128 Mathematics for Finance
Exercise 6.2
Suppose that the price of stock on 1 April 2000 turns out to be 10%
lower than it was on 1 January 2000. Assuming that the risk-free rate
is constant atr=6%,what is the percentage drop of the forward price
on 1 April 2000 as compared to that on 1 January 2000 for a forward
contract with delivery on 1 October 2000?Remark 6.1
In the case considered here we always haveF(t, T)=S(t)er(T−r)>S(t). The
differenceF(t, T)−S(t), which is called thebasis, converges to 0 ast↗T.
Remark 6.2
Under periodic compounding the forward price is given by
F(0,T)=S(0)(1 +r
m)mT.In terms of zero-coupon bond prices, this formula becomes
F(0,T)=S(0)B(0,T)−^1.The last formula is in fact more general, requiring no assumption about con-
stant interest rates.
Including Dividends.We shall generalise the formula for the forward price
to cover assets that generate income during the lifetime of the forward contract.
The income may be in the form of dividends or a convenience yield. We shall
also cover the case when the asset involves some costs (called the cost of carry),
such as storage or insurance, by treating the costs as negative income.
Suppose that the stock is to pay a dividend div at an intermediate timet
between initiating the forward contract and delivery. At timetthe stock price
will drop by the amount of the dividend paid. The formula for the forward
price, which involves the present stock price, can be modified by subtracting
the present value of the dividend.
Theorem 6.2
The forward price of a stock paying dividend div at timet,where0<t<T,is
F(0,T)=[S(0)−e−rtdiv]erT. (6.3)