Data Analysis and Related Applications 3

9Constructing Trinominal Models Based on Cubature Method on Wiener Space: Applications to Pricing Financial Derivatives

This contribution deals with an extension to our developed novel cubature methods of degrees 5 on Wiener space. In our previous studies, we studied cubature formulae that are exact for all multiple Stratonovich integrals up to dimension equal to the degree. In fact, cubature method reduces solving a stochastic differential equation to solving a finite set of ordinary differential equations. Now, we apply the above methods to construct trinomial models and to price different financial derivatives. We will compare our numerical solutions with the Black’s and Black–Scholes models’ analytical solutions. The constructed model has practical usage in pricing American options and American-style derivatives.

9.1. Introduction and outline of this chapter

In mathematical finance, it is common to describe the random changes in risky asset prices by stochastic differential equations (SDEs). SDEs can be rewritten in their integral forms. However, it is not possible to calculate all stochastic integrals in closed form. Therefore, proper numerical methods should be used to estimate the value of such stochastic integrals.

One of the most popular numerical methods to estimate stochastic integrals is the Monte Carlo method (estimate). In particular, according to Nohrouzian et al. (2022), cubature methods and consequently cubature formulae construct a probability measure with ...