Consider a forward contract, where the buyer is under the obligation to purchase one unit of stock at some future time T , for an agreed price K. Consider first the case of no dividends or borrow costs. Show, that by shorting one unit of stock at a price S0 today, and investing the proceeds in zero coupon bonds paying unity at T , the fair strike for the contract is given by:

where P(T0, T ) is the value of the zero coupon bond paying unity at expiry. In the case where rates rt are deterministic, show that the bond price is given by:

(Hint: consider shorting the bond and investing the proceeds in a cash account accruing interest at rate r(t).)
Now consider the case where the stock pays a continuous dividend yield, such that the dividend paid out to a stock holder over the interval [t, t + dt] is given by Stq(t)dt. By considering a total return strategy, where the dividend is continuously reinvested into stock, show that the starting short position in stock consists of shares. Use this result to show, in the case where both the dividend yield and rates are deterministic, that the fair strike of the forward contract is given by:

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K = So P(To,T)