Affine multivariate GARCH models

Non-Affine GARCH Option Pricing Models, Variance Dependent Kernels, and Diffusion Limits

Downside Risk Neutral Probabilities

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.

Empirical risk aversion functions-estimates and assessment of their reliability

The establishment of the fractional Black–Scholes option pricing model is under a major condition with the normal distribution for the state price density (SPD) function. However, the fractional Brownian motion is deemed to not be martingale with a long memory effect of the underlying asset, so that the estimation of the state price density (SPD) function is far from simple. This paper proposes a convenient approach to get the fractional option pricing model by changing variables. Further, the option price is transformed as the integral function of the cumulative density function (CDF), so it is not necessary to estimate the distribution function individually by complex approaches. Finally, it encourages to estimate the fractional option pricing model by the way of nonparametric regression and makes empirical analysis with the traded 50 ETF option data in Shanghai Stock Exchange (SSE).

Purpose – The purpose of this paper is to understand that option pricing is the response of option implied volatility (IV) to macroeconomic announcements. Design/methodology/approach – The authors use high-frequency data on ASX SPI 200 index options to examine the response of option IV, as well as higher moments of the underlying return distribution, to macroeconomic announcements. Additionally, the authors identify the response of the moments as a function of moneyness of the options. Findings – The findings suggest that in-the-money and out-of-the money options have difference characteristics in their responses, leading to the conclusion that heterogeneity in investor beliefs and preferences affect option IV through the state price density (SPD) function. Originality/value – The research contributes to the literature that examines whether IV captures the beliefs of market participants about the likelihood of future states together with the preferences of market participants towards these states. In particular, the authors relate changes in option IV to changes in macroeconomic announcements, through the impact of these announcements on the moments of the SPD function.

Implied volatility is an important element in risk management and option pricing. Black-Scholes model assumes a constant volatility, however, the evidence from financialmarket shows that the volatility is not constant but change with strike and time tomaturity. In this paper, the time to maturity is fixed and we will construct the implied volatility function of strike or moneyness. We can use regression method for estimation, but the data from financial market contains some noise and we need to apply smoothing techniques to estimate this implied volatility function. The standard non- and semiparametric regression methods don’t guarantee the resulting IV functions are arbitrage free, so we will insert our estimation result to Black and Scholes model and calculate the state price density (SPD). In a Black-Scholes model it is lognormal distribution with constant volatility parameter. In practice as volatility changes the distribution deviates from log-normality. We estimate volatilities and SPDs using EUREX option data on the DAX index by using different smoothing techniques. Our estimation will be carried out through the strike direction and moneyness direction. We will briefly introduce Local polynomials as one method. The most important smoothing techniques we will applied in this paper is B-splines, with the usage of roughness penalty, which allows a flexible choice on the degree of smoothness, and is promising for future research work on the arbitrage free constraint of implied volatility.

Downside Risk Neutral Probabilities

‘Risk Aversion, background Risk, and the Pricing Kernel’ looks in more detail at utility functions and their effect on the shape of the pricing kernel. The authors discuss the meaning of risk aversion and, in particular, ‘relative risk aversion’ and show that if relative risk aversion is constant at different levels of wealth, then the pricing kernel exhibits constant elasticity. They then show that the introduction of ‘background risk’, that is, non-hedgeable risks, causes the pricing kernel to exhibit declining elasticity. This effect on the pricing kernel is particularly significant for the pricing of options.

We introduce a pricing kernel with time-varying volatility risk aversion to explain the observed time variations in the shape of the pricing kernel. When combined with the Heston-Nandi GARCH model, this framework yields a tractable option pricing model in which the variance risk ratio (VRR) emerges as a key variable. We show that the VRR is closely linked to economic fundamentals, as well as sentiment and uncertainty measures. A novel approximation method provides analytical option pricing formulas, and we demonstrate substantial reductions in pricing errors through an empirical application to the S&P 500 index, the CBOE VIX, and option prices.

Smile from the past: A general option pricing framework with multiple volatility and leverage components

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with nonparametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by nonparametric regression. The subjective density is estimated nonparametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures. Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.

‘Forward and Futures Prices of Contingent Claims’ applies the multi-period rational expectations model to analyse the pricing of forward and futures contracts and derives an expression for the difference between the prices of the two, which is commonly known as the convexity adjustment. These are illustrated using an example where the asset price, the pricing kernel and the bond prices are joint lognormal variables. The authors then consider the futures and forward prices of contingent claims, such as call options. Futures prices of options can be significantly different from the forward prices of the contingent claims, especially when the options are on interest-rate related assets such as bonds or swaps. Forward prices of long-term bond can be obtained by taking appropriate expected values of future spot pries under the risk-neutral measure.