Pricing derivatives using Monte Carlo Techniques Essay

Pricing derivatives using Monte Carlo Techniques - Essay Example 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 funct ion 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 functi on 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