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For the case where $K^* \leq 0$, the option becomes trivial - the option payoff is 0 for a put and a linear function of the underlying values for a call. The valuation is then also trivial. For coding, you need to check the cases whether $K^* \leq 0$ or not, and then treat them differently.


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Note that \begin{align*} \int_0^T\ln S_u du &= \int_0^T\Big[\big(r-\frac{1}{2}\sigma^2\big)u + \sigma W_u \Big] du\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + \sigma\int_0^T\int_0^u dW_s \,du\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + \sigma\int_0^T\int_s^T du \,dW_s\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + ...



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