Estimating Heston stochastic volatility model parameters - Musya, Martin (Strathmore University, 2022)
The Black Scholes model is widely favoured for pricing derivatives such as the European put and call options. This model while having the benefit of ease of application has some restrictive assumptions. First there is the assumption that volatility of asset returns is constant. This assumption is easily violated in the volatility smile that is widely documented in literature as well as observed in option market data, another assumption is that asset returns are normally distributed. The assumption of normal distribution is reasonable for long term horizons but not for shorter horizons. Market are rarely if ever complete. There always exists informational asymmetry where some investors know more about the market than others. It is also well known that a single asset is insufficient to hedge away risk. The Heston Model improves on previous assumption of constant variance (homoskedasticity) by allowing correlation between volatility and the price of the underlying. This research sought to estimate the Heston model parameters over various periods using maximum likelihood method. This was to compare the performance of both the Heston model and the Black Scholes model in periods of market uncertainty and in relatively stable periods where markets are performing well. The three periods, also called epochs, in this study are during the 2008 financial crisis, the Covid-19 pandemic and the relatively stable years between the two periods. The data for this thesis was obtained from the S&P 500 volatility index (VIX). Both the option data and the underlying data was available. A¨ıt-Sahalia (2002) constructed the sequence of approximations to the transition probability for a diffusion process. The first step involved standardizing the diffusion function using the Lamperti transform in order to remove some state dependent term in order to get rid of boundary conditions and correlation structures resulting in simple diffusion terms. This allowed for parameter estimation for the differentiable unit diffusion function. After this, the pseudo normalized” increment for random variable of the diffusion function was obtained and maximum likelihood estimation was done on this transformed variable to get rid of the issue of transition probabilities getting peaked when the step size gets small. The state variables for the Heston diffusion function were those of the underlying process and the diffusion process. The option price and that of the underlying in the data were first organized into a matrix. The determinant of that matrix which was the Jacobian term was then used to estimate the Heston and Black-Scholes parameters. A hypothesis test on the output for the three periods showed that at the 95% significance level the Heston model parameters are significant unlike the Black-Scholes parameters for all periods under study. The Heston model performed better even in times of financial instability such as during the 2008 crisis and the Covid-19 pandemic.
Submitted in partial fulfillment for the Degree of Masters of Science in Mathematical Finance at Strathmore University