Valuing a Stock Forward and Distinguishing Its Price from Its Value

# Pricing forward contract on a stock


# Pricing forward contract on a stock












Please tell me where I've gone wrong (if I did in fact make a mistake). I'm pricing a long forward on a stock. The usual setup applies:

- This has payoff $S(T) - K$ at time $T$.

- We are at $t$ now.

- $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t)+\sigma(W(T)-W(t))}$.

- $W(t)$ is a Wiener process.

- $K \in \mathbb{R}_+$.

- $Q$ is the risk-neutral measure.

- $\beta(t) = e^{rt}$ is the domestic savings account, a tradable asset. $r$ is the constant riskless rate.

My Attempt:

$f(t,S) = E^Q[\frac{\beta(t)}{\beta(T)}(S(T)-K)|\mathscr{F}_t]$

$ = E^Q [\frac{\beta(t)}{\beta(T)}S(T)|\mathscr{F}_t] - E^Q [\frac{\beta(t)}{\beta(T)}K|\mathscr{F}_t]$

$ = E^{P_S}[\frac{\beta(t)}{\beta(T)}S(T) \frac{\beta(T)S(t)}{\beta(t)S(T)}|\mathscr{F}_t] - \frac{\beta(t)}{\beta(T)}K$

$ = S(t) - K\frac{\beta(t)}{\beta(T)}$

$ = S(t) - Ke^{-r(T-t)}$

This isn't graded homework or assignment. (It is ungraded homework)

## Answer by Richi Wa (score 2, accepted)

https://quant.stackexchange.com/a/4613

In my mind you are simply right: you arrive at $$ f(t,S) = S(t) - K e^{-r(T-t)}. $$ Assume that $t=0$, so we are at the inception of the contract, then $$ f(0,S) = S(0) - Ke^{-r T}. $$ If you choose $K = S(0) e^{r T}$ then the contract value at inception is zero. This simply means that the fair price for the forward is given by $K= S(0) e^{r T}$ which is the formula that you find in text books. Does this answer your question?

## Answer by Fab (score 2)

https://quant.stackexchange.com/a/10827

Richard nails it.

One needs to distinguish the forward price (or just "forward"), which is a number that denotes at which strike you can now enter a forward without upfront payment, and the value of a forward contract, which is typically zero at inception (if the strike chosen is indeed the forward price), but then varies over time, and ends up as $S(T) - K$ at T, with whatever strike K was chosen.

So, if there are no dividends and other cost of carry besides rates r, the forward price at 0 for expiry T is indeed $K = S(0) e^{rT}$, and thus the value at time $t$ of a forward contract expiring at time $T$ that was entered at time 0 is

$S(t) - S(0)e^{rt}$

which, incidentally, shows nicely that a forward has a delta of 1, at least in the absence of dividends and other distractions (which is why, incidentally, I think delta-one desks should be renamed to gamma-zero... :-)