- Forward and Futures Contracts 131
Next, suppose that a stock pays dividends continuously at a raterdiv>0,
called the (continuous)dividend yield. If the dividends are reinvested in the
stock, then an investment in one share held at time 0 will increase to become
erdivTshares at timeT.(The situation is similar to continuous compounding.)
Consequently, in order to have one share at timeTwe should begin with e−rdivT
shares at time 0. This observation is used in the arbitrage proof below.
Theorem 6.3
The forward price for stock paying dividends continuously at a raterdivis
F(0,T)=S(0)e(r−rdiv)T. (6.6)Proof
Suppose that
F(0,T)>S(0)e(r−rdiv)T.
In this case, at time 0
- enter into a short forward contract;
- borrow the amountS(0)e−rdivTto buy e−rdivTshares.
Between time 0 andTcollect the dividends paid continuously, reinvesting them
in the stock. At timeTyou will have 1 share, as explained above. At that time
- sell the share forF(0,T),closing out the short forward position;
- payS(0)e(r−rdiv)Tto clear the loan with interest.
The final balanceF(0,T)−S(0)e(r−rdiv)T >0 will be your arbitrage profit.
Now suppose that
F(0,T)<S(0)e(r−rdiv)T.
If this is the case, then at time 0
- take a long forward position;
- sell short a fraction e−rdivTof a share investing the proceedsS(0)e−rdivT
risk free.
Between time 0 andTyou will need to pay dividends to the stock owner, raising
cash by shorting the stock. Your short position in stock will thus increase to 1
share at timeT. At that time
- buy one share forF(0,T) and return it to the owner, closing out the long
forward position and the short position in stock; - receiveS(0)e(r−rdiv)Tfrom the risk-free investment.
Again you will end up with a positive amountS(0)e(r−rdiv)T−F(0,T)>0,
contrary to the No-Arbitrage Principle.