Consider a market with one stock and one bond, with the following assumptions on

Question

Consider a market with one stock and one bond, with the following assumptions on parameters: mu and sigma > 0 are constant but the interest rate is r(t) = a + bt for some constants a and b. Find the value at t = 0 for the European call option with the strike price K and maturity T. Note that this is the same option as in the usual Black-Scholes formula but the assumptions on the interest rate are different. Your answer has to be explicit in the sense that one could feed it into a computer program, just like the Black-Scholes formula. The formula may contain integrals. It is not necessary to simplify the formula.

Explanation / Answer

Assume price of claim is P(0) at time t=0.

Let's assume strike price was = q

there will be profit if P(T1) is greater than K1 and payoff will be = K2- q

if P(T1) is less than K1 then payoff is 0.

By using arbitrage opportunity we can find the price.

suppose there is an initial wealth x which is geater than 0 a pair (k1,k2) belongs to A (x).

then at T=0

x+aP(0) will be less than 0. where a is a constant which can be either 1 or -1.

x will be a function of K1,K2

SO x can be denoted by Xx,K1,K2

So equation will be

Xx,K1,K2 (0) + a P(0) 0

Less than sign represent that there is no orbitrage opportunity in the expiration time of the option (contingent claim).

From this you can calculate the value of P(0).

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