## Archive for **November 2011**

## Markov Chain for historical volatility

In previous post we used Markov Chain to discover a behavior of historical volatility and find out the 3/2 rule for ups and downs of random variable.

Now let’s construct more complicated model with the following volatility changes as states in chain:

- Less than -25%: denote it as -100.
- From -25% to -15% : -20.
- From -15% to -5% : -10.
- From -5% to 5% : 0.
- From 5% to 15% : 10.
- From 15% to 25% : 20.
- More than 25% : 100.

## Markov chain for Geometric Brownian Motion parameters

A Markov chain is a discrete-time random process with Markov property. Its components are states and probability transitions between them. Markov property states that the probability of next states depends only on the current state.

So Markov chain is a set of states and all transition probabilities between states.

## Simple chain for drift

Let’s assume that estimation of drift parameter might lead to the following 2 states:

- Positive, i.e. drift is greater or equal zero
- Negative, i.e. drift is less than zero

So, one could construct Markov chain for these states as it shown below.

## Maximum Likelihood Estimation of Stochastic Process Parameters

Maximum Likelihood estimation (MLE) is a method of parameter estimations of statistical model. The base idea is to establish joint density probability for observations and to maximize its value by model’s parameters. To say it differently, we are looking for the most probable explanation of observed data.

### Problem setup

Assume that we’re given a one-dimensional stochastic process:

where and are some functions of arguments .

We observe this process by measuring $\latex S_i(t_i)$ where . For sake of simplicity assume that observations are equidistant in time, i.e. .

So, let’s estimate parameters.

## SABR model calibration – attempt 2

The SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for “Stochastic Alpha, Beta, Rho”, referring to the parameters of the model. It was developed by Patrick Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.

The SABR model describes a single forward F, such as a LIBOR forward rate, a forward swap rate, or a forward stock price. The volatility of the forward F is described by a parameter σ. SABR is a dynamic model in which both F and σ are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations:

Constant parameters should satisfy the condition

Here, and are two correlated Wiener processes with correlation coefficient . For simplicity sake, we assume that , therefore, we put :

## Correlation trading on FX

Correlation trading is based on few simple ideas:

- Correlations are changing with time
- Correlations of pairs of 3 currencies are bounded by strict equation
- Correlation is bound in the interval

Therefore, one may try to “buy correlation” at -1 and to “sell it” at +1.