Question

Question 1 Consider the derivation of the Black-Scholes model of option pricing. Let S=S(t) be the underlying stock price at time t and let f=f(S, t) be the option price at time t.

a) Write down the value P of the portfolio defined in the Black-Scholes model. [2 marks]

b) Use Itô’s lemma to find an expression for the change Δf in the discrete time Δt. [5 marks]

c) Use the expression you have found in point b) to find an expression for the discrete change in the value of the Black-Scholes portfolio. [5 marks]

d) Find the number of shares ‘Delta’ so that the random component is eliminated from the discrete change in the value of the Black-Scholes portfolio? [3 marks]

e) Given the choice you indicated for the Black-Scholes ‘Delta’, now derive the BlackScholes partial differential equation. [15 marks]

Answer 1

a)The Black Scholes model, also known as the Black-Scholes-Merton (BSM) model, is a mathematical model for pricing an options contract. In particular, the model estimates the variation over time of financial instruments such as stocks, and using the implied volatility of the underlying asset derives the price of a call option.

Black-Scholes model makes certain assumptions:

• The option is European and can only be exercised at expiration.
• No dividends are paid out during the life of the option.
• Markets are efficient (i.e., market movements cannot be predicted).
• There are no transaction costs in buying the option.
• The risk-free rate and volatility of the underlying are known and constant.
• The returns on the underlying are normally distributed