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]