Schmitz Abe, Klaus and Giles, M. B. (2006) Pricing exotic options using strong convergence properties? In: ECMI 2006.

PDF
209kB 
Abstract
In finance, the strong convergence properties of discretisations of stochastic differential equations (SDEs) are very important for the hedging and valuation of exotic options. In this paper we show how the use of the Milstein scheme can improve the convergence of the multilevel Monte Carlo method, so that the computational cost to achieve an accuracy of O(e) is reduced to O() for a Lipschitz payoff. The Milstein scheme gives first order strong convergence for all 1−dimensional systems (one Wiener process). However, for processes with two or more Wiener processes, such as correlated portfolios and stochastic volatility models, there is no exact solution for the iterated integrals of second order (Lévy area) and the Milstein scheme neglecting the Lévy area gives the same order of convergence as the EulerMaruyama scheme. The purpose of this paper is to show that if certain conditions are satisfied, we can avoid the calculation of the Lévy area and obtain first convergence order by applying an orthogonal transformation. We demonstrate when the conditions of the 2−Dimensional problem permit this and give an exact solution for the orthogonal transformation. We present examples of pricing exotic options to demonstrate that the use of both the orthogonal Milstein scheme and the Multilevel Monte Carlo give a substantial reduction in the computation cost.
Item Type:  Conference or Workshop Item (Paper) 

Subjects:  D  G > Game theory, mathematical finance, economics, social and behavioral sciences 
Research Groups:  Mathematical and Computational Finance Group 
ID Code:  581 
Deposited By:  Eprints Administrator 
Deposited On:  27 Mar 2007 
Last Modified:  29 May 2015 18:24 
Repository Staff Only: item control page