Lookback option to find stock price

Question

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock at current time.

I've done some research and this option seems similar to the floating strike option. But I don't know how to approach this problem yet. Any help would be appreciated.

Answer 1

Below I assume that you meant: $\psi (T) = \max (S_t - S_T, 0) $ which constitutes the payout of a forward start rather than a lookback option. If not please clarify your question...

If you are looking for the option price $V_0$, assuming a Black-Scholes diffusion (GBM + constant interest rates), you have

\begin{align*} V_0 &= P(0,T) E[ \psi (T) \vert \mathcal {F}_0] \\ & = P (0,T) E \left[ (S_t - S_T)^+ \vert \mathcal {F}_0 \right] \\ & = P (0,T) E \left[ E [ (S_t - S_T)^+ \vert \mathcal {F}_t ] \vert \mathcal {F}_0 \right] \\ & = P (0,t) E \left[ P (t,T) E [ (S_t - S_T)^+ \vert \mathcal {F}_t ] \vert \mathcal {F}_0 \right] \\ &= P (0,t) E \left[ P_{BS}(S_t; S_t, T-t) \vert \mathcal {F}_0 \right] \\ &= P (0,t) E [ S_t P_{BS}(1; 1, T-t) \vert \mathcal {F}_0 ] \\ &= P (0,t) F (0,t) P_{BS}(1; 1, T-t) \end{align*}

Where $P(s,t) = e^{-r(t-s)}$ denotes a generic discount factor and each the successive equalities come from:

1. Option premium as a risk-neutral expectation

2. Definition of the option's payout

3. Tower property of conditional expectation

4. Composition of discount factors

5. Price of a put option which, under the modelling assumptions, is given by the BS formula (first argument denotes stock price, second strike level and last time to maturity)

6. Space homogeneity of BS formula

7. Forward price as a risk-neutral expectation ($P_{BS} (1;1,T-t)$ is deterministic)

Note that because the forward price $F (0,t) $ is directly proportional to the spot price $S_0$ you can obviously infer the spot value if you know the option premium $V_0$. Actually assuming no dividends: $V_0 = S_0 P_{BS}(1; 1, T-t) $

However, I don't see the point of doing that, except for some pure academic fun.