prices from iv calibration in the Merton model (red)

38 calibration 4-15 Results: parameters But the parameters in the Bates model are unstable. The parameters in the other two models are stable

39 options 5-1 Options We have considered the following six types of barrier options: - d&o put with maturity 1 year, 80% barrier and 110% strike - d&o put with maturity 2 years, 60% barrier and 120% strike - u&o call with maturity 1 year, 120% barrier and 90% strike - u&o call with maturity 2 years, 140% barrier and 80% strike - forward start (1 year) d&o put with maturity 1 year, 80% barrier and 110% strike - forward start (1 year) u&o call with maturity 1 year, 120% barrier and 90% strike 0 0.4

40 options 5-2 Monte Carlo simulation We have computed the prices and greeks of these options by Monte Carlo simulations. We have found that butterfly spreads give a good variance reduction as control variates

41 options 5-3 correlation time to maturity Figure 17: Correlation of the 1 year d&o put barrier and control variates: Black Scholes barrier (black), underlying (blue), European put (green), butterfly spread (red) and option with final barrier payoff (cyan)

42 options 5-4 Prices The prices of the puts differ across the models. The prices of the calls on the other hand are similar for all models. Hence, Schoutens results can be confirmed partly. But we conclude more precisely that there are also classes without significant price differences

43 options year down-and-out put price per notional*e years Figure 18: Prices of 1y dop in the Bates model (blue), the Heston model (green) and the Merton model (red)

44 options years down-and-out put 0.04+price per notional*e years Figure 19: Prices of 2y dop in the Bates model (blue), the Heston model (green) and the Merton model (red)

45 options year up-and-out call price per notional*e years Figure 20: Prices of 1y uoc in the Bates model (blue), the Heston model (green) and the Merton model (red)

46 options years up-and-out call price per notional*e years Figure 21: Prices of 2y uoc in the Bates model (blue), the Heston model (green) and the Merton model (red)

47 options 5-9 forward start down-and-out put 0.01+price per notional*e years Figure 22: Prices of fs dop in the Bates model (blue), the Heston model (green) and the Merton model (red)

48 options 5-10 forward start up-and-out call price per notional*e years Figure 23: Prices of fs uoc in the Bates model (blue), the Heston model (green) and the Merton model (red)

49 hedging 6-1 Hedging We have considered three hedging methods: - delta hedging - vega hedging - delta hedging with minimum variance 0 0.4

50 hedging 6-2 Bates Heston Merton cumulative hedging error cumulative hedging error cumulative hedging error Figure 24: Hedging results for 1y dop

51 hedging 6-3 Bates Heston Merton cumulative hedging error*e cumulative hedging error*e cumulative hedging error*e-2 Figure 25: Hedging results for 2y dop

52 hedging 6-4 Bates Heston Merton cumulative hedging error*e cumulative hedging error*e cumulative hedging error*e-2 Figure 26: Hedging results for 1y uoc

53 hedging Bates cumulative hedging error Heston cumulative hedging error Merton cumulative hedging error*e-2 Figure 27: Hedging results for 2y uoc

54 conclusions 7-1 Conclusions Bakshi: We have concluded that the Heston model gives the best calibration results with respect to fit and stability of parameters. Moreover, hedging in the Heston model does not perform worse than in the other models. These findings correspond to Bakshi s results

55 conclusions 7-2 Conclusions II Schoutens: We have found in our study that the prices of some exotic options differ among various models although the models are all calibrated to the same plain vanilla ivs. But we have also seen examples where these price differences are only small. Hence, we can not support Schoutens results fully. It seems that there are classes with price differences and other classes without

56 bibliography 8-1 Reference Bakshi, G., Cao, C. and Chen, Z. Empirical Performance of Alternative Pricing Models The Journal of Finance, 1997, 5: Schoutens, W., Simons, E. and Tistaert, J. A Perfect Calibration! Wilmott magazine

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