Fixed Income with Equity

The first day of class in a financial engineering course generally starts with the following:

Let S1 be the price of  a stock today,  F1 the the value of an equity forward contract, K the strike price of such a contract and R the risk free rate. What is the non-arbitrage price of K?

(Side note: There are not that many good ideas in finance. I would actually summarize them in three: 1. Time value of money. 2. Non-arbitrage and 3. Risk neutral pricing (state contingent pricing is probably better) which is what Black, Scholes and Merton did).

Note, the value of F1 today is equal to F1=S1-K, the question is, what is the value of this K then or how much should you pay for it? The class continues in the following way:

Assume you sell (short) a stock, you could invest the proceeds at the rate R and in the next period get the amount of S1*(1+R). Now, imagine that at the same time that you sold S1 you opened with your favorite broker a forward contract where you agreed to buy stock S in the following period (you eventually have to deliver the stock you sold). The market value of your forward in the future will be F2=S2-K. Got it? No? Well, just look at the profit equation at time 2

Profit= S1*(1+R) + (S2-K) -(S2) The first term is what you invested at the bank, the second one the market value of your forward and the third the cost you incur for giving back the stock you shorted. Clearly (regardless of S2), if K is less than S1*(1+R) you can make yourself rich without investing any money, people will start selling S1 until K=S1*(1+R). You can apply a similar logic when K is greater than S1*(1+R).

Why all this? Well, note that the  payoff does not depend upon the final value of the stock, that is, we have managed to transform equity trades into a fixed income transaction. That is financial engineering, although I still prefer the name quantitative finance.