A Time Series Approach to Option Pricing

An equity option is a financial asset that gives its buyer the right (but not the obligation) to buy or sell a certain quantity of stocks or financial instruments on or before specified dates at a predefined price. To a certain extent, an option is similar to an equity future, except for the fact that the buyer has no commitment to buy or sell anything at the due date. Actually, options fall into two main classes (see e.g. Hull, Options, futures and other derivatives, 8th edn. Prentice Hall, Boston, 2011): vanilla and exotic that differ in exercise styles and payoff values. As their respective names make it rather clear, vanilla options are standard financial assets with a simple type of guaranty whereas exotic ones have more complex financial structures (e.g. a Barrier option structured such that the underlying stock has to reach a certain level to be active or inactive). To avoid specific technical and numerical problems, in this book we will focus on the vanilla type as it has been used in the academic literature as a benchmark for comparing empirical performances of pricing models. Basically, we wish to help readers to understand the value that the time series methodologies presented in this book can add to this well known basis before turning to securities with a more complex payoff.

The evaluation of financial risks and the pricing of financial derivatives are based on statistical models trying to encompass the main features of underlying asset prices. From the seminal works of Bachelier (Ann Sci Ecole Norm Supér 17:21–86, 1900) based on Gaussian distributions, the random walk hypothesis for the returns or the log-returns has frequently been suggested. Its remarkable mathematical tractability, in particular in the multidimensional case, was the keystone of nice financial theories like Markowitz’s (Portfolio selection: efficient diversification of investments. Wiley, New York, 1959) portfolio management or Black and Scholes (J Polit Econ 81:637–659, 1973) option pricing model, among others. Nevertheless, during the last decades, the explosion of computational tools efficiency has allowed researchers to pay more attention to the analysis of financial datasets and the test of models assumptions. It is now well-documented that in spite of their huge heterogeneity concerning the nature of financial assets (stocks, commodities, interest rates, currencies…), the frequency of observations or the multiplication of financial centers, financial time series exhibit common statistical regularities (called stylized facts) that make satisfactory models difficult to obtain. A major attempt in this direction was done during the 1980s by Engle (Econometrica 50:987–1007, 1982) and Bollerslev (J Econ 31:307–327, 1986) through the ARCH/GARCH approach. After a brief reminder of the classical stylized facts observed for the daily log-returns of financial indices, the aim of the chapter is to present the main features of the GARCH modelling approach and its recent extensions.

In the perfect and unrealistic Black and Scholes (J Polit Econ 81:637–659, 1973) world, the dynamics \((S_{t})_{t\in [0,T]}\) of the risky asset, under the historical probability \(\mathbb{P}\), is given by the following stochastic differential equation: where \((W_{t})_{t\in [0,T]}\) is a standard Brownian motion under \(\mathbb{P}\). In this case, there is no ambiguity in the definition the arbitrage-free price of any European contingent claim with maturity T. In fact, in this complete market which is set in continuous time, this value is none other than the value of any replicating portfolio. Moreover, prices may be expressed in terms of conditional expectations under a unique equivalent martingale measure Q whose density with respect to the historical probability is given by the Girsanov theorem where r is the constant and continuously compound risk-free rate. Unfortunately, as we have seen in Sect. 2.​1, the restrictive underlying hypotheses (constant volatility, independent increments, Gaussian log-returns, etc…) are questioned by many empirical studies and GARCH models appear as excellent alternative solutions to potentially overcome some well-documented systematic biases associated with the Black and Scholes model.

Having in mind the previously developed models (Chap. 2) and pricing approaches (Chap. 3), this chapter intends to provide the reader with an empirical comparison of them. Even though part of the arguments in this book can seem rather theoretical, the underlying intention behind all of this is to provide a pricing technology that is based on realistic models of financial markets. In this chapter, we will be using a dataset of European options on the S&P500 to test and compare those models. We will not only compare the time series approaches that have been discussed earlier, but compare them to various “standards” of the industry, namely the calibrated and estimated versions of the Heston (1993) model (or at the very least their Heston and Nandi (2000) discrete time counterparts). By doing so, we aim at putting a greater emphasize on a key point most of quantitative analysts around the world are familiar with: rough scales of magnitude for option pricing errors that anyone could expect from a good option pricing model.