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Consider the equation (such that all variables are positive real numbers)

$$S=S_0e^{t\left(\mu-\frac{1}{\omega}\right)-\frac{t\sigma^2}{2}}-\frac{2\Lambda\left(1- e^{\frac{2t\left(\mu-\frac{1}{\omega}\right)-t\sigma^2}{2}}\right)}{\omega\left(2\left(\mu-\frac{1}{\omega}\right)-\sigma^2\right)}.$$

Is there an explicit solution for $\omega$ (possibly in Lambert form)? Otherwise, could a suitable approximation be made (since Newton's method seems tedious in this case)? Any help would be much appreciated. For context/derivation see Determining Properties of Stochastic Differential Equation ...

Simply Beautiful Art
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UNOwen
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  • I made an edit to my answer in the linked question. This could take a lot of pressure out of the problem here. – Kurt G. Sep 10 '21 at 14:41
  • Thank you for your reply. I was wondering if a general solution existed (i.e. assuming $S$ follows the above, can $\omega\in\mathbb{R}^+$ be isolated). – UNOwen Sep 10 '21 at 14:53
  • Looks pretty hopeless. Also: you are assuming $W_t=0$ which drastically 'simplifies' your original problem. – Kurt G. Sep 11 '21 at 06:01
  • Doesn't setting $W_t$ to $0$ yield the expected value of $S$, since $W_t$ is normally distributed? I could try numerical methods (i.e. similar to calculating implied volatility for options). So, what could be an initial estimate to start Newton's method (i.e. similar to https://quant.stackexchange.com/questions/7761/a-simple-formula-for-calculating-implied-volatility)? – UNOwen Sep 11 '21 at 11:28
  • Currently, I have the approximation $\omega\approx K\tanh^{-1}\left(\frac{\left(\Lambda-S\right)e^{\frac{t\sigma^{2}}{2}}}{Le^{\frac{t\sigma^{2}}{2}}-S_{0}e^{t\mu}}\right)$ but can't determine $K$ (which is probably a function of $\sigma,\mu,t$). – UNOwen Sep 11 '21 at 11:57
  • If you do not wish to differentiate, why not use a simple derivative-free method like the secant method? – Simply Beautiful Art Sep 13 '21 at 13:08
  • Equivalently you want a closed form for $e^x=A+B/(x+C)$ for some constants $A,B,C$ where $x=t(\mu-1/\omega)-t\sigma^2/2$. If $A\ne0$ we can't have a closed form even with Lambert $W$. – TheSimpliFire Sep 13 '21 at 14:10

1 Answers1

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As stated in the comments, an approximation of the form

$$\omega=K\tanh^{-1}\left(\frac{\left(\Lambda-S\right)e^{\frac{t\sigma^{2}}{2}}}{\Lambda\ e^{\frac{t\sigma^{2}}{2}}-S_{0}e^{t\mu}}\right)$$

seems sufficient.

UNOwen
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