The Delta Of An Arithmetic Asian Option Via The Pathwise Method
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In the slides present a structured description of the methods that can be used to calculate the delta for an asian option. Only European options are considered. The reference list has added as the last slide. Enjoy the presentation!
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The convenience yield implied by quadratic volatility smiles presentation [...
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Hull White shore interest rate model slide.
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二.真实使馆公证(即留学回国人员证明,不成功不收费)
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4:这个认证书并且可以归档倒地方
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一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
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3:国家专业人才认证中心颁发入库证书
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5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
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4:这个认证书并且可以归档倒地方
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The Delta Of An Arithmetic Asian Option Via The Pathwise Method
- 1. THE DELTA OF AN ARITHMETIC ASIAN OPTION VIA THE PATHWISE METHOD Anna Borisova University of Bocconi 1/12/2014
- 2. Assignment 1. Compute the Monte Carlo simulation for the price of an Asian option on a lognormal asset (with descrete monitoring at dates t1, t2, … , tM); 2. Provide the pathwise estimate of the Delta of these options.
- 3. Structure ❏ Mathematical model; ❏ The code features; ❏ Output.
- 4. The price of an Asian option by MCS An Asian option (call) has discounted payoff: Y = e-rT [S - K]+ S = 1 m S(ti ) i=1 m å For fixed dates 0<t1<…<tm<T Since for Monte Carlo simulation we describe the risk-neutral dynamics of the stock price, we need to use the stochastic differential equation for modeling the price movement of the underlying asset S(T) = S(0)exp([r - 1 2 s 2 ]T +sW(T)) Where W(T) is the random variable, normally distributed with mean 0 and variance T.
- 5. S(T) = S(0)exp([r - 1 2 s 2 ]T +sW(T)) Monte Carlo simulation of the lognormal asset price movement Model: S = 100, r = 0.045, σ=15%, trials = 100000 For fixed dates 0<t1<…<tm<T
- 6. Constructing the Brownian motion The properties of Brownian motion: • W(0) = 0; • The increments {W(t1)-W(t0), W(t2)-W(t1), … , W(tk)-W(tk-1)} are independent; • W(t) – W(s) is normally distributed N(0, t-s) for any 0 ≤ s < t ≤ T. S(t) = S(0)exp([r -0.5s 2 )t + ts Zi i=0 t å
- 7. 0.5697 -1.0916 -1.3929 -0.1540 -1.7215 -0.1277 -0.9524 -1.9553 -0.5648 Numberofsimulations Number of averaging periods + + + + + + -0.5367 -4.7684 -2.0854 0.4157 -2.8131 -1.5206 0.5697 -1.0916 -1.3929 S(t) = S(0)exp([r -0.5s 2 )t + ts Zi i=0 t å
- 8. Matlab code
- 9. Model: S = 100, r = 0.045, σ=15% 12 averaging points 100simulations5000simulations 1000simulations10000simulations
- 10. Option value estimation and its confidence level The sample standard deviation sC = 1 n-1 (Yi - ˆYn )2 i=1 n å 1-δ quantile of the standard normal distribution zdConfidence interval: ˆYn ± zd/2 sC n ˆYn = 1 n Yi 1 n å E[ ˆYn ]= Y ˆYn -Y sC / n Þ N(0,1) The estimation of the option value is unbiased As number of replications increases, the standardized estimator converges in distribution to the standard normal
- 11. Matlab code
- 12. TRADE OFF: The value of the option, confidence interval and computational time Number of simulations 100 1000 3000 5000 7000 10000 Value of the option 0.9306 1.3199 1.125 6 1.3533 1.3347 1.2666 1.1797 1.1634 1.2084 1.2256 1.2575 1.1560 Comput. time 0.15 0.11 0.13 0.17 0.42 0.47 0.76 0.85 1.13 1.16 1.67 2.14 95% c.l. lower bound 0.4123 0.929 8 1.2043 1.0898 1.1318 1.1917 95% c.l. upper bound 1.4490 1.321 4 1.4652 1.2696 1.2850 1.3233 99% c.l. lower bound 0.3496 1.1336 1.1454 1.0755 1.1472 1.0932 99% c.l. upper bound 2.2901 1.5730 1.3878 1.2512 1.3040 1.2188
- 13. The pathwise estimator of the option delta This estimator has great practical value • This estimator is unbiased; • mean(S) is simulated in estimating the price of the option also, so finding the delta requires just a little additional effort; • This method reduces variance and computing time compared to finite-difference. dY dS(0) = dY dS dS dS(0) = e-rT 1{S > K} S S(0) dS dS(0) = 1 m dS(ti ) dS(0) = 1 m S(ti ) S(0) = i=1 m å i=1 m å S S(0) dY dS(0) = e-rT 1{S > K} S S(0)
- 14. Matlab code
- 15. Model: Asian call option S(0) = from 0 to 200 with step 1; r = 4,5%; sigma = 1; K=50; T = 1 with 24 averaging points (two times a month); Trials = 10000.
- 16. Matlab code
- 17. References: • Glasserman “Monte Carlo Methods in Financial Engineering” • Mark Broadie, Paul Glasserman “Estimating Security Price Derivatives Using Simulations” • John C. Hull “Options, futures and other derivatives” • Huu Tue Huynh, Van Son Lai, Issouf Sourmare “Stochastic Simulation and Applications in Finance with MATLAB® Programs”