波动率模型——Local Vol and Dupire PDE(2)

上一期我整理了第一种证明Dupire PDE的方法，这一期我从Fokker Plank Equation用另一种方法证明

Let&#39;s consider a Vanilla European Call with a Strike K and Maturity T. Its payoff function is as below:

Assume the underlying asset follows GBM process:

The Fokker Planck Equation with respect to this process is:

Integrate both side of with respect to :

where is the density function of at time
Take the derivative of with respect to

replace using :

We can use Integration by part on rhs separately. For the first part of rhs:

For the second part of , we also do integration by parts. However, note that there is a second order differential term, so we have to do integration by parts twice. Here I just skip the calculation:

With and , can be written as :

Note that can be written as:

Also, we take the derivative of with respect to K:

Then is:

Finally, is:

(15) is the Dupire PDE.

相比上一篇文章中的方法，用Fokker Planck Equation证明涉及到两次分部积分，相对比较繁琐一些。关于Fokker Plank的证明方法，小伙伴可以参考随记：Fokker-Planck 方程的导出 - 知乎 (zhihu.com) 。月底了工作有点忙，可能下一次更新会在五月。

Reference

Stochastic Volatility Modeling by Bergomi, Lorenzo

2. The Volatility Surface A Practitioners Guide (Wiley Finance) by Jim Gatheral, Nassim Nicholas Taleb
3. Local Volatility Model: Dupire PDE and Valuation/Pricing PDE Derivations and Comparisons - YouTube