I'm looking to price a call option with an exotic feature. The price I'm trying to calculate at time $t=0$ is
\begin{equation} C = E^\mathbb{Q}[(S_T-K_T)^+] \end{equation}

where $S_t$ is the stock price with dynamics \begin{equation} dS_t =\sigma S_t dW_t \end{equation}

The strike price $K_T$ is a stochastic variable given by \begin{equation} K_T = \min\{K_{max}, \max\{K_{min}, \lambda A\}\}, \quad \lambda < 1 \end{equation}

i.e., $K_T\in[K_{min},K_{max}]$, where $A$ is the average price during some future interval $[t_1,t_2]$ \begin{equation} A = \frac{1}{t_2-t_1}\int_{t_1}^{t_2} S_t dt \end{equation}

What is a good and fast way to calculate this and also solve for the implied volatility from market prices?



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