 # A forward contract valuation problem

Suppose there is a forward contract that is cash settled at time T. It uses the difference between the asset price at T-1 (ie S_(T-1)) and delivery price K as the pay-off. Then what is the value of the forward contract at time 0?

ymc, the value of most forward contracts is zero at initiation. There are also off-market FRAs, meaning the contract has a non-zero value and payment must be made by the the long to the short (or vice versa) at initiation. The CFAI volumes clearly address this and provide the equations for no-arbitrage pricing and valuation of forwards at initiation, during their lifetimes, and at termination. Check out the first reading in Volume VI.

But I think you can sell your contract to someone else before the settlement date for a certain price, right? In my problem, time 0 refers to time T before the settlement date. It doesn’ t mean the contract was initiated at time 0.

ymc, a little clarification: yes, a forward can be sold prior to maturity, but I don’t think this is probable because forwards (unlike futures) are contracts customized to meet the needs of the parties involved and therefore may not be attractive to third parties. However, there are other good reasons to know the value of forwards during their lives, like assessing credit risk, for example. The reason your question wasn’t clear initially is because you’re using notations that differ from CFAI. They use: time zero = initiation time t = some period after initiation but before maturity time T = maturity The general equation for what you want is V_t(0,T) = S_t - F(0,T)/(1+r)^(T-t) But if the underlying has cash flows, the PV of those have to be subtracted out too. There’s also discrete vs. continuous compounding to be considered. Reading 62 provides all the equations for forwards on equities, fixed income, interest rates and currencies.

I know the equatons and I know the value of the forward is S_0 + K*exp(-r*T) if the payoff function is the difference between S_T and K. But the problem is that the payoff function is now the difference between S_(T-1) and K. That’s what messed up my mind…

I think I get the answer now. It goes like this At time 0, Borrow S_0*exp(-r) at rate r Buy exp(-r) shares short a cash settled forward contract that has delivery price K and uses S_(T-1) to settle at time T. Denote f_0 as the proceed (ie contract value) you receive from shorting this contract. Put f_0 to earn interest r At time T-1, Sell all the shares you hold at time T-1, pay off your loan. The proceed is S_(T-1)*exp(-r) - S_0*exp(r*(T-2)). Put the proceed to earn interest r for one period. At time T, proceed from time T-1 grows by exp® receive pay off from forward contract K - S_(T-1) The arbitrage profit is K - S_(T-1) + [S_(T-1)*exp(-r) - S_0*exp(r*(T-2))*exp® + f_0*exp(r*T) To be arbitrage-free, this whole thing should be 0. So f_0 = (S_0*exp(r*(T-1) - K)*exp(-rT) Do you think my math is right? Thanks!