Solving Black-Scholes PDE using Laplace transform

Question

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?

Answer 1

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."