Emese Lazar

Option Valuation with Normal Mixture GARCH Models

The class of mixture GARCH models introduced by Haas, Mittnik and Paollela (2004) and Alexander and Lazar (2006) provides a better alternative for fitting financial data than various other GARCH models driven by the normal or skewed t-distribution. In this paper we propose different option pricing methodologies when the underlying stock dynamic is modeled by an asymmetric normal mixture GARCH model with K volatility components. Since under GARCH models the market is incomplete there are an infinite number of martingale measures one can use for pricing. For our mixture setting we analyze the impact of three risk-neutral candidates: a generalized local risk neutral valuation relationship, an Esscher transform and an extended Girsanov principle. We investigate the out-of-sample performance of an asymmetric GARCH model with a mixture density of two normals for Call options written on the S&P 500 Index. The performance under all three transformations is quite impressive when compared to the benchmark GARCH model with normal driving noise. The overall improvement is explained not only by the skewness and leptokurtosis exhibited by the innovation mixture distribution, but also by the richer parametrization used in modeling the dynamics of the multi-component conditional volatility.

Modelling Regime-Specific Stock Price Volatility

Single-state generalized autoregressive conditional heteroscedasticity (GARCH) models identify only one mechanism governing the response of volatility to market shocks, and the conditional higher moments are constant, unless modelled explicitly. So they neither capture state-dependent behaviour of volatility nor explain why the equity index skew persists into long-dated options. Markov switching (MS) GARCH models specify several volatility states with endogenous conditional skewness and kurtosis; of these the simplest to estimate is normal mixture (NM) GARCH, which has constant state probabilities. We introduce a state-dependent leverage effect to NM-GARCH and thereby explain the observed characteristics of equity index returns and implied volatility skews, without resorting to time-varying volatility risk premia. An empirical study on European equity indices identifies two-state asymmetric NM-GARCH as the best fit of the 15 models considered. During stable markets volatility behaviour is broadly similar across all indices, but the crash probability and the behaviour of returns and volatility during a crash depends on the index. The volatility mean-reversion and leverage effects during crash markets are quite different from those in the stable regime.

Information entropy and measures of market risk

In this paper we investigate the relationship between the information entropy of the distribution of intraday returns and intraday and daily measures of market risk. Using data on the EUR/JPY exchange rate, we find a negative relationship between entropy and intraday Value-at-Risk, and also between entropy and intraday Expected Shortfall. This relationship is then used to forecast daily Value-at-Risk, using the entropy of the distribution of intraday returns as a predictor.

Analytic Moments For GARCH Processes

Conditional returns distributions generated by a GARCH process, which are important for many problems in market risk assessment and portfolio optimization, are typically generated via simulation. This paper extends previous research on analytic moments of GARCH returns distributions in several ways: we consider a general GARCH model -- the GJR specification with a generic innovation distribution; we derive analytic expressions for the first four conditional moments of the forward return, of the forward variance, of the aggregated return and of the aggregated variance -- corresponding moments for some specific GARCH models largely used in practice are recovered as special cases; we derive the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments; and we demonstrate empirically that some excellent approximate predictive distributions can be obtained from these analytic moments, thus precluding the need for time-consuming simulations.

Analytic Approximations to GARCH Aggregated Returns Distributions with Applications to VaR and ETL

It is widely accepted that some of the most accurate predictions of aggregated asset returns are based on an appropriately specified GARCH process. As the forecast horizon is greater than the frequency of the GARCH model, such predictions either require time-consuming simulations or they can be approximated using a recent development in the GARCH literature, viz. analytic conditional moment formulae for GARCH aggregated returns. We demonstrate that this methodology yields robust and rapid calculations of the Value-at-Risk (VaR) generated by a GARCH process. Our extensive empirical study applies Edgeworth and Cornish-Fisher expansions and Johnson SU distributions, combined with normal and Student t, symmetric and asymmetric (GJR) GARCH processes to returns data on different financial assets; it validates the accuracy of the analytic approximations to GARCH aggregated returns and derives GARCH VaR estimates that are shown to be highly accurate over multiple horizons and significance levels.

Rethinking Capital Structure Arbitrage: A Price Discovery Perspective

The capital structure arbitrage strategy exploits the discrepancies between the credit default swap and equity markets. It assumes that both markets instantaneously react to new information, so it fails to take into account the lead-lag relationships between the prices in the two markets and their form of cointegration. Here we introduce three new alternative strategies that exploit the information provided by the time-varying price discovery of the equity and credit markets and the cointegration of the two markets. We implement the strategies for both US and European obligors and find that these outperform traditional arbitrage trading during the financial crisis. Furthermore, the returns of the new strategies have lower correlation with market returns than the standard capital structure arbitrage.