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The Stochastic Differential Equation that models the change in an asset price is

$$ dS = (12S-sin(S))dt+\frac{\sigma S}{S^2+1}dX $$

where dX's are random variables drawn from standard normal distribution. What is the Black –Scholes-ish partial differential equation that must be solved in order to price derivatives of this asset

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  • $\begingroup$ Just to clarify: Is $dX$ truly a standard normal variate or is it rather a Brownian motion? If it is the latter, then I think you can proceed with the usual Black-Scholes-Merton approach, i.e. set up the portfolio, assume a Delta-Hedge etc; no? $\endgroup$ – Kermittfrog Nov 9 '20 at 7:46
  • $\begingroup$ It's a brownian motion. I would really appreciate of you can show the steps too. Thanks! $\endgroup$ – Cassey_Adams Nov 9 '20 at 7:51
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    $\begingroup$ The BS PDE is $O_t + rSO_S + \frac{1}{2}O_{SS}(dS)^2-rO$. In your case, $(dS)^2$ equals $\frac{\sigma^2S^2}{\left(1+S^2\right)^2}$, so the PDE should be $O_t + rSO_S + \frac{1}{2}O_{SS}\frac{\sigma^2S^2}{\left(1+S^2\right)^2}-rO$ $\endgroup$ – Kermittfrog Nov 9 '20 at 8:39

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