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# 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 Delta Of An Arithmetic Asian Option Via The Pathwise Method

1. 1. THE DELTA OF AN ARITHMETIC ASIAN OPTION VIA THE PATHWISE METHOD Anna Borisova University of Bocconi 1/12/2014
2. 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. 3. Structure ❏ Mathematical model; ❏ The code features; ❏ Output.
4. 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. 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. 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. 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. 8. Matlab code
9. 9. Model: S = 100, r = 0.045, σ=15% 12 averaging points 100simulations5000simulations 1000simulations10000simulations
10. 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. 11. Matlab code
12. 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. 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. 14. Matlab code
15. 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. 16. Matlab code
17. 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”