Overview

A forward is a derivative contract which specifies some number of units of an underlying asset to be delivered at some maturity date.

Forwards and Arbitrage

The most common way to value a forward is through the use of an arbitrage arbitrage. In particular, it is asked how much money the issuer of the forward should ask for in order to be able to hedge her risk entirely.

• Forward on a Stock (or other spot asset) - in order to fully hedge a forward position on a stock, the issuer of the forward must actually own the stock. That is, she must borrow money to buy the stock today, for delivery at the maturity date. The price that she then must be paid is price of the stock today, compounded at the rate present in the fixed income market for a zero coupon maturing at the maturity date of the forward.
• Forward Rate Agreement A forward rate agreement is an agreement to enter a fixed income contract at some time, say {% T %} in the future, for maturity at another later time, say {% S %}. That is, the purchaser wishes to invest a set amount of money at time {% T %} to expire at time {% S %}, and wants to lock in the interest rate today. The forward rate agreement is typically constructed so as not to have a price. That is, price is zero.
  
  To hedge this risk, the issuer borrows money today, to mature at time time {% T %}. The amount of money that she borrows today is the discounted value of the dollar amount to be recieved at time {% T %}, that is
  {% u = e^{-r_T T} %}
  where we assume that the forward agreement is for a single dollar to be invested. {% r_T %} is the current (continuous) discount rate for maturity at {% T %}, and {% u %} is the amount borrowed today.
  
  Next the issuer invests the {% u %} dollars today to mature at time {% S %}.
  
  At time {% T %}, the issuer receives a dollar from the purchaser, which shen then uses to pay off her loan.
  
  At time {% S %}, the issuer receives {% u e^{r_S S} = e^{r_S S - r_T T} %} dollars from the investment she made at time {% 0 %}. This money is then given to the purchaser.
  
  To compute the rate that the purchaser received on the dollar invested in the forward rate agreement, we set
  {% e^{r(S-T)} = e^{r_S S - r_T T} %}
  where {% r %} is the rate of the forward agreement.
  
  This implies that the arbitrage free rate is
  {% r = \frac{r_S S - r_T T}{S-T} %}
  Note, this is quoted in terms of continuous compounding. If simple compounding is used, we have
  {% e^{r_S S - r_T T} = 1 + R (S-T) %}
  {% R = \frac{1}{S-T}(e^{r_S S - r_T T} -1) %}