Question

(a) What is the Black-Scholes Partial Differential Equation for the price f(t,St) at time t of a European derivative security on a stock with price St? Specify the meaning of the terms or symbols in the equation.

(b) Note that the Black-Scholes Partial Differential Equation does not specify whether the derivative is a call option, a put option, or some other derivative. How do you incorporate the derivative payoff when using Black-Scholes PDE to price a derivative?

Answer 1

Derivative is omething whose value is based on the value of one or more underlying asset. Examples of derivatives are stock, bonds, commodities, currencies, interest rate, market indices etc.

Black Scholes is an important concept  of finance today. This pricing model is considered appropriate one while valuing stocks.Black Scholes is basically a mathematical model developed for finance market. Its partial differentiation equation governs the prices of European call option or put option. So, here we undwerstand this model is basically used in case of European derivatives.This model is also called Black Scholes Merton Model. It is said that Fisher Black had started the working of this model and his purpose was to create valuation model for stock warrants. Later on Myron Scholes joined him in the developing process and it resulted in Black Scholes pricing model. The model assumes that price of highly traded assets folow a brownian motion with a constant shift and unpredictability. When this pricing model is used to a stock option, it helps in knowing constant price variation, time value of money, the option strike rate as well as the end of an option or say expiry time of an option.

Assumptions of Black Scholes price differentiation Model:

There are basically 6 assumptions and they are:

a) Stock pays no dividend.

b) Option can only be traded till expiry.

c) Market movement is not predictable.

d) Commission are not added upto transactions.

e) Interest rates remains constant.

f) Profitability or return on stock is evenly distributed.

Now coming to the question,

S = Stock Price or current value , t= time to maturity,P= value of portfolio,V(s,t) = option price, K= striker rate and r = risk free rate.σ = price volatility.