In this dissertation we consider the problem of calibrating the probability density of price returns on the market and individual assets through the skew effects of stochastic volatility. We will consider, in the first part of the dissertation, the problem of deriving a specific closed form approximation for the probability density of price returns under a fast mean reverting stochastic volatility model, which we hypothesize accurately reflects the price returns of the market. We next deal with the problem of calibrating the parameters of the approximated density, and propose several approaches for undertaking this calibration using only the observed price returns and not the unobservable volatility. These approaches will include the classic method of moments and maximum likelihood estimation techniques as well as a unique call option method of moments technique developed for the first time here. Next we compare the performance of our different calibration approaches both theoretically and numerically with the results showing the ideal data regimes for each approach. Finally we calibrate our approximated density on S&P 500 price returns and show it’s superiority to a Gaussian density in calculating return risk through the skew of returns.

In the second part of this dissertation we will show how the approximated density of market returns developed in the first part can be used to calibrate, through the use of the stock’s beta parameter, the expected returns of individual stocks. We will first develop a continuous time Capital Asset Pricing Model, and show how to approximate option prices on the market and individual stocks under this model. We will then show how the beta parameter can be estimated through the parameters of these options prices. This allows us to calibrate the beta parameter from forward looking option prices. Finally we compare the performance of our forward looking beta calibration technique against the calibration of the beta parameter using historic prices for a sample of S&P 500 stocks. We show that the beta estimate from our forward looking beta calibration technique is more accurate that the beta estimate calibrated on historic price returns in a majority but not all stock cases.