Hessian Matrix and its Applications in Financial Risk
11th September 2014
[Note: This article is excerpted with lecture notes for Financial and Engineering Mathematics (FEM) course delivered by CFE School, Risk Latte.]
In many areas of quantitative finance, we encounter a Hessian matrix. A Hessian matrix is useful in analyzing saddle points which we encounter while analyzing the second order risk of financial derivatives. Also, a Hessian matrix is used in analyzing the value at risk of portfolios.
The second derivative of an option price with respect to the asset is known as a gamma. This gamma has a saddle point. Many derivatives traders, especially those new to the business, get quite surprised when they encounter the fact that gamma does not necessarily decreasing with longer time to maturity. The maximum value of gamma for a given strike price first decreases until the saddle gamma point is reached and then it increases again.
Hessian matrix is an important concept in mathematics which combines the use of matrix operations and differential calculus. In particular, it is very useful in estimating local maxima, local minima and saddle points using partial derivatives for multivariable functions. Say, we have a function, that depends on two variables, and . If we plot such a function we'll get a three dimensional surface with valleys and peaks.
A Hessian matrix is a matrix that arranges partial derivatives of in a particular fashion. The Hessian will be given by
In the above matrix, terms such as , , , etc. are partial derivatives (second mathematical derivatives) of with respect to and .
The determinant of the Hessian matrix is given by
And, the trace of the Hessian matrix is given by
If both the determinant and the trace of the Hessian matrix is positive then the point is a local maxima. If determinant is positive while the trace is negative then we would get local minima on the surface. However, if the determinant is negative then regardless of whether the trace is positive or negative, the point on the surface is a saddle point.
- Bhattacharya, Rahul, The Book of Greeks, CFE School Publishing, Draft Edition 2014
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