# Heston modellen: Udledning, egenskaber og praktisk anvendelse

Martin Bech Rasmussen

Student thesis: Master thesis

## Abstract

The main subject of this master thesis is to derive and analyse a closed-form solution for the stochastic volatility model developed by Stephen L. Heston in 1993. The main properties of the model are illustrated and the practical use of the model tested in various situations. The motivation for developing a stochastic volatility model is that the basic option pricing model developed by Black & Scholes  is not consistent with observed market prices. One of the main characteristics of the Black-Scholes model is that the underlying asset follows a geometric Brownian motion with constant volatility. However empirical studies show that the volatility is not constant over time, so Heston  suggested an extension to the Black-Scholes model, where the volatility follows a stochastic process and thus varies over time. The technique used by Heston to derive the closed-form solution for option prices under stochastic volatility is based on solving two partial differential equations for the characteristic functions associated with the probabilities. Then the connection to the probabilities in the model is calculated using an inverse Fourier transformation. The solution technique is presented in detail in chapter 3 of this thesis, and it can be used to derive closed-form solutions to many different problems. The derivation of the partial differential equations for the characteristic functions is shown in appendix B. Another version of the Heston model, known as the Displaced Heston model, is also introduced. The main difference between the two models is that the process for the underlying asset is defined differently in the Displaced Heston model, which gives the model the advantage of allowing the correlation between the underlying asset and the variance to be zero without losing one of the important properties of the stochastic volatility model. In some applications, this proves to be an advantage. The main effect that causes the Heston model to differ from the Black-Scholes model, is its ability to generate skewness and kurtosis in the probability density function. In the Heston model, skewness is generated by the correlation parameter, and kurtosis is generated by the volatility of volatility parameter. In the Displaced Heston model the correlation was set to zero, but this model still has the ability to generate skewness through the displaced parameter. The effects on the probability density function lead to another important property of the model – its ability to generate a volatility smile that is similar to the volatility smiles observed in the market. In the volatility smile, the skew is generated by the correlation parameter, and the smile is generated by the volatility of the volatility parameter. All the other model parameter effects on the volatility smile are analysed in chapter 4 in this thesis. A practical use is introduced in chapter 5 where the two stochastic volatility models are used to price the interest-rate derivative known as a cap. The model parameters are calibrated to observed market prices by minimising the sum of the squared pricing error for all the maturities and exercise prices. Various tests show that the Displaced Heston model is the best model to use on the data in this thesis, and that it performs very well in describing the market data. Another type of practical use is introduced in chapter 6 where the Heston model simulation abilities are tested in a Monte Carlo setup. A simple Euler scheme for simulating the two processes is described, and the pricing ability of the method is compared to the analytical Heston formula. It is shown that when enough simulations are used and the discrete time steps in each simulation are small then the simulated price is very close to the analytical price. Finally, the simulation setup is used in a real-life example where a structured stock obligation is priced using the Euler scheme and a calibration setup similar to the one used in chapter 5. The two models and the calculations made throughout the thesis are implemented in the mathematical programming language Matlab, and the source code for the programs is shown in appendix C.