Floating Strike Lookback Call Option

Question

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate $r$, stock drift $\mu$ and volatility $\sigma$).

If $r=\frac{\sigma^2}{2}$. Compute the price at time $t = 0$ of the lookback call option with maturity $T$, that is the option with payoff $S_T-min_{t\in[0,T]}S_t$ at time $T$.

If I follow the PDE approach to computing the price of the lookback call with floating strike, how do I make use of the condition $r=\frac{\sigma^2}{2}$? Do they mean that then $S_t=S_0e^{\sigma W_t}$? This is what I got when I calculated $\mathbb{E}[min_{[0,T]}S_t]=2\mathbb{E}[S_T]\Phi(-\sigma\sqrt{t})$. How do I further use this result in computing the price of the floating lookback?

Would really appreciate all the help I can get!

Answer 1

Under the condition $r=\frac{\sigma^2}{2}$, it is true that $S_t = S_0e^{\sigma W_t}$. Since \begin{align*} E\Big( S_T - \min_{0 \le t \le T} S_t\Big) = E\big( S_T\big) - E\Big(\min_{0 \le t \le T} S_t\Big), \end{align*} what you need is the expectation $E\big(\min_{0 \le t \le T} S_t\big)$. Note that \begin{align*} \min_{0 \le t \le T} S_t = S_0e^{\sigma \min_{0 \le t \le T} W_t}. \end{align*} Using the density function of $\min_{0 \le t \le T} W_t$, the expectation $E\big(\min_{0 \le t \le T} S_t\big)$ can then be computed directly.