Determination of Multi-Asset Black-Scholes Differential Equation Using Backward Stochastic Differential Equations (BSDEs)
Aimmatul Ummah Alfajriyah (*), Endah RM Putri, Kistosil Fahim

Department of Mathematics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Jl. Raya ITS, Sukolilo, Surabaya, 60111, Indonesia.
(*)aimmatulummahff[at]gmail.com

Abstract

Multi-asset Black-Scholes differential equation is widely used in option pricing. One method of option pricing is backward stochastic differential equations (BSDEs). A BSDE has important applications in the field of mathematical finance, since the BSDE can be used in pricing financial products on incomplete markets. A primary objective of this study is to determine the multi-asset Black-Scholes differential equation using the BSDE. This research begins with constructing a multi-asset portfolio in the form of BSDE. Relationship between BSDE and partial differential equations (PDEs) is given by Feynman-Kac theory. Using the Feynman-Kac theory, we derive BSDE of multi-asset portfolio such that it has a unique solution. It is also a solution of multi-asset Black-Scholes differential equation. Then, by deriving BSDE, the multi-asset portfolio can be transformed into the multi-asset Black-Scholes differential equation. We also obtain the exact solution of the multi-asset option price on the basket option, by transforming multi-asset Black-Scholes differential equation into a diffusion equation. As an application, some simulations of multi-asset option prices are conducted.

Keywords: Basket Option- Black-Scholes Equation- BSDEs- Feynman-Kac- Multi Asset Option- Option Pricing

Topic: MATHEMATICS AND STATISTICS