波动率模型——Local Vol and Dupire PDE(1)

最近工作接触了一些波动率模型，自己开始看The Volatility Surfacey以及Stochastic Volatility Model这两本书，但是苦于书上的推导实在过于简单，对于我这种数学白痴来说读起来属实有点难顶，于是自己也在网上找了一些资料，整理了下面这些笔记
Dupire PDE 推导
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:

Take the derivative of the payoff function with respect to , using Ito&#39;s Lemma：

Let&#39;s take a look at rhs:

Similarly, we take the derivative of with respect to :

Then can be rewritten as below:

Replace and and simplify:

Take the Expectation of both side. Note that the second part of the rhs is a martingale:

Let&#39;s first consider the first term of the rhs

Note that the first term of the rhs of is the undiscounted payoff of call option:

Then we look at the second part. Using ,we can have:

Then can be written as

Remember we are now try to work on and so far we have finished the first term of the rhs of . Now we will work on the second term:

Using Double expectation theorem, we have:

Then can be written as

Using , can be written as:

Then can be written as

So far, we have figured ou both part of . All taken in to :

The lhs of can be rewritten as ,where is the undiscounted payoff of the call option:

This is called Depire PDE and local vol can be written as

这套证明方法和Bergomi的《Stochatic Volatility Model》书中类似，可惜书中比较简略，我结合了Youtuber quantpie的视频整理了上述的推导过程，觉得有用的hxd可以掯个赞，下一期可能会更新基于 Fokker Planck Equation的推导方法，也就是Jim Gatheral的《Volatiility Surface》一书中的证明方法。

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