Take the 2-minute tour ×
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

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)

share|improve this question
    
I am increasing in confidence that this is correct because I get the same answer when I work with measure $P^*$ associated with taking the growth optimal portfolio as the numeraire. –  Jase Nov 23 '12 at 14:45
    
You should probably make clear what $r$ (a constant, I guess) and $r(t)$ is. The easiest would be constant interest- ie. $r$ everywhere, right? –  Richard Nov 23 '12 at 20:34
add comment

2 Answers

up vote 2 down vote accepted

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?

share|improve this answer
add comment

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... :-)

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.