### Parameter Estimation of Mean Reversion and Related Class of Stochastic Differential Equations with Slowly Varying Coefficients.

Mean Reversion is widely used to model trading strategies in several financial markets. A trader has to use great judgment and trading skills to recognize various phases of mean reversion patterns and cycles. It is many times very difficult to determine when the spread will start to converge. We are working with parameter estimation and filtering of slowly time varying parameters of the mean reversion and statistical arbitrage class of equations. One example is given

The filtering of slowly varying parameters of the above stochastic differential equation has never been discussed in non-linear filtering literature. This new technology has many advantages towards decision making in statistical arbitrage strategies. We know that a mean reversion cycle has a divergence and convergence phase that usually follow each other though we cannot tell with absolute certainty whether the pair has started converging. We also cannot tell from historical relationships whether a certain pair will ever converge because the historical convergence relationships might break down due to a large number of reasons. Our parameter estimation and filtering techniques help overcome the problems related with inference of phases of mean reversion cycles. For example, knowledge of these parameters will help us clearly determine whether the pair is still diverging or has started to converge. Any unusual change in pattern of mean reversion or statistical arbitrage cycles will become obvious when the trader has a complete knowledge of the parameters of the equation.

To give the technical reader a better idea about the project, we state the crux of our methodology here. By using our state of the art parameter estimation technology which employs non-linear filtering and Bayesian likelihood methods, we find the two dimensional joint densities of the slowly moving parameters θ(*t*)and mean reversion speed *k*(*t*) in the above equation. The current states of these parameters help the trader in better prediction towards the future movements of the spread and aids in objective decision making regarding their mean reversion strategy.

There is no published work on small variance parameter estimation of the above stochastic differential equation. We invite the client firms to set objective benchmarks for the performance of our method. We also hand over the code for the method with a brief explanation to our clients. The projected cost of this project to our clients would be $250,000.

### Determination of Data Generating Stochastic Differential Equation from the Observed Data

This is a more general project. We ask our client firms to give us the time series data and we will describe the stochastic differential equations that best capture the statistical properties of the data generating process. If there is enough data, we might be able to find the true data generating process. This project is not limited to financial derivatives or statistical trading and we offer our services to other scientific disciplines including medical sciences. The projected cost of this project to our clients would be $400,000.

### Stochastic Local Volatility Project

We are working with a new and extremely general stochastic-local volatility(SLV) model that nests the most existing SLV models. Work on the project is almost complete. The dynamics of the asset price are given either by a lognormal or SABR model with an extremely general correlated mean reverting stochastic volatility. We employ one dimensional transition probabilities framework and use proprietary techniques to reduce the two dimensional problem to a single dimensional problem on the variance grid. Some of the details specific to SLV models using this method have never been published. Our techniques are promised to be faster than those of the most competitive peers in the industry. We calculate all greeks and related risk management parameters on a single dimensional grid in real time. The projected cost of this project to our clients would be $125,000.