Financial innovation has been the force driving global finance to greater economic efficiency since the late nineteenth century. Among all the innovations, derivative securities, such as options, futures, bonds, etc, have caused some of the most dramatic changes in the financial world.
Although option agreements have been made for many centuries, a theoretically consistent frame work of option pricing did not become available until the publications on option pricing by Black and Scholes (1973) and Merton (1973b). Despite its biases and defects, it has received many plaudits, including the Nobel Price in economics for Scholes and Merton in 1997. Since then, researchers and scholars have spared no effort to improve the Black-Scholes (hereafter, B-S) model. Many limitations of the B-S model have been relaxed, and a variety of extensions have been developed which are applicable to various financial derivatives.
Based on the assumption of constant volatility, the B-S formula is the universal benchmark of option pricing. However, as the non-flat implied surface has been noticed and become more pronounced (Wilmott, 1998), researchers have attempted to extend the B-S model to relax its limitations and to explain empirical observations. The first attempt was made by Merton (1973b), who allows the volatility to be a deterministic function of time. This model is able to explain different implied volatilities for different times to maturity. Nevertheless, it fails to explain the smile and skew shape for different strike prices. Further trials were made by Dupire (1994), Derman and Kani (1994) and Rubinstein (1994), whereby volatility is allowed to depend on not only the time, but also state variables. This local volatility approach yields a complete market model and allows the local volatility surface to be fitted to market observations. However, these still cannot explain the persistent smile and skew shape which remains as time passes.
Following this local volatility approach, researchers made new suggestions that the volatility per se could be a random process. The pioneering work of stochastic volatility includes Hull and White (1987) and Heston (1993). Hull and White (1987) study the option pricing of a European call on a stock having a stochastic volatility. The works from Merton (1973b), Garman (1976), and Cox, Ingersoll, and Ross (1985) have shown the governing differential equation for options with stochastic volatilities. Hull and White produce a solution in series form, the sum of a sequence of terms, when the stock price is instantaneously uncorrelated with the volatility. In their paper, they also assume that the volatility is not a traded asset. They find the option price is lower than the B-S price for close to the money options, whereas higher for deep in or out of the money options leading to an implied volatility smile. The exercise prices overpriced by B-S are within about ten percent of the security price, which is the range of exercise prices over which most option trading takes place. As such, it is reasonable to expect the B-S model to overprice options. This effect is more obvious when the time to maturity increases. Hull and White argue that if the B-S equation is used to determine the implied volatility of a close to the money option, the longer the time to maturity the lower the implied volatility.
While Hull and White require volatility to be uncorrelated with the stock price, Heston (1993) relaxes this constraint by allowing arbitrarily correlation between volatility and asset prices. Heston’s model stands out for two reasons. First, the diffusion process for stochastic volatility is non-negative and mean-reverting, which satisfies the market observations. Secondly, there exists a closed-form solution for European call options, which greatly simplifies the computation (Weron and Wystup, 2005). Heston’s model is also one of the first models that is able to explain the smile and skew shape, and also allows practical applications in real markets. However,little literature has been presented on Heston’s model applied to early-exercise options. One possible reason for this might be that a closed-form solution for American options is not provided, which indicates large-scale computation might be required to implement Heston’s model. Furthermore, this computational complication also increases the difficulty of calibration. The first part of this thesis intends to address these problems by providing an accurate and efficient numerical solution to Heston’s model for American options.
Contents
Abstract
Declaration
Copyright Statement
Acknowledgements
1 Introduction
2 Extension of Heston’s Model to American Options
2.1 A Brief Review of Heston’s Model for European Call Options
2.2 The Application of Heston’s Model to American Options
2.3 Volatility Smiles
- 2.3.1 American Options
2.3.2 A Comparison between American and European Options
2.4 A Comparison between Heston’s Model and the B-S Model
2.5 Calibration of Heston’s Model
- 2.5.1 Calibration Method
2.5.2 Data Description
2.5.3 The Results of Equity Options
2.5.4 The Results of S&P 100 Index Options
2.6 A Comparison between Market Prices and Modelled Prices after Calibration
3 Stochastic Interest Rate Models
3.1 Some Well-known Models
3.2 Extensions of Heston’s (1993) Model to Bonds
3.3 US Treasury Securities
3.4 A Comparison between Market Price and Modelled Price
4 Singular Perturbations Applied to Option Pricing
4.1 Singular Perturbations for One-asset Options
- 4.1.1 European Options
4.1.2 American Options
4.2 Singular Perturbations for Multiple-asset Options
- 4.2.1 European Multi-asset Call Options
4.2.2 European and American Put Options
4.3 Calculation of Implied Volatilities and Correlation Coefficients
5 Conclusion
A Finite-Difference and its Applications to Finance
References
