134 Mathematics for Finance
Such a contract may have non-zero value initially. In the case of a stock paying
no dividends
VX(0) = [F(0,T)−X]e−rT=S(0)−Xe−rT. (6.10)For a stock paying one dividend between times 0 andTthe initial value of the
contract is
VX(0) =S(0)−div 0 −Xe−rT,
div 0 being the value of the dividend discounted to time 0. For a stock paying
dividends continuously at a raterdiv, the initial value of the contract is
VX(0) =S(0)e−rdivT−Xe−rT.Exercise 6.7
Suppose that the price of a stock is $45 at the beginning of the year, the
risk-free rate is 6%, and a $2 dividend is to be paid after half a year.
For a long forward position with delivery in one year, find its value after
9 months if the stock price at that time turns out to be a) $49, b) $51.6.2 Futures .................................................
One of the two parties to a forward contract will be losing money. There is
always a risk of default by the party suffering a loss. Futures contracts are
designed to eliminate such risk.
We assume for a while that time is discrete with steps of lengthτ, typically
aday.
Just like a forward contract, afutures contractinvolves an underlying asset
and a specified time of delivery, a stock with pricesS(n)forn=0, 1 ,...and
timeT, say. In addition to the usual stock prices, the market dictates the so-
calledfutures pricesf(n, T) for each stepn=0, 1 ,...such thatnτ≤T.These
prices are unknown at time 0, except forf(0,T), and we shall treat them as
random variables.
As in the case of a forward contract, it costs nothing to initiate a futures
position. The difference lies in the cash flow during the lifetime of the contract.
A long forward contract involves just a single paymentS(T)−F(0,T)atde-
livery. A futures contract involves a random cash flow, known asmarking to
market. Namely, at each time stepn=1, 2 ,...such thatnτ≤Tthe holder of
a long futures position will receive the amount
f(n, T)−f(n− 1 ,T)