This tutorial is continued from the previous GMM Tutorial

Here we try to see the importance of the optimal weight matrix. ( Its role is similar to variance covariance matrix employed in estimation of Generalized least squares )

Recall that the objective of GMM is to choose parameter vector so as to make the sample moment as close as possible to population moment of zero. This mimimization is achieved by performing iteratations until the gmm estimate converges

I have written this simple code to demonstrate the concept of Generalized method of moments.

Follwoing are the ideas behind this tutorial/code:
1.We simulate random variables in T distribution with a given degrees of freedom.
2.Just for comparision purposes, we use maximum likelihood estimation to determine the estimation of the degrees of freedom parameter.
3.We use the property of the 2nd moment analytical formula, and solve the equation to find DF estimiate.
4.We use the property of the 4th moment analytical formula, and solve the equation to find DF estimiate. Note that this will be different from value result of (3)
5.We assume weight matrix as indentity matrix, and solve both the moments together by criteria function as mentioned in lecture notes Generalized Method of Moments - lecture notes by Stanislav Radchenko

Note that W matrix used is an arbitrary choice and may be suboptimal. I will continue more about the optimization in next tutorial

This is a simple tutorial or demo for understanding the concept of maximum likelihood estimation.

We generate a sequence of random variables for a given degrees of freedom parameter. Then we go back and maximize the log likelihood function to get back the paramater used for generation of the random variables.

This file calculates price of an American put option using Gauss siedel method.
The main benefit of SOR method is its simplicity in comparision to other finite difference methods.
It is not even necessary to form a matrix equation.
I have simply used the fully implicit discretization of Black scholes equation, and taken V_i(k+1)
as a function of
V_{i+1}(k+1),V_{i-1}(k+1),V_i(k)

where V_i(k) denotes option value at stock axis node i and time axis node k

This file contains MATLAB code to perform Monte Carlo simulation for getting price of an European swaption under the Libor Market Model (LMM) framework.
It is based on Milstein scheme discretization. For equation, pl refer to
Fixed income Lecture notes on page 82