Pricing stock options under stochastic volatility and interest rates with efficient method of moments estimation

While the stochastic volatility (SV) generalization has been shown to improve the explanatory power over the Black-Scholes model, empirical implications of SV models on option pricing have not yet been adequately tested. The purpose of this paper is to first estimate a multivariate SV model using the efficient method of moments (EMM) technique from observations of underlying state variables and then investigate the respective effect of stochastic interest rates, systematic volatility and idiosyncratic volatility on option prices. We compute option prices using reprojected underlying historical volatilities and implied stochastic volatility risk to gauge each model’s performance through direct comparison with observed market option prices. Our major empirical findings are summarized as follows. First, while theory predicts that the short-term interest rates are strongly related to the systematic volatility of the consumption process, our estimation results suggest that the short-term interest rate fails to be a good proxy of the systematic volatility factor; Second, while allowing for stochastic volatility can reduce the pricing errors and allowing for asymmetric volatility or leverage effect does help to explain the skewness of the volatility smile, allowing for stochastic interest rates has minimal impact on option prices in our case; Third, similar to Melino and Turnbull (1990), our empirical findings strongly suggest the existence of a non-zero risk premium for stochastic volatility of stock returns. Based on implied volatility risk, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information in the options market in pricing options; Finally, both the model diagnostics and option pricing errors in our study suggest that the Gaussian SV model is not sufficient in modeling short-term kurtosis of asset returns, a SV model with fatter-tailed noise or jump component may have better explanatory power.

An Asymptotic Expansion with Push-Down of Malliavin Weights

This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multi-dimensional stochastic volatility models. In particular, the integration-byparts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in stochastic volatility environment. Some numerical examples are also shown.

An Asymptotic Expansion with Push-Down of Malliavin Weights

This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multi-dimensional stochastic volatility models. In particular, the integration-byparts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in stochastic volatility environment. Some numerical examples are also shown.

Pricing of long-dated commodity derivatives: Do stochastic interest rates matter?

© 2017 Elsevier B.V. Does modelling stochastic interest rates, beyond stochastic volatility, improve pricing performance on long-dated commodity derivatives? To answer this question, we consider futures price models for commodity derivatives that allow for stochastic volatility and stochastic interest rates and a correlation structure between the underlying variables. We examine the empirical pricing performance of these models on pricing long-dated crude oil derivatives. Estimating the model parameters from historical crude oil futures prices and option prices, we find that stochastic interest rate models improve pricing performance on long-dated crude oil derivatives, when the interest rate volatility is relatively high. Furthermore, increasing the model dimensionality does not tend to improve the pricing performance on long-dated crude oil option prices, but it matters for long-dated futures prices. We also find empirical evidence for a negative correlation between crude oil futures prices and interest rates that contributes to improving fit to long-dated crude oil option prices

Maximum Likelihood Estimation of Stochastic Volatility Models

We develop and implement a new method for maximum likelihood estimation in closed-form of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by the implied volatility of a short dated at-the-money option. We find that the approximation results in a negligible loss of accuracy. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine model of Heston (1993) and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.

Identifying Volatility Risk Premium from Fixed Income Asian Options

We provide approximation formulas for at-the-money asian option prices to extract volatility risk premium from a joint dataset of bonds and option prices. The dynamic model generates stochastic volatility and a time-varying volatility risk premium, which explicitly depends on the average cross section of bond yields and on the time series behavior of option prices. When estimated using a joint dataset of Brazilian local bonds and asian options, the model generates bond risk premium strongly correlated (89%) with a widely accepted emerging markets benchmark index, and a negative volatility risk premium implying that investors might be using options as insurance in this market. Volatility premium explains a significant portion (32.5%) of bond premium, confirming that options are indeed important to identify risk premium in dynamic term structure models.

Option Pricing models with Stochastic Volatility and Jumps

Exotic equity options are specialized instruments which are typically traded over the counter. Their prices are primarily determined by option pricing models which should be able to price exotic options consistently with the market prices of corresponding vanilla options. Additionally, option pricing models should have intuitive dynamics which are able to capture real world behavior (such as stochastic volatility effects and jumps in the price of the underlying). This dissertation tackles the question of which option pricing model to use; it compares diffusion, pure jump and jump-diffusion models. All models are fitted to one-day price data on S&P500 European vanilla options; the models with the best fit exhibit the smallest error in pricing between model prices and market prices. The stochastic volatility with jumps (SVJ) models are found to perform the best. The SVJ-DE model, a new variant of this type of model (which is based on Heston-type stochastic volatility and Kou-type double exponential jumps in the log price), is presented and tested. The Heston SV model is ranked third best. There is a significant performance gap between the SV/SVJ models and the remaining models. The variance-gamma model with stochastic time is found to be the best performing model from the pure jump and simple jump-diffusion categories. The Kou jump-diffusion model with double exponential jumps and constant diffusion volatility ranks next, followed by the Merton jump-diffusion model and the variance-gamma pure jump model. On comparison of model and market implied volatility surfaces, the pure jump and simple jump-diffusion models are found to be efficient at generating volatility smile effects, but not volatility skew effects. The converse holds for the Heston SV model. The SVJ models exploit this behavior in an attempt to use the jump component to generate the smile effects on the short end of the volatility surface and the stochastic volatility diffusion component to generate the skew effects on the long end of the volatility surface. The application of the SV and SVJ models is demonstrated by computing the prices of barrier options via Monte Carlo simulation. Both of the SVJ models give similar barrier option prices. Diffusion processes and jump processes are the two main building blocks of any option pricing model. This research finds that simple jump-diffusion models and pure jump models are unable to demonstrate good performance when fitting to a complete grid of market option prices. The Heston stochastic volatility pure diffusion model gives better performance compared to these jump models. The SVJ models which have both a stochastic volatility diffusion component and a jump component are found to give the best performance. The SVJ-DE model has the added advantage of being able to generate upward and downward jumps from different exponential distributions, versus the Bates model which generates jumps from a normal distribution

Short and Long Term Smile Effects: The Binomial Normal Mixture Diffusion Model

This paper extends the normal mixture diffusion (NMD) local volatility model of Brigo and Mercurio (2000, 2001a,b, 2002) so that it explains both short-term and long-term smile effects. Short-term smile effects are captured by a local volatility model where excess kurtosis in the price density decreases with maturity. This follows from the central limit theorem and concords with the ‘stylised facts’ of econometric analysis of ex-post returns of different frequencies. We introduce a term structure for option prices in the NMD model by assuming there is a fixed probability of each volatility state occurring in every time interval Dt, and we show that with this assumption the mixing law for the price density is the multinomial density. This very parsimonious model can easily be calibrated to observed option prices. However, smile effects in currency options often persist into fairly long maturities, and to capture at least some part of this it is necessary to introduce stochastic volatility. The last part of this paper considers only two possible volatility states in each Dt with probabilities l and (1 - l). If l were fixed, the binomial mixing law model would only apply to short-term smile effects. But by making l stochastic, longer-term smile effects that arise from uncertainty in volatility are also captured by the model. The results are illustrated by calibrating the model with and without stochastic l, to a currency option smile surfaceLocal volatility, stochastic volatility, smile consistent models, term structure of option prices, normal variance mixtures

The Continuous Limit of GARCH Processess

Contrary to popular belief, the diffusion limit of a GARCH variance process is not a diffusion model unless one makes a very specific assumption that cannot be generalized. In fact, the normal GARCH(1,1) prices of European call and puts are identical to the Black-Scholes prices based on the average of a deterministic variance process. In the case of GARCH models with several normal components – and these are more realistic representations of option prices and returns behaviour – the continuous limit is a stochastic model with uncertainty over which deterministic local volatility governs the return. The risk neutral model prices of European options are weighted averages of Black-Scholes prices based on the integrated forward variances in each state. An interesting area to be considered for application of this model is path dependent options. Since both marginal and transition price densities are lognormal mixtures the mixture GARCH option pricing model is not equivalent to the mixture option pricing models that have previously been discussed by several authors.GARCH diffusion, normal mixture, stochastic volatility, time aggregation