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I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process.

The well known Black scholes PDE is given by $$ \frac{1}{2}\sigma(x)^2x^2\frac{\partial^2}{\partial x^2}C(x,\tau)+\mu x\frac{\partial}{\partial x}C(x,\tau)-rC(x,\tau)-\frac{\partial}{\partial \tau}C(x,\tau)=0. $$ where the initial condition $C(x,0)=max(x-K,0)$ and $\sigma(x)=\delta x^\beta$.

Taking the Laplace transform with respect to $\tau$, we obtain the following ODE: $$ \frac{1}{2}\delta x^{2\beta+2}\frac{\partial^2}{\partial x^2}\hat{C}(x,\lambda)+\mu x\frac{\partial}{\partial x}\hat{C}(x,\lambda)-(\lambda+r)\hat{C}(x,\lambda)=-max(x-K,0). $$ where $\hat{C}(x,\lambda)=\int_0^\infty e^{-\lambda \tau}C(x,\tau)d\tau$

and the initial condition is transformed to $$ \hat{C}(x,\lambda)=\int_0^\infty e^{-\lambda \tau}C(x,0) d\tau=max(x-K,0)/\lambda $$(is this right??? it seems wrong..)

Then, $\hat{C}(x,\lambda)$ can be analytically formulated by the case $x>K$ and $x\leq K$.

How to get explicit formula for $\hat{C}(x,\tau)$? I can't proceed from this stage.

I know one paper, "(2001 Dmitry) Pricing and Hedging Path-Dependent Options under the CEV", related to this question. However, there's big jumps for me to understand readily. Could you explain it step by step?

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    $\begingroup$ It's perfectly legitimate to use the Laplace transform (and vonjd's linked paper does a fine job), but I've personally always preferred to solve the PDE by changing variables until the PDE turns into the standard diffusion equation. $\endgroup$ – Brian B Sep 18 '14 at 15:39
  • $\begingroup$ Welcome to Quant Stackexchange :-) Thank you for your interesting first question. If the answer was helpful you could upvote and accept it :-) $\endgroup$ – vonjd Sep 21 '14 at 6:38
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The following paper gives you really all of the missing steps in a very detailed form:

A Complete Solution to the Black-Scholes Option Pricing Formula by Ravi Shukla and Michael Tomas

From the paper:

"This presentation is purely for pedagogical purposes. In the course of doing work on option pricing, we found no complete solution for the Black-Scholes model. By complete, we mean from the assumptions of the stochastic process to the closed form of the Black-Scholes model with significant algebraic steps for the reader to follow. Many texts derive the partial differential equation from the stochastic process and then make a leap to the closed form solution. [One text presents] steps based on the method of Laplace transform [...] Here we present the complete solution filling in all the necessary algebraic details."

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    $\begingroup$ The link is broke. $\endgroup$ – Hans Sep 18 '18 at 19:03
  • $\begingroup$ @Hans: Thank you. I cannot find another version on the web. I will contact the authors... $\endgroup$ – vonjd Sep 19 '18 at 6:56
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    $\begingroup$ @Hans: Ravi emailed me the current link, I updated the answer. Thank you again for the heads-up. $\endgroup$ – vonjd Sep 20 '18 at 6:33

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