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### Megh's slides

1. 1. Option Pricing with Long Range Dependence Megh Shah Thesis Supervised by Dr. Andriy Olenko Department of Mathematics and Statistics La Trobe University Masters in Statistical Science, 2011 Megh Shah Option Pricing with Long Range Dependence
2. 2. Long Range Dependence Deﬁnition of Long Range Dependence Long range dependency for a stationary process is deﬁned as ∞ γl = ∞. l=1 Long range dependency means that events that happened a long time ago would still have an impact on the present or future values of the process. In contrast, short range dependency presupposes that the autocovariance decays fast enough to be summable. Megh Shah Option Pricing with Long Range Dependence
3. 3. Autocorrelation in Stock Returns ACF plot of S&P 500 Returns from 4/1/1990 to 31/8/2011 1.0 0.8 0.6 ACF 0.4 0.2 0.0 0 50 100 150 Lag Megh Shah Option Pricing with Long Range Dependence
4. 4. Long Range Dependence in Squared Stock Returns ACF plot of S&P 500 Squared Returns from 4/1/1990 to 31/8/2011 1.0 0.8 0.6 ACF 0.4 0.2 0.0 0 50 100 150 Lag Megh Shah Option Pricing with Long Range Dependence
5. 5. Call Option Payoﬀ Call Option: The option contract that gives the right but not the obligation to buy the underlying contract (currency, stocks, interest rates, commodity, bonds etc) is termed a call option. The payoﬀ for a European call option C with a given strike price K and stock price s at expiry is given as C = Max (s − K , 0) . Payoff of the European Call Option at expiry 50 Stock price=100 In the money Calls 40 Out of the Money Calls Call option price 30 At the money Call 20 10 0 60 80 100 120 140 Strike price Megh Shah Option Pricing with Long Range Dependence
6. 6. Fractional Brownian Motion Fractional Brownian motion is capable of capturing long range dependence. Properties of fractional Brownian motion H B0 = 0 E BtH = 0 ∀ t∈R. 1 E BtH Bs = H 2 | t |2H + | s |2H − | t − s |2H , ∀ t,s∈R. When H = 1 the process has independent increments and corresponds to 2 Brownian motion. But when 1 < H ≤ 1 the process is said to have long 2 range dependence or long memory. Megh Shah Option Pricing with Long Range Dependence
7. 7. Arbitrage Arbitrage is a strategy such that you make a “riskless proﬁt” beyond the risk free rate. This strategy must be self-ﬁnancing. The change in the portfolio is because of the change in the value of the asset without money being withdrawn or added to the portfolio. Arbitrage strategy for a portfolio Vt 1 V0 = 0, the initial value of this strategy is 0. 2 ∃ t such that P(Vt ≥ 0) = 1 which states that the portfolio would have a value greater than 0 almost surely. P(Vt > 0) > 0, which means that we win with non zero probability. Megh Shah Option Pricing with Long Range Dependence
8. 8. Arbitrage in Fractional Brownian Markets Simulation of Shiryayev’s Arbitrage 10 8 6 Portfolio value 4 2 0 0 0.04 0.098 0.16 0.218 0.28 0.338 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Time Megh Shah Option Pricing with Long Range Dependence
9. 9. Los and Jaimdee Model The option price for stock price s, strike price K , time left for maturity t, volatility σ and Hurst exponent H as is C0 = sSD d1 − ke −rt SD d2 , where s ln K + rt + 1 σ 2 t 2H 2 d1 = , σt H s ln + rt − 1 σ 2 t 2H K 2 d2 = . σt H In the expression above SD() is the cumulative distribution function of Stable distribution. Megh Shah Option Pricing with Long Range Dependence
10. 10. Hu and Øksendal Model For stock price s, strike price K , time left for maturity t, volatility σ and Hurst exponent H the European call option price is given as C0 = sN d1 − ke −rt N d2 where s ln K + rt + 1 σ 2 t 2H 2 d1 = , σt H s ln K + rt − 1 σ 2 t 2H 2 d2 = . σt H Megh Shah Option Pricing with Long Range Dependence
11. 11. Long Range Dependencies in Asset Prices using FractalActivity Time Model (FATGBM) The subordinator model describes stock price St dynamics as St = S0 e µt+θTt +σB(Tt ) , where Tt is a positive non-decreasing random process with stationary but not necessarily independent increments, denoted over unit time by τt = Tt − Tt−1 . µ, θ and σ > 0 are all constants. Features of FATGBM Model Skewess and leptokurtosis in returns. ACF for returns would not display long memory but squared or absolute returns would. Stochastic volatility in returns. Returns can be modelled using heavy tailed or semi-heavy tailed distribution. Aggregational gaussianity in real returns. Arbitrage would not be possible under an appropriate change of probability measure. Megh Shah Option Pricing with Long Range Dependence
12. 12. FATGBM Models The distribution of stock returns Xt in FATGBM model is 1 d Xt = log (St ) − log (St−1 ) = µ + θτt + στt2 B (1) . Student t FATGBM Model If τt is Inverse Gamma (RΓ) distributed with parameters (α, β) then this results in Xt having marginal (skew) t distribution with v degrees of freedom where v = 2α. Variance Gamma FATGBM Model If τt is gamma (Γ) distributed with parameters (α, λ) then this results in Xt having a marginal (skew) variance gamma distribution. Megh Shah Option Pricing with Long Range Dependence
13. 13. FATGBM Models The distribution of stock returns Xt in FATGBM model is 1 d Xt = log (St ) − log (St−1 ) = µ + θτt + στt2 B (1) . Student t FATGBM Model If τt is Inverse Gamma (RΓ) distributed with parameters (α, β) then this results in Xt having marginal (skew) t distribution with v degrees of freedom where v = 2α. Variance Gamma FATGBM Model If τt is gamma (Γ) distributed with parameters (α, λ) then this results in Xt having a marginal (skew) variance gamma distribution. Megh Shah Option Pricing with Long Range Dependence
14. 14. Option Pricing in FATGBM model Option pricing in Student t FATGBM Model ∞ ln( St )+rt+ 1 σ 2 u ln( St )+rt− 1 σ 2 u C (t, K ) = 0 St N K √ 2 σ u − Ke −rt N K √ 2 σ u × u−t+t −H v v −2 t −H fRΓ tH ; 2, 2 du. Option pricing in Variance Gamma FATGBM Model ∞ ln( St )+rt+ 1 σ 2 u ln( St )+rt− 2 σ 2 u 1 C (t, K ) = 0 St N K √ 2 σ u − Ke −rt N K √ σ u × u−t+t −H v v t −H fΓ tH ; 2, 2 du. Megh Shah Option Pricing with Long Range Dependence
15. 15. Option Pricing in FATGBM model Option pricing in Student t FATGBM Model ∞ ln( St )+rt+ 1 σ 2 u ln( St )+rt− 1 σ 2 u C (t, K ) = 0 St N K √ 2 σ u − Ke −rt N K √ 2 σ u × u−t+t −H v v −2 t −H fRΓ tH ; 2, 2 du. Option pricing in Variance Gamma FATGBM Model ∞ ln( St )+rt+ 1 σ 2 u ln( St )+rt− 2 σ 2 u 1 C (t, K ) = 0 St N K √ 2 σ u − Ke −rt N K √ σ u × u−t+t −H v v t −H fΓ tH ; 2, 2 du. Megh Shah Option Pricing with Long Range Dependence
16. 16. Calibrating Option Prices Loss functions compute the diﬀerence in the model price and observed market price of the option. n 1 \$RMSE (θ) = ek (θ)2 where ek = Ck − C (θ). n k=1 By minimizing these loss functions using an optimization routine we can calibrate the pricing model. Megh Shah Option Pricing with Long Range Dependence
17. 17. Calibrated Option Prices in Black Scholes Model BS calibrated Price vs Market prices 50 x BS Price Market Price November December 40 \$RMSE=\$6.29 contracts contracts January April contracts contracts x x 30 x x Option prices x x x x x x x x x 20 x x x x x x x x x x x x xx x x x x x x 10 xx x x x xx x x x x xx x xx xx xx xx xx x x xx xx 0 95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150 Strike prices Megh Shah Option Pricing with Long Range Dependence
18. 18. Calibrated Option Prices in Hu and Øksendal Model Hu and Oksendals model calibrated Price vs Market prices 50 x Hu and Oksendal Price Market Price November December \$RMSE=2.25 40 contracts contracts January April contracts contracts 30 x Option prices x x x x x x 20 x x x x x x x x xx x x 10 x x x x xx x x x x x x xx x xx xx x xx xx xx xxx x x x xx x x x x xx x xx x 0 95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150 Strike prices Megh Shah Option Pricing with Long Range Dependence
19. 19. Calibrated Option Prices in Student t FATGBM Model Student t FATGBM model calibrated Price vs Market prices 50 x Student t FATGBM Price Market Price November December 40 \$RMSE=\$2.24 contracts contracts January April contracts contracts 30 x Option prices x x x x x x 20 x x x x x x x x xx x x 10 x x x xx x x x x x x x xx x xx xx x xx xx xx xxx x x x xx x x x x xx x x xx 0 95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150 Strike prices Megh Shah Option Pricing with Long Range Dependence
20. 20. Calibrated Option Prices in Variance Gamma FATGBMModel Variance Gamma FATGBM model calibrated Price vs Market prices 50 x Variance Gamma FATGBM Price Market Price November December 40 \$RMSE=\$2.23 contracts contracts January April contracts contracts 30 x Option prices x x x x x x 20 x x x x x x x x xx x x 10 x x x x xx x x x x x x xx x xx x xx xx xx xx x x xx x x x xx x x x xx x xxx 0 95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150 Strike prices Megh Shah Option Pricing with Long Range Dependence
21. 21. Calibrated Parameters and \$RMSE Values Parameters Models σ H v Black Scholes 0.7669898 Hu and Øksendal 0.424934 0.51000 Student t FATGBM 0.4102277 0.8781472 44.57739 Variance Gamma FATGBM 0.422940 0.851147 53.028287 Model \$RMSE Error Black Scholes 6.290929 Hu and Øksendal 2.250464 Student t FATGBM 2.244872 Variance Gamma FATGBM 2.236839 Megh Shah Option Pricing with Long Range Dependence
22. 22. Contribution My contribution in this thesis: Applied the modiﬁed ITo’s formula to develop a portfolio strategy which demonstrates arbitrage in fractional Brownian motion setting with derivation and simulation. Critically reviewed Jamdee,S. & Los, C. (2007) Long memory options: LM evidence and simulations. Research in International Business and Finance 21(2), Pages 260-280. Justiﬁcation for the measure change from real world measure to skew corrected martingale measure is given for FATGBM models along with detailed proof for pricing European style options in the FATGBM models. R codes to calibrate and compare four models versus market prices. Megh Shah Option Pricing with Long Range Dependence