The Approximate Formula Of The Solution To The Option Pricing Problem Of A Basket Of Currencies

In the field of finance,a basket currencies option is an important branch of multi-asset European option.Mathematically,this kind of option can be described by the famous multidimensional Black Scholes equation（see reference [1]）.The multidimensional Black Scholes equation is the following second order partial differential equation of the formwhere Siis the exchange rate of ith international currency,V =V（S1,S2,· · ·,Sn,t）is the option price based on the exchange rate of international currencies S1,S2,· · ·,Sn,the constant qi∈（0,+∞）is dividend rate of original asset Si,the constant r ∈（0,+∞）is risk-free rate,and the following expression formula is the coefficient of the second derivative termwhere σ_ijis volatility,m and n are two given positive constants.Furthermore,when t = T which means the maturity date of the option arrives,the expression formula of a basket currencies option iswhere the constant K ∈（0,+∞）is weighed average exchange rate depended on different international currencies trade fees,the constant αi∈（0,1）is the proportion of the ith exchange rate of Siin the basket currencies option,which has the following propertiesClearly,the rank of the coefficient matrix of second derivatives of BlackScholes equationequals or is less than n.We discuss the following two situations respectively.Situation I:The rank of matrix（aij）n×nequals n.At the same time,matrix（aij）n×nis a symmetric positive definite matrix,and det（aij）n×n> 0.Situation II:The rank of matrix（aij）n×nis less than n.At the same time,matrix（aij）n×nis a symmetric positive semidefinite matrix,and det（aij）n×n= 0.For the situation I,i.e.det（aij）n×n> 0,Professor Lishang Jiang（see [1]-[2]）finds the solution of a basket currencies option pricing problem Black-Scholes formula.This Black-Scholes formula is a n multiple integral with singular integrand.Then Professor Lishang Jiang（see [1]-[2]）in remark illustrates “ Any multiple asset European option pricing problem has an explicit expression,but it is an n multiple integral with singular integrand function.If there are too many original assets,that is to say,the multiplicity of integral is high,it’s a difficult problem to compute the integral.Therefore it’s important to find a more efficient way to calculate.In this sense,for multiple asset European option pricing problem,it’s the very first step to find an explicit expression.To fundamentally solve this problem,we have to try our best to find a way to simplify the problem for each specific problem.”Next,Professor Lishang Jiang（see [1]-[2]）points out（from reference [1],page225）:“According to the portfolio theory,the volatility of a basket of risk assets is generally relatively small.” The empirical results of “the volatility of the exchange rate of a basket of currencies is relatively small” can also be found in the reference[3],page 20-22.It can be found in reference [3]（see pages 20-22 of reference [3]）.Based on the above reasons,this thesis explores the approximate formula of the solution of a basket currencies option pricing problem in the case of small volatility to simplify the calculation.To be specific,In Chapter 1,we introduce research background an research meaning of a basket currencies option pricing.Besides,we introduce some fundamental knowledge.In Chapter 2,we discuss situation I,i.e.det（aij）n×n> 0.Under this circumstance,multidimensional Black-Scholes equation is a backward parabolic equation,so we can have multidimensional Black-Scholes formula.First,we lead volatility parameter.Second,we apply several techniques,including smoothing operator,overcome the difficulty caused by unbounded terminal value function and discontinuousness of first degree of derivative,find an approximate formula of a basket currencies option pricing problem as small volatility,simplify multidimensional Black-Scholes formula,and then to simplify the calculation.In Chapter 3,we discuss situation II,i.e.det（aij）n×n= 0,under this circumstance,Black-Scholes equation is no longer a backward parabolic equation.Obviously,Black-Scholes formula for a basket currencies option pricing problem doesn’t exist,either the classical solution.First,we lead a volatility parameter.Second,we apply smoothing operator and the technique of uniform estimation.We overcome the difficulty caused by not only unbounded terminal value function and discontinuousness of first degree of derivative,but also no more backward parabolic equation.We define a set of generalized solution,prove the existence and uniqueness of generalized solution.At the same time,we also got an approximate formula for the generalized solution to simplify the calculation.In Chapter 4,we give an example for numerical calculation and an empirical analysis.These numerical calculation and empirical analysis fully show that the approximate formula obtained in this paper has a certain reliability,so there is applicable meaning for our research.All in all,there is not only certain financial meaning,but also the theoretical significance and application value in the paper.It can also gives some advice for investors.