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 being 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 again implies 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 with natural competitors in order to test its efficiency. More generally, our empirical investigations analyse the ability of specific parametric innovations to reproduce market prices in the context of an exponential affine specification of the stochastic discount factor.     

Diagnosing affine models of options pricing: Evidence from VIX

Affine jump-diffusion models have been the mainstream in options pricing because of their analytical tractability. Popular affine jump-diffusion models, however, are still unsatisfactory in describing the options data and the problem is often attributed to the diffusion term of the unobserved state variables. Using prices of variance-swaps (i.e., squared VIX) implied from options prices, we provide fresh evidence regarding the misspecification of affine jump-diffusion models, as variance-swap prices are affine functions of the state variables in a broader class of models that do not restrict the diffusion term of the state variables. We apply the nonparametric methodology used by Aït-Sahalia (1996b), supplemented with bootstrap tests and other parametric tests, to the S&P 500 index options data from January 1996 to September 2008. We find that, while the affine diffusion term of the state variables may contribute to the misspecification as the literature has suggested, the affine drift of the state variables, jump intensities, and risk premiums are also sources of misspecification.     

Profitability and investment‐based factor pricing models

Ex ante predictors of stock returns must exhibit explanatory power across the feasible set of investments. But empirical results of factor pricing models that incorporate firm investment and profitability cannot explain the apparently high returns of US small stocks with very high investment levels and very low profitability. Whilst these stocks comprise only a small fraction of US data sets, this is not the case across global markets. Using a data set that is concentrated with stocks that exhibit high investment despite low profitability, we demonstrate that such factor models are limited in their explanatory power over these stocks.     

The intertemporal capital asset pricing model with dynamic conditional correlations

The intertemporal capital asset pricing model of Merton (1973) is examined using the dynamic conditional correlation (DCC) model of Engle (2002). The mean-reverting DCC model is used to estimate a stock’s (portfolio’s) conditional covariance with the market and test whether the conditional covariance predicts time-variation in the stock’s (portfolio’s) expected return. The risk-aversion coefficient, restricted to be the same across assets in panel regression, is estimated to be between two and four and highly significant. The risk premium induced by the conditional covariation of assets with the market portfolio remains positive and significant after controlling for risk premia induced by conditional covariation with macroeconomic, financial, and volatility factors.     

Stochastic duration and fast coupon bond option pricing in multi-factor models

Generalizing Cox, Ingersoll, and Ross (1979), this paper defines the stochastic duration of a bond in a general multi-factor diffusion model as the time to maturity of the zero-coupon bond with the same relative volatility as the bond. Important general properties of the stochastic duration measure are derived analytically, and the stochastic duration is studied in detail in various well-known models. It is also demonstrated by analytical arguments and numerical examples that the price of a European option on a coupon bond (and, hence, of a European swaption) can be approximated very accurately by a multiple of the price of a European option on a zero-coupon bond with a time to maturity equal to the stochastic duration of the coupon bond. This revised version was published online in June 2006 with corrections to the Cover Date.     

An empirical comparison of GARCH option pricing models

Recent empirical studies have shown that GARCH models can be successfully used to describe option prices. Pricing such contracts requires knowledge of the risk neutral cumulative return distribution. Since the analytical forms of these distributions are generally unknown, computationally intensive numerical schemes are required for pricing to proceed. Heston and Nandi (2000) consider a particular GARCH structure that permits analytical solutions for pricing European options and they provide empirical support for their model. The analytical tractability comes at a potential cost of realism in the underlying GARCH dynamics. In particular, their model falls in the affine family, whereas most GARCH models that have been examined fall in the non-affine family. This article takes a closer look at this model with the objective of establishing whether there is a cost to restricting focus to models in the affine family. We confirm Heston and Nandi's findings, namely that their model can explain a significant portion of the volatility smile. However, we show that a simple non affine NGARCH option model is superior in removing biases from pricing residuals for all moneyness and maturity categories especially for out-the-money contracts. The implications of this finding are examined. JEL Classification G13     

The pricing of sentiment risk in European stock markets

This paper studies whether sentiment is rewarded with a significant risk premium in the European stock markets. We examine several sentiment proxies and identify the Economic Sentiment Indicator from the EU Commission as the most relevant sentiment proxy for our sample. The analysis is performed for the contemporaneous excess returns of EA-11 stock markets in the period from February 1999 to September 2015. We apply a conditional multi-beta pricing model in order to track the variation of the sentiment risk premium over time. The results demonstrate a positive significant relationship between sentiment and contemporaneous excess returns which is consistent with previous studies. The calculated sentiment risk premium is significant as well but negative implying that an investment in EA-11 countries over the examined time period – that is bearing sentiment risk – would have been unattractive to the investors on average.     

The reduced-rank beta in linear stochastic discount factor models

In a linear stochastic discount factor model, failure of the full-rank conditions affects the standard statistical inference of coefficients. We propose a novel risk measurement, the reduced-rank beta, which is the risk sensitivity to the effective part of factors for the full-rank covariance matrix. Our reduced-rank beta is a generalisation of the standard beta when the full-rank condition is not satisfied. By considering the Fama–French five-factor (FF5) model for the US equity market, the failure of the full-rank condition is found to affect beta estimates. We demonstrate the reduced-rank beta has important empirical implications for model reductions and anomaly explanations.     

Modelling spikes and pricing swing options in electricity markets

Most electricity markets exhibit high volatilities and occasional distinctive price spikes, which result in demand for derivative products which protect the holder against high prices. In this paper we examine a simple spot price model that is the exponential of the sum of an Ornstein–Uhlenbeck and an independent mean-reverting pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of the spot price process at maturity T. Hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulae for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilized which in turn uses approximations to the conditional density of the spot process.     

The characteristics of informed trading: Implications for asset pricing

This paper investigates the linkage of microstructure, accounting, and asset pricing. We determine the relationship between firm characteristics as captured by accounting and market data and a firm's probability of private information-based trade (PIN) as estimated from trade data. This allows us to determine what types of firms have high information risk. We then use these data to create an instrument for PIN, the PPIN, which we can estimate from firm-specific data. We show that PPINs have explanatory power for the cross-section of asset returns in long sample tests. We also investigate whether information risk vitiates the influence of other variables on asset returns. We develop a PPIN factor and show that it dominates the Amihud factor in asset returns. Our results provide strong support for information risk affecting asset returns in long sample tests.     

The unintended consequences of grouping in tests of asset pricing models

We identify a number of unintended consequences of grouping when the capital asset pricing model is true and when it is false. When the model is true, grouping may cause fundamental problems with the most basic capital asset pricing and cross-sectional regression relationships. For example, with traditional grouping, the market portfolio is super-efficient––unless securities in each group are value weighted. Yet, when the model is grossly false, grouping may cause the model to appear to be absolutely correct. Ironically, the only way this can occur is when securities in each group are value weighted. To make matters worse, when the model is false, the slope of a cross-sectional regression of expected returns on betas fitted to grouped data may be either steeper or flatter than when the regression is fitted to ungrouped data. In other words, grouping may exacerbate the very problem it was meant to alleviate.     

New bond pricing models with applications to Japanese data

In this paper, the cross-sectional bond pricing model for individual bonds Kariya (1993) proposed by formulating stochastic discount function (term structure) is first applied to Japanese Government bond (JG-bond) data. The model performs very well as it stands. Second, we generalize the cross-sectional model to two types of time-dependent Markov models (TDM's) with the term structure of discount rates of each bond att being dependent on the one att−1, and apply them to the same data to find significantly improved results over those of the cross-sectional model. In fact, almost all the differences between actual prices and model values are less than 0.5 yen in each month over 12 years, implying that the error rate is less than 0.5%. On the basis of our analysis, we propose a TDM as a model for JG-bond trading.     

On option pricing models in the presence of heavy tails

We propose a modification of the option pricing framework derived by Borland which removes the possibilities for arbitrage within this framework. It turns out that such arbitrage possibilities arise due to an incorrect derivation of the martingale transformation in the non-Gaussian option models which are used in that paper. We show how a similar model can be built for the asset price processes which excludes arbitrage. However, the correction causes the pricing formulas to be less explicit than the ones in the original formulation, since the stock price itself is no longer a Markov process. Practical option pricing algorithms will therefore have to resort to Monte Carlo methods or partial differential equations and we show how these can be implemented. An extra parameter, which needs to be specified before the model can be used, will give market makers some extra freedom when fitting their model to market data.     

Selection and monetary non-neutrality in time-dependent pricing models

For a given frequency of price adjustment, monetary non-neutrality is smaller if older prices are disproportionately more likely to change. Selection for the age of prices provides a complete characterization of price-setting frictions in time-dependent models. Selection for older prices is weaker and non-neutralities are larger if the hazard function of price adjustment is less strongly increasing. Selection is weaker if there is heterogeneity in price stickiness. Finally, selection is weaker if durations of price spells are more variable. In particular, the Taylor (1979) model exhibits maximal selection for older prices, whereas the Calvo (1983) model exhibits no selection.     

Implications of state-dependent pricing for dynamic macroeconomic models

State-dependent pricing (SDP) models treat the timing of price changes as a profit-maximizing choice, symmetrically with other decisions of firms. Using quantitative general equilibrium models which incorporate a “generalized (S,s) approach,” we investigate the implications of SDP for topics in two major areas of macroeconomic research, the early 1990s SDP literature and more recent work on persistence mechanisms. First, we show that state-dependent pricing leads to unusual macroeconomic dynamics, which occur because of the timing of price adjustments chosen by firms as in the earlier literature. In particular, we display an example in which output responses peak at about a year, while inflation responses peak at about 2 years after the shock. Second, we examine whether the persistence-enhancing effects of two New Keynesian model features, namely specific factor markets and variable elasticity demand curves, depend importantly on whether pricing is state dependent. In an SDP setting, we provide examples in which specific factor markets perversely work to lower persistence, while variable elasticity demand raises it.     

Non-linear Gaussian sovereign CDS pricing models

Prior literature indicates that quadratic models and the Black–Karasinski model are very promising for CDS pricing. This paper extends these models and the Black [J. Finance 1995, 50, 1371–1376] model for pricing sovereign CDS’s. For all 10 sovereigns in the sample quadratic models best fit CDS spreads in-sample, and a four factor quadratic model can account for the joint effects on CDS spreads of default risk, default loss risk and liquidity risk with no restriction to factors correlation. Liquidity risk appears to affect sovereign CDS spreads. However, quadratic models tend to over-fit some CDS maturities at the expense of other maturities, while the BK model is particularly immune from this tendency. The Black model seems preferable because its out-of-sample performance in the time series dimension is the best.     

Option pricing based on hybrid GARCH-type models with improved ensemble empirical mode decomposition

The exploration of option pricing is of great significance to risk management and investments. One important challenge to existing research is how to describe the underlying asset price process and fluctuation features accurately. Considering the benefits of ensemble empirical mode decomposition (EEMD) in depicting the fluctuation features of financial time series, we construct an option pricing model based on the new hybrid generalized autoregressive conditional heteroskedastic (hybrid GARCH)-type functions with improved EEMD by decomposing the original return series into the high frequency, low frequency and trend terms. Using the locally risk-neutral valuation relationship (LRNVR), we obtain an equivalent martingale measure and option prices with different maturities based on Monte Carlo simulations. The empirical results indicate that this novel model can substantially capture volatility features and it performs much better than the M-GARCH and Black–Scholes models. In particular, the decomposition is consistently helpful in reducing option pricing errors, thereby proving the innovativeness and effectiveness of the hybrid GARCH option pricing model.     

A perturbative approach to Bermudan options pricing with applications

In this paper we address the problem of the valuation of Bermudan option derivatives in the framework of multi-factor interest rate models. We propose a solution in which the exercise decision entails a properly defined series expansion. The method allows for the fast computation of both a lower and an upper bound for the option price, and a tight control of its accuracy, for a generic Markovian interest rate model. In particular, we show detailed computations in the case of the Bond Market Model. As examples we consider the case of a zero coupon Bermudan option and a coupon bearing Bermudan option; in order to demonstrate the wide applicability of the proposed methodology we also consider the case of a last generation payoff, a Bermudan option on a CMS spread bond.