The Black-Scholes-Merton model (also called the Black-Scholes model) is proposed for option pricing in 1973 by Fischer Black, Myron Scholes and Robert Merton. It is the model which is the first analytic option pricing model[13]. Myron Scholes and Robert Merton (Black died already in 1995) received the 1997 Nobel Memorial Prize in Economic Sciences for their work, as stated in [7]. Although the Black-Scholes formula is often quite successful in explaining the stock option prices, it does have known biases [14]. It is important to have fair trading in a well-functioning market where portfolios with sure profits are not involved. The great disparity of the model is constant volatility.
On the other hand, to overcome the disparity of the Black-Scholes model, some stochastic volatility models are introduced by many researchers. Among them, Hull and White [11], Johnson and Shanno [15] and Wiggins [16] contributed a lot. Steven Heston proposed a stochastic volatility model as a generalization of the Black- Scholes model in 1993[3]. In this model, the randomness of the volatility of the underlying asset is considered. It describes stock options and other derivatives and it has also a closed-form solution. The solution of the technique is based on characteristic functions. In this model, a stock price St and variance vt follow a Black-Scholes type stochastic process and a CIR process respectively.
In this project, we have discussed the European call option prices for the Black-Scholes model and Heston model. The simulation paths are observed on both models and the effects of changing the input parameters of option pricing are also presented. Besides, the Volatility smile in option pricing of the Black-Scholes model and Heston model are represented in the project. Furthermore, for log-returns in these models, we are assuming an arbitrage-free market since that is the only well-functioning type of market. Log returns for both models are also presented here. Additionally, for observing the skewness and kurtosis of the Heston model, we have figured out the effect of correlation(ρ) and the effect of Volatility of Variance(σ). These diagrams give us a better idea of the tails of distributions. Finally, returns, squared returns and absolute returns are observed for getting the idea that sequences are independent or not for both models.