**Mortgage Cash Flow**

We model the closed monthly cash flows from a pool of mortgage. Here cash flows consist of principal and interest payments. Principal payments arise from the regular amortization of principal, as well as from scheduled and unscheduled principal pre-payments.

2. **Trinomial Tree Algorithm**

A trinomial tree can be used for pricing particular types of barrier options. We consider particular types of single barrier and double barrier options. Each method for pricing a particular type of barrier option is based on a combination of techniques, that is, a tree generation technique and an appropriate backward induction pricing technique.

3. **Lookback Call Option**

A model is presented for pricing a European lookback call option on a stock index with guaranteed exchange rate (LBCGER). Closed-form formulas exist for pricing certain types of lookback options. For example, assuming that the underlying security earns no dividend.

4. **Trinomial Tree Construction**

A trinomial tree based method is presented for pricing exotic options. The model is based on a combination of techniques. that is, a tree generation technique and an appropriate backward induction pricing technique.

Since the volatility parameter in the SDE is of a piecewise constant form, the tree generation techniques may, in some cases, construct trees that are non- recombining. In the worst case, then, the space complexity of the tree generation techniques is proportional to the exponential of the number of time slices in the tree.

5. **Partial Barrier Option**

A model is presented for pricing certain types of European, continuously monitored partial barrier options. The method is based on certain analytical formulas, for pricing such options.

6. **Conditional Probability of Hitting Barrier**

A model is developed for evaluating the conditional probability of hitting an upper barrier before a lower barrier, and vice versa, for a tied down geometric Brownian motion with drift. The method produces an analytical value for this probability, assuming that the barrier levels are constant and continuously monitored.

7. **Monte Carlo Short Rate**

The Monte Carlo Multi-factor Short Rate Mode has been used extensively in pricing a variety of interest rate derivative securities. The model assumes that short rates at reset dates are lognormally distributed; the short rate at a reset time arises as the limiting spot value from a corresponding forward rate process, which is a geometric Brownian motion with drift. The short rate model is, by construction, arbitrage free, and numerical test results bear this out.

8. **Delta Gamma Vega Value at Risk**

The Delta Gamma Vega (DGV) methodology is developed to estimate Value-at-Risk (VaR) for portfolios of equities and equity options in order to comply, in regard to market risk measurement. The model can accurately estimate over-night VaR for portfolios with non-zero convexity or linear risk.

9. **Asian Swap**

A model is developed for valuing a swap1 between party A and party B. Here party A receives a fixed amount and makes a single variable payment at swap maturity. The payment amount can be modeled as the value of a European discrete Asian call option on a basket of indices. Here the basket price consists of an arithmetic average of various stock and bond indices. The call option payoff at maturity is equal to maximum of zero and an arithmetic average of basket values at certain points in time prior to option expiry less a fixed strike.

10. **Swap with Better-of Cliquet Option**

A model is developed for pricing a swap with better of cliquet option. The floating amount payer makes semi-annual payments based on USD-LIBOR-BBA minus a spread. The fixed rate payer makes a single payment at swap maturity based on the arithmetic average of the S&P 500 Index price over certain pre-specified windows of ten consecutive trading days.