Hedging Exotic Options


 Thomas Evans
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1 Kai Detlefsen Wolfgang Härdle Center for Applied Statistics and Economics HumboldtUniversität zu Berlin Germany
2 introduction 11 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 12 Bakshi (1997) Bakshi et al. compared stochastic volatility models with jumps and stochastic interest rates by  insample fit  stability of parameters  hedging performance of European options 0 0.4
4 introduction 13 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 14 Schoutens (2004) Schoutens et al. studied modern option pricing models that  all led to good insample 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 15 Overview 1. introduction 2. data 3. calibration 4. Monte Carlo simulation 5. hedging 6. conclusion 0 0.4
7 data 21 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 22 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 23 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 25 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 26 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 29 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 31 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 32 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 34 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 squareroot 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 35 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 37 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 41 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 42 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 43 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 44 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  BroydenFlechterGoldfarbShanno algorithm  simulated annealing algorithm
28 calibration 45 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 46 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 47 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 48 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*E2 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 415 Results: parameters But the parameters in the Bates model are unstable. The parameters in the other two models are stable
39 options 51 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 52 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 53 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 54 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 downandout 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 downandout 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 upandout 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 upandout 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 59 forward start downandout 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 510 forward start upandout 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 61 Hedging We have considered three hedging methods:  delta hedging  vega hedging  delta hedging with minimum variance 0 0.4
50 hedging 62 Bates Heston Merton cumulative hedging error cumulative hedging error cumulative hedging error Figure 24: Hedging results for 1y dop
51 hedging 63 Bates Heston Merton cumulative hedging error*e cumulative hedging error*e cumulative hedging error*e2 Figure 25: Hedging results for 2y dop
52 hedging 64 Bates Heston Merton cumulative hedging error*e cumulative hedging error*e cumulative hedging error*e2 Figure 26: Hedging results for 1y uoc
53 hedging Bates cumulative hedging error Heston cumulative hedging error Merton cumulative hedging error*e2 Figure 27: Hedging results for 2y uoc
54 conclusions 71 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 72 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 81 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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