Matrix Calculus

What's the partial derivatives (w.r.t. μ and Σ) of this function?

       

Yes, it's a beautiful formula: log-likelihood function of mvn distribution.

The answer can be found on page 40 of this book: The Matrix Cookbook

You will find equation (81), (57) and (61) are useful to get the partial derivatives.


The partial derivatives are used in Vibrato Monte Carlo method, which is a Path-wise/LRM hybrid method.

Note that there are a few alternative approaches to valuate financial derivatives which have non-differentiable payoff functions.

• Likelihood Ratio Method (LRM) 
• Mallianvin Calculus (Stochastic Calculus of Variations)
• "Vibrato" Monte Carlo Method