# Floating Strike Lookback Call Option

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!

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.