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No. Actually "risk neutral pricing" does not make assumptions on the risk preferences of the agents. Securities are priced as if agents were risk neutral (that is to say as a straight expectation of discounted payoffs) but where probabilities of states of the world are not the true ones but they have been adjusted to reflect preferences. The math: Say ...


I managed to figure out another way of doing it via change of measure as follows... We know that the dynamics of $S_t^2$ is given by, $$ \begin{align} S_t^2&=S_0^2 \text{exp}\left( \left( 2r-2q-\sigma^2\right)t+2\sigma^2 W_t^Q \right)\\ \Rightarrow \frac{1}{S_t^2}&=\frac{1}{S_0^2} \text{exp}\left( -\left( 2r-2q-\sigma^2\right)t-2\sigma^2 W_t^Q ...


Let $$I= \mathbb{E}_t^\mathbb{Q}\left[\text{exp}(-2\sigma W_{T-t}) \cdot\mathbb{1}_{S_T\ge K}\right] = \frac{1}{\sqrt{2\pi}} \int_{\hat{d}_2}^{\infty} e^{-2\sigma x} e^{-x^2/2} dx.$$ So $$ I = \frac{1}{\sqrt{2\pi}} \int_{\hat{d}_2}^{\infty} e^{-(x-2\sigma)^2/2} dx \, e^{2\sigma^2}. $$ Change variables $y = x-2\sigma$ and you are done.


Essentially the question is importance sampling $$ \int f(S_T) \psi_{r}(S_T) dS_T = \int f(S_T) \psi_{\alpha}(S_T) \frac{\psi_r}{\psi_\alpha}(S_T) dS_T $$ Here $\psi_{\mu}$ denotes the log-normal density with drift $\mu.$ So when you simulate with drift $\alpha$ each sample used is $$ f(S_T) \frac{\psi_r}{\psi_\alpha}(S_T) $$ instead of $f(S_T).$ You ...

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