Pricing derivatives using Monte Carlo Techniques - Essay Example

Running head: Monte Carlo Pricing derivatives using Monte Carlo Techniques ___________ ________________________ ________________ Pricing derivatives using Monte Carlo Techniques Introduction Monte Carlo simulation has been identified as an important computer run statistical technique for setting up prices of complex derivative products. It also has immediate use in valuations of various categories of option products. The simulation's capabilities to handle several variables in one construct makes it an ideal technique to address issues involved in analysis of market risk and credit risk and thereby it assumes an important place in overall risk management architecture. In the following paragraphs we report a literature survey to find the limitations of Monte' Carlo simulation and the possible ways to get around them. Limitations, Solutions, Development & Implementation The two main limitations of the Monte Carlo method are its sluggishness and its inefficient pricing of American options. Joy et al. (1996)[1] and, Boyle, Kolkiewic and Tan (2000)[2] suggest that variance reduction techniques using specially selected deterministic points, known as quasi-random points, instead of the usual random points, can increase the speed and improve the efficiency of the method. In practice generic Monte Carlo pricing engines face computational problems in the presence of discontinuous payoffs options, because of above stated time consumption limitation but also due to poor convergence with its finite difference estimates and brute force perturbation. Benhamou (2001)[3] following Fourni et al. (1999)[4] use Malliavin calculus to smoothen the simulation function. Benhamou(2001)[3] assumes that the functions are smooth enough to be able to perform the different computation following technical assumptions enunciated earlier, in particular the assumption regarding uniform ellipticity of the volatility operator, in Benhamou (2000-i)[5] (2000-ii)[6] and Fourni et al. (2001)[7]. Benhamou (2001)[3] further states when using finite difference approximation for the Greeks, with jumped price and taking the sensitivity issues into account, errors on numerical computation of the expectation via the Monte Carlo, and another one on the approximation of the derivative function occur. Analysis ends up approximating the second order derivative of the payoff function .An approximation is obviously very inefficient for very discontinuous payoffs like for binary, range accrual, barrier and other type of digital options. To reduce this inefficiency, Broadie and Glasserman (1996)[8] suggested using the likelihood ratio method. Benhamou says," All Greeks can be written as the expected value of the payoff times a weight function and thee weight functions are independent from the payoff function implying that for a general pricing engine, such as Monte Carlo, using certain (numerical) criteria of smoothness, one can branch on the appropriate method. Because it is in a sense independent from the payoff function, the general implementation is simpler that the one of variance reduction technique that only apply to very specific payoff (like the use of control variate).Also no extra computation is required for other payoff function as long as the payoff is a function of the same points of the Brownian trajectory. This can be cached in memory to make it efficient Benhamou (2001).Thus Mallavian calculus promises savings in terms of computations, complexity, cache memory and in time though it may produce some noise. The formidable amount of literature exists which intends to suggest analytical pricing formulae for single asset American options. It includes Carr(1998)[9], Grant et al(1997)[10], Bunch and Johnson(2000)[11],Huang et al(1996)[12], Geske and Johnson(1984)[13] and Barone Adesi and Whaley(1987)[14].One can even refer to older constructs like the binomial model of Cox et al(1979)[15].Many of these constructs deploy elaborate mathematical tools, like recursive integration schemes or Richardson extrapolation. The Binomial trees with tens of thousand nodes used in some of these studies are used even now to obtain rapid and approximate pricing with the assistance of little or no coding. Multiple assets involved pricing were first attempted by Tilley (1993)[16] and Bossaerts (1989)[17]. However Tilley and the genre authors that followed could not produce algorithms predicting with accuracy. Tilly's methods were developed further in Barraquand and Martineau (1995)[18] using Stratified State Aggregation along Price however these were proven non-convergent (Coyle and Yang, 1999)[19] with dimensionality problem still remaining more or less unresolved. Boyle et al. (1997)[20] gave another method which was restricted by exponential complexity. Broadie and Glasserman (1997-i)[21] tried to get about this problem. They proved for a large class of problems the simulation estimator to be biased. Their alternative simulated tree approach produced two simulation estimators, one with high bias high and the other with low bias low. These estimators were shown to converge asymptotically and give out confidence intervals for the true value. This algorithm, however, proved to be computationally tedious as the number of exercise points or underlying assets grew. Broadie and Glasserman (1997-ii)[22] then presented the stochastic mesh method where exponential complexity was addressed but this model turned out to be having an extremely gradual simulation convergence. Taking cue from the stochastic mesh method, Boyle et al. (2000)[23] exhibited that with an appropriate choice of mesh density, the stochastic mesh method can be used in conjunction with the quasi-Monte Carlo technique to get a substantial bias reduction of the high-biased estimator. This work was extended further by them by considering both low and high-biased estimators to set up an interval for true simulation solution. Earlier Avramidis and Hyden (1999)[24] had produced similar study though in the context of a stochastic mesh only. An efficient solution to American option pricing came in the form of the construct developed by Longstaff and Schwartz (2001)[25].This construct specifically showed how efficient estimate by regression could be arrived based on the continuation value conditionally on optimal exercise strategy. Now we have latest issues that suggest variance reduction techniques and use of Malliavin calculus to improve the Monte Carlo simulations in terms of making it faster, with sharper convergence and relatively efficient pricing of difficult products like single and multiple assets American options. Among implementation issues it is at once clear that due to the fact that Monte Carlo methods are computer intensive one must use an adaptable and rich programming language with accompanying applications in compatible software. While C language can be used for writing code applications can be built in Visual Basic(VB).C is faster than VB.Therefore any latest C compiler can be used to code the model. Users interfaces can be built in a combination of Microsoft excel and VB. VB can be linked to external programs easily using dynamically linked libraries (dll). The built-in C language random number generator has unsatisfactory performance (Press et al, 2002)[26]. The randomization of uniform deviates is essential in Monte Carlo simulation and stochastic modeling. Therefore an alternative random number generator could be used that produces robust uniform deviates randomization. Works Cited 1. Joy C., Boyle, P.P. and Tan, K.S. 1996. Quasi-Monte Carlo methods in numerical finance. Management Science, 42(6):926-938. 2. Boyle, P.P., Kolkiewicz, A. and Tan K.S. 2000. Pricing American style options using low discrepancy mesh methods. Technical report, IIPR 00-07, University of Waterloo, 3. Benhamou, Eric .Smart Monte Carlo: January, 2001. Various tricks using Malliavin calculus. 4. Benhamou E. June 2000-i. Application of Malliavin Calculus and Wiener Chaos to Option Pricing Theory, Ph.D. Thesis, London School of Economics. 5. Benhamou E. December 2000-ii. Optimal Malliavin Weighting Function for the Computation of the Greeks, Proceedings of the Monte Carlo Congress, Monte Carlo June 2000. 6. Fourni E., Lasry J.M., Lebuchoux J., Lions P.L. and Touzi N. 1999. Applications of Malliavin Calculus to Monte Carlo methods in Finance, Finance and Stochastics. 3. 391-412. 7. Fourni E., Lasry J.M., Lebuchoux J. and Lions P.L. 2001. Applications of Malliavin Calculus to Monte Carlo Methods in Finance. II., Finance and Stochastics. 8. Broadie, M., Glasserman, P.1996.Estimating security price derivatives using simulation, Manag. Sci. 42. 269-285. 9. Carr, P. 1998. Randomization and the American put. Review of Financial Studies, 11.597-626. 10. Grant, D., Vora, G., and Weeks, D. 1997. Path dependent options and early exercise: extending the Monte Carlo simulation method. Management Science. 43.1589-1602. 11. Bunch, D. and Johnson, H. 2000. The American put option and its critical stock price. Journal of Finance. 55.2333-2356. 12. Huang, J., Subrahmanyam, M., and Yu, G. 1996. Pricing and hedging American options: A recursive integration method. Review of Financial Studies. 9(1).277-300. 13. Geske, R. and Johnson, H. 1984. The American put option valued analytically. Journal of Finance.39, 1511-1524. 14. Barone Adesi, G. and Whaley, R. 1987.Efficient analytic approximation of American option values. Journal of Finance. 42.301-320. 15. Cox, J. C., Ross, S. A., and Rubinstein, M. 1979. Option pricing: a simplified approach. Journal of Financial Economics.7.229-263. 16. Tilley, J. 1993. Valuing American options in a path simulation model. Transactions of the Society of Actuaries, 45.83-104. 17. Bossaerts, P. 1989. Simulation estimators of optimally early exercise. Technical report, Graduate School of Industrial Admistration.Carnegie Mellon University. 18. Barraquand, J. and Martineau, D.1995. Numerical valuation of high dimensional multivariate American securities. Journal of Financial and Quantitative Analysis. 30.383-405. 19. Coyle, L. and Yang, J.1999. Analysis of the SSAP method for the numerical valuation of high-dimensional multivariate American securities. Algoritmica. 25.75-98. 20. Boyle, P., Broadie, M., and Glasserman, P. 1997.Monte Carlo methods for security pricing. Journal of Economics Dynamics and Control.21.1267-1321. 21. Broadie, M. and Glasserman, P. 1997-i.Pricing American-style securities using simulation. Journal of Economic Dynamics and Control. 21(8-9).1323-1352. 22. Broadie, M. and Glasserman, P.1997-ii. A stochastic mesh method for pricing. high-dimensional American options. Technical report, Columbia University Working Paper. 23. Avramidis, A.N. and Hyden, P. 1999. Efficiency improvement for pricing Amer- ican options with a stochastic mesh. In P.A. Farrington, H.B. Nembhard, 24. D.T. Sturrock, and G.W. Evans, editors, Proceedings of the 1999 Winter Conference.344-350. 25. Longstaff, F. and Schwartz, E. 2001. Valuing American options by simulation: a simple least-squares approach. Review of Financial Studies.14 (1).113-147. 26. Press, William H., Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery.2002.Numerical Recipes in C++.Cambridge: Cambridge University Press. Read More