The volatility of financial assets is an important parameter in risk management, portfolio trading and option pricing. The variability in price changes displays the uncertainty of the market and thus volatility is mainly determined by trading itself. There is much literature on volatility models mainly on stochastic models but during the last years more and more work is done on implied volatilities resulting from the well known Black-Scholes (BS) formula for option pricing, when the option price is known.
The Black-Scholes model, introduced by Black and Scholes [4] and Merton [35] in 1973, gives us a clear and easy model to price options on financial assets like stocks and stock indexes under only few, well-interpretable assumptions, such as log normality of asset price returns, constant trading and a constant volatility parameter. Last one is the crucial point in the model, as the volatility of underlying asset prices cannot be directly observed at the market. The model assumes a constant volatility across time, strike prices and option maturity. This assumption turned out to be wrong in practice. When determining the implied volatility from observed option prices according to the BS formula, we get a volatilitiy function that is curved in strike and maturity dimension, i.e. the volatility smile and term structure, respectively. Implied volatility time series vary in trend and variance. Consequently the log normal assumption for asset returns cannot be correct and in fact the distribution of asset prices turns out to have fatter tails than the log normal distribution in practice.
In the past two decades much work has been done on the improvement of the Black-Scholes option pricing model. Degrees of freedom in the model are increased by introducing stochastic volatility, additional stochastic diffusion coefficients or jump intensities and amplitudes, see for example Cox et al. [10], Hull and White [26], Stein and Stein [48], Heston [24], Rubinstein [43], Bates [2], Derman [11], Barndorff Nielsen [1] and Dumas et al. [12]. Although it is defective, the Black-Scholes model has become very popular and is largely applied in the financial world. Therefore, we do not want to follow the approaches to improve the model but to regard the resulting implied volatility as a state variable that reflects current market situations and thus is interesting by itself.
Contents
1 Introduction
2 Selected Topics of Functional Data Analysis
2.1 Local Polynomial Kernel Estimation
2.2 Local Polynomial Kernel Estimation of Functional Data
2.3 Principal Component Analysis
3 Functional Variance Processes
3.1 The Model
3.2 Estimation of model components
3.3 Asymptotics
4 Financial Options and Implied Volatilities
4.1 Basic Properties of Stock Options
4.2 Black-Scholes Option Pricing
4.3 Implied Volatility
5 Analysis of Implied Volatilities
5.1 Introduction of the Data Set
5.2 Functional Data Analysis of the IVS
- 5.2.1 First steps of data analysis
5.2.2 Principal component analysis
5.3 Application of Functional Variance Processes
5.3.1 Index IV example: DAX30X
- 5.3.2 Stock IV example: VOLKSWAGEN
5.3.3 Swap IV example: JPY
5.4 Time series of functional variance processes
6 Summary and Open Questions
