On a method of solving the Black-Scholes Equation

Abstract

The paper proposes a different method of solving a simplified version of the Black-Scholes equation. In the first part of the paper, the Black-Scholes equation is transformed into ordinary differential equation to get a solution similar to the solution of the Euler equation. The second part of the paper focuses on partial differential equation. Separation of variables method is used to solve the Black-Scholes equation. Plots corresponding to put and call options are also given.

Numerical Approximations of a Nonlinear Volatility Model with European Options

“…Black-Scholes model creates an adjustment between stocks and options [22]. The Black-Scholes equation is one kind of parabolic equation [17,31] and at the same time it depends on two independent variables, first is the time and the second is the stock price which follows a random path [1,2]. Basically Black-Scholes model equation represents a partial differential equation (PDE) [2,7,9,20,23,24] and it is very important to note that if we wish to solve a PDE, then we may get infinite number of solutions [9].…”

“…Black-Scholes equation is the most preferable model in the field of option pricing as it can calculate the price of an option more correctly [1,2,4,17,18]. A large number of researchers and investors have shown their interests in the study of Black-Scholes model in these days as it is one of the most important tools to compute the value of an option [1,31]. Fisher Black, Myron Scholes and Robert Merton, these three great economists [31] contributed to build-up the model in 1970 [1,2,4,19].…”

“…A large number of researchers and investors have shown their interests in the study of Black-Scholes model in these days as it is one of the most important tools to compute the value of an option [1,31]. Fisher Black, Myron Scholes and Robert Merton, these three great economists [31] contributed to build-up the model in 1970 [1,2,4,19]. It is possible to banish risks that does exist in markets through Black-Scholes model that can create a risk-less hedging portfolio [18,20,21].…”

“…Options are hugely applied for hedging [11]. The fixed price is called the strike price or exercise price and it is denoted by E [1,4,11,31]. The fixed time is called the expiration time or the maturity time and it is denoted by T [1,4,11,31].…”

“…The fixed price is called the strike price or exercise price and it is denoted by E [1,4,11,31]. The fixed time is called the expiration time or the maturity time and it is denoted by T [1,4,11,31]. Basically call option and put option are exercised in real market [11].…”