Pricing of options using copula functions

Abstract

When we look at the mathematical models applied to evaluate prices of various assets, all quantitativemodels are based on assumptions vis-à-vis the markets on which they are to be applied. Standard pricesquoted for many financial products are often based on “normal" conditions. This may be interpreted in a more economic sense, or more specially referring to the distributional (i.e. normal, Gaussian) behaviour of some underlying data. Hence the classical literature is full of deviations from the so-called random walk (Brownian motion) model and heavy tails appear prominently. So, that is where the notion of Copula comes in; to construct models which allow to go beyond normal dependence and tackle problems like spill over, the behaviour of correlations under extreme market movements, the pros and contras of linear correlation as a measure of dependence, the construction of risk measures for functions of dependent risks. In the evolution of the derivatives markets, we have seen numerous innovations used in designing and pricing of various instruments. The innovation arises out of the need to serve clientele with a niche demand. In our attempt to continue the same, we have in this work priced multi asset digital options and index options using Copula functions, famous for their utility in pricing of CDOs and calculating the tail risk. We move on from there to price Vanilla Options but with dynamic volatility. The GARCH(1,1) model has been previously used by many to estimate the changing volatility. We use the expected value of this model as the mean of the variance in a chi square distribution to arrive at the variance at each point of time.