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)