The method for evaluating the conditional probability of hitting an upper barrier before a lower one appears to produce the correct analytical value, which is expressed as an infinite series of exponentially decreasing terms. In practice, however, this infinite series will need to be truncated to a finite sum; if more accuracy is required, then more terms in the series should be included.

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.

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.

The arithmetic average of the resulting approximate basket price process is further approximated, based on a different analytical moment matching technique, using a shifted lognormal random variable. The call option price is then computed as a discounted expected value of the maximum of zero and the shifted lognormal random variable value less the fixed strike. Here, relevant defining parameters for the shifted, lognormal random variable are computed.