126 Mathematics for Finance
delivery areS(T)−F(0,T) for a long forward position andF(0,T)−S(T)for
a short position; see Figure 6.1.
Figure 6.1 Payoff for long and short forward positions at deliveryIf the contract is initiated at timet<Trather than 0, then we shall write
F(t, T) for the forward price, the payoff at delivery beingS(T)−F(t, T)fora
long forward position andF(t, T)−S(T) for a short position.
6.1.1 Forward Price......................................
The No-Arbitrage Principle makes it possible to obtain formulae for the forward
prices of assets of various kinds. We begin with the simplest case.
Stock Paying No Dividends.Consider a security that can be stored at no
cost and brings no profit (except perhaps for capital gains arising from random
price fluctuations). A typical example is a stock paying no dividends. We shall
denote byrthe risk-free rate under continuous compounding and assume that
it is constant throughout the period in question.
An alternative to taking a long forward position in stock with delivery at
timeTand forward priceF(0,T) is to borrowS(0) dollars to buy the stock
at time 0 and keep it until timeT. The amountS(0)erTto be paid to settle
the loan with interest at timeTis a natural candidate for the forward price
F(0,T).The following theorem makes this intuitive argument formal.
Theorem 6.1
For a stock paying no dividends the forward price is
F(0,T)=S(0)erT, (6.1)whereris a constant risk-free interest rate under continuous compounding. If
the contract is initiated at timet≤T,then
F(t, T)=S(t)er(T−t). (6.2)