Saturday, May 7, 2016

Matrix Calculus

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

       \ln(L)= -\frac{1}{2} \ln (|\boldsymbol\Sigma|\,) -\frac{1}{2}(\mathbf{x}-\boldsymbol\mu)^{\rm T}\boldsymbol\Sigma^{-1}(\mathbf{x}-\boldsymbol\mu) - \frac{k}{2}\ln(2\pi)

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

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