12 Mathematics for Finance
Assumptions 1.1 to 1.5 as well as the No-Arbitrage Principle extend readily to
this case.
The forward priceF is determined by the No-Arbitrage Principle. In par-
ticular, it can easily be found for an asset with no carrying costs. A typical
example of such an asset is a stock paying no dividend. (By contrast, a com-
modity will usually involve storage costs, while a foreign currency will earn
interest, which can be regarded as a negative carrying cost.)
A forward position guarantees that the asset will be bought for the forward
priceF at delivery. Alternatively, the asset can be bought now and held until
delivery. However, if the initial cash outlay is to be zero, the purchase must be
financed by a loan. The loan with interest, which will need to be repaid at the
delivery date, is a candidate for the forward price. The following proposition
shows that this is indeed the case.
Proposition 1.2
Suppose thatA(0) = 100,A(1) = 110, andS(0) = 50 dollars, where the risky
security involves no carrying costs. Then the forward price must beF =55
dollars, or an arbitrage opportunity would exist otherwise.
Proof
Suppose thatF> 55 .Then, at time 0:
- Borrow $50.
- Buy the asset forS(0) = 50 dollars.
- Enter into a short forward contract with forward priceFdollars and delivery
date 1.
The resulting portfolio (1,−^12 ,−1) consisting of stock, a risk-free position, and
a short forward contract has initial valueV(0) = 0. Then, at time 1:
- Close the short forward position by selling the asset forFdollars.
- Close the risk-free position by paying^12 ×110 = 55 dollars.
The final value of the portfolio,V(1) =F− 55 > 0 ,will be your arbitrage
profit, violating the No-Arbitrage Principle.
On the other hand, ifF<55, then at time 0:
- Sell short the asset for $50.
- Invest this amount risk-free.
- Take a long forward position in stock with forward priceF dollars and
delivery date 1.
The initial value of this portfolio (− 1 ,^12 ,1) is alsoV(0) = 0.Subsequently, at
time 1: