Numerical Methods for Option Pricing

1. Numerical Methods for Option Pricing Prof: Olivier Pironneau Kimiya Minoukadeh Ecole Polytechnique M2 Mathématiques Appliquées, OJME

2. Agenda • Introduction to Monte-Carlo method • Heston stochastic volatility model using M-C • Basket option using Monte-Carlo • Accuracy of Monte-Carlo methods • Variance Reduction methods • Conclusion

3. Monte-Carlo Method I • Based on the expectation of a random variable X, given • N samples {X1,X2,…,XN} • Price of a European Call option is therefore calculated as where: is the ith estimate of the stock price at time T, the time of maturation, r is the risk free interest rate and K is the strike price.

4. Monte-Carlo Method II • The stock price Stfollows the stochastic differential • Equation (SDE) • where • is the drift term • is the volatility

5. Agenda • Introduction to Monte-Carlo method • Heston stochastic volatility model using M-C • Basket option using Monte-Carlo • Accuracy of Monte-Carlo methods • Variance Reduction methods • Conclusion

6. Heston Stochastic Volatility I • Studies have shown that the volatility , if held constant, does not reproduce observed market data. We therefore consider the model suggested by Heston volatility of stock rate of mean reversion volatility mean volatility of volatility The cost of the call at time t = 0

7. Heston stochastic volatility II • Results are consistent with the a priori lower bounds known for call options.

8. Heston stochastic volatility III • Barrier options pose the constraint that a certain asset is never allowed to reach outside a certain interval [a,b]. Expectation of payoff considerably reduced Price of option reduced a = 0 b = 130

9. Agenda • Introduction to Monte-Carlo method • Heston stochastic volatility model using M-C • Basket option using Monte-Carlo • Accuracy of Monte-Carlo methods • Variance Reduction methods • Conclusion

10. Basket options I • Sometimes a derivative may be based on more than one underlying asset. {S(1),S(2),…,S(p)} • The Black-Scholes equation becomes p-dimensional. We consider the case of two underlying assets: p = 2, and once again the Brownian motions have a correlation • Payoff is based on the sum of the two stocks at time T

11. Basket options II • Suppose we use • L starting prices of each of the two stocks • N samples of the estimated stock prices • M intervals for the calculations of the stock prices using explicit Euler’s method • Complexity of the program would be O(L2NM). • To reduce this by a factor M to O(L2N), we use Ito’s Lemma with Yi = log(S(i)) to obtain the explicit solution to the SDE

12. Basket options III By using the explicit solution we can observe that we get desirable results, accuracy similar to using the Explicit Euler’s method, however time performance improved dramatically. TIME PERFORMANCE ERROR ANALYSIS

13. Basket options IV • Letting K = K1 + K2 for the respective quasi strike prices of stocks S(1) and S(2), we observe the following results By choosing S0(1) = K1 = 100, we observe that results resemble that of a standard European call option with one underlying asset

14. Agenda • Introduction to Monte-Carlo method • Heston stochastic volatility model using M-C • Basket option using Monte-Carlo • Accuracy of Monte-Carlo methods • Variance Reduction methods • Conclusion

15. Accuracy of Monte-Carlo method The central limit theorem shows that the accuracy of the Monte-Carlo method is controlled by Thus to halve the error we would need to quadruple the number of samples N used in the Monte-Carlo simulation.

16. Agenda • Introduction to Monte-Carlo method • Heston stochastic volatility model using M-C • Basket option using Monte-Carlo • Accuracy of Monte-Carlo methods • Variance Reduction methods • Conclusion

17. Variance Reduction Methods I IDEA: Reduce the variance of the random process X. For an independent random process Y, we note that The variance is then given by therefore we have

18. Variance Reduction Methods II Need to choose a random variable Y such that it is closely correlated with X. We adapt a method suggested by P. Pellizzari [1] for variance reduction of basket options [1] P. Pellizzari. Efficient Monte-Carlo pricing of basket options. Finance, EconWPA, 1998

19. Variance Reduction Methods III We see that we considerably improve the accuracy of the Monte-Carlo method when using variance reduction technique. With variance reduction, we obtain with N = 2000 samples, results as accurate as the normal Monte-Carlo method with N=10000 samples.

20. Agenda • Introduction to Monte-Carlo method • Heston stochastic volatility model using M-C • Basket option using Monte-Carlo • Accuracy of Monte-Carlo methods • Variance Reduction methods • Conclusion

21. Conclusion The Monte-Carlo method is intuitive and extremely easy to implement It can be used to calculate call prices when an analytic solution of a PDE does not exist Data is consistent with observed data For well estimated expectations we need many sample simulations. To double accuracy, number of samples must quadruple. IMPROVEMENT: When analytic solutions do not exist and we are obliged to use Monte-Carlo methods, variance reduction can improve the performance of the calculation.