Models for indices - Yee, Z (oxford university;mathematical institute, 2011)
We consider a market index model for a large portfolio of risky assets traded in the stock market where the correlation is due to a market factor. By taking the limit of a simple systems of stochastic differential equations (SDEs), we obtain a limit stochastic differential equation (SDE) for the index price. We also investigated the limit empirical measure for the infinite system. The density evolves according to a stochastic partial differential equation (SPDE) which we also solve in some special cases. Using our limit SDE, we also try to compare its accuracy numerically with the “real index” calculated from summing the individual stock prices. Lastly, we make use of our limit SDE and empirical density to price different derivatives such as the European call & put option on the index and the European call & put option on the maximum of the constituent stock in the index.