Christophe Chorro

Option Pricing for GARCH-Type Models with Generalized Hyperbolic Innovations

In this paper, we provide a new dynamic asset pricing model for plain vanilla options and we discuss its ability to produce minimum mispricing errors on equity option books. Given the historical measure, the dynamics of assets are modeled by Garch-type models with generalized hyperbolic innovations and the pricing kernel is an exponential affine function of the state variables, we show that the risk neutral distribution is unique and implies again a generalized hyperbolic dynamics with changed parameters. We provide an empirical test for our pricing methodology on two data sets of options respectively written on the French CAC 40 and the American SP 500. Then, using our theoretical result associated with Monte Carlo simulations, we compare this approach to natural competitors in order to test its efficiency. More generally, our empirical investigations analyze the ability of specific parametric innovations to reproduce market prices in the context of an exponential affine specification of the stochastic discount factor.

Testing for Leverage Effect in Financial Returns

This article questions the empirical usefulness of leverage effects to describe the dynamics of equity returns. Using a recursive estimation scheme that accurately disentangles the asymmetry coming from the conditional distribution of returns and the asymmetry that is related to the past return to volatility component in GARCH models, we test for the statistical significance of the latter. Relying on both in and out of sample tests we consistently find a weak contribution of leverage effect over the past 25 years of S&P 500 returns, casting light on the importance of the conditional distribution in time series models.This article questions the empirical usefulness of leverage effects to describe the dynamics of equity returns. Relying on both in and out of sample tests we consistently find a weak contribution of leverage effects over the past 25 years of S&P 500 returns. The skewness in the conditional distribution of the return’s time series model is found to explain most of the returns’ distribution’s asymmetry. This conclusion holds both at the index level and for 70% of the individual stocks constituents of the equity index.

The Time Series Toolbox for Financial Returns

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 Super 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.

From Time Series of Returns to Option Prices: The Stochastic Discount Factor Approach

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:

Martingalized Historical approach for Option Pricing

\displaystyle{ dS_{t} =\mu S_{t}dt +\sigma S_{t}dW_{t} }

A time series approach to option pricing: Models, Methods and Empirical Performances

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

A Time Series Approach to Option Pricing

\displaystyle{ \frac{dQ} {d\mathbb{P}} = e^{-\frac{\mu -r} {\sigma } W_{T}-\left (\frac{\mu -r} {\sigma } \right )^{2} \frac{T} {2} } }