Option pricing under a stressed-beta model

Empirical studies have concluded that stochastic volatility is an important component of option prices. We introduce a regime-switching mechanism into a continuous-time Capital Asset Pricing Model which naturally induces stochastic volatility in the asset price. Under this Stressed-Beta model, the mechanism is relatively simple: the slope coefficient—which measures asset returns relative to market returns—switches between two values, depending on the market being above or below a given level. After specifying the model, we use it to price European options on the asset. Interestingly, these option prices are given explicitly as integrals with respect to known densities. We find that the model is able to produce a volatility skew, which is a prominent feature in option markets. This opens the possibility of forward-looking calibration of the slope coefficients, using option data, as illustrated in the paper.     

Option pricing under a Gamma-modulated diffusion process

We study a Gamma-modulated diffusion process as a long-memory generalization of the standard Black-Scholes model. This model introduces a time dependent volatility. The option pricing problem associated with this type of processes is computed.     

Option pricing under regime switching

Abstract

This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an efficient algorithm for the computation of American option prices for the general case.     

Option pricing under short-lived arbitrage: theory and tests

Models in financial economics derived from no-arbitrage assumptions have found great favour among theoreticians and practitioners. We develop a model of option prices where arbitrage is short lived. The arbitrage process is Ornstein–Uhlenbeck with zero mean and rapid adjustment of deviations. We find that arbitrage correlated with the underlying can have sizeable impact on option prices. We use data from five large capitalization firms to test implications of the model. Consistent with the existence of arbitrage, we find that idiosyncratic factors significantly effect arbitrage model parameters.     

Option pricing under fast-varying and rough stochastic volatility

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.     

Option pricing and Esscher transform under regime switching

Summary We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

Option pricing with quadratic volatility: a revisit

This paper considers the pricing of European options on assets that follow a stochastic differential equation with a quadratic volatility term. We correct several errors in the existing literature, extend the pricing formulas to arbitrary root configurations, and list alternative representations of option pricing formulas to improve computational performance. Our exposition is based entirely on probabilistic arguments, adding a fresh perspective and new intuition to the existing PDE-dominated literature on the subject. Our main tools are martingale methods and shifts of probability measures; the fact that the underlying process is typically a strict local martingale is carefully considered throughout the paper.     

Option pricing: A simplified approach

This paper presents a simple discrete-time model for valuing options. The fundamental economic principles of option pricing by arbitrage methods are particularly clear in this setting. Its development requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-Scholes model, which has previously been derived only by much more difficult methods. The basic model readily lends itself to generalization in many ways. Moreover, by its very construction, it gives rise to a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.     

Option pricing under time-varying risk-aversion with applications to risk forecasting

We present a two-factor option-pricing model, which parsimoniously captures the difference in volatility persistences under the historical and risk-neutral probabilities. The model generates an S-shaped pricing kernel that exhibits time-varying risk aversion. We apply our model for two purposes. First, we analyze the risk preference implied by S&P500 index options during 2001–2009 and find that risk-aversion level strongly increases during stressed market conditions. Second, we apply our model for Value-at-Risk (VaR) forecasts during the subprime crisis period and find that it outperforms several leading VaR models.     

Option pricing: A general equilibrium approach

This article examines option valuation in a general equilibrium framework. We focus on the general equilibrium implications of price dynamics for option valuation. The general equilibrium considerations allow us to derive an alternative option valuation formula that is as simple as the Black and Scholes formula, and that exhibits different behavior with respect to the exercise price and time to expiration. They also help us clarify comparative-statics properties of option valuation formulas in general and of the Black and Scholes model in particular.     

Option pricing model with sentiment

In this paper, we develop a closed-form option pricing model with the stock sentiment and option sentiment. First, the model shows that the price of call option is amplified by bullish stock sentiment, and is reduced by stock bearish sentiment, and the price of put option is in the opposite situation. Second, the price of call option is more sensitive to bullish stock sentiment; the price of put option is more sensitive to bearish stock sentiment. Third, the price of call option increases substantially with respect to the stock sentiment and the option sentiment. The price of put option decreases substantially with respect to the stock sentiment, increases substantially with respect to the option sentiment. Fourth, our models also reveal that the option volatility smile is steeper (flatter) when the stock sentiment becomes more bearish (bullish). Finally, stock sentiment and option sentiment lead to the option price deviating from the rational price. The model could offer a partial explanation of some option anomalies: option price bubbles and option volatility smile.     

Option pricing under linear autoregressive dynamics,heteroskedasticity, and conditional leptokurtosis

Daily returns of financial assets are frequently found to exhibit positive autocorrelation at lag 1. When specifying a linear AR(1) conditional mean, one may ask how this predictability affects option prices. We investigate the dependence of option prices on autoregressive dynamics under stylized facts of stock returns, i.e. conditional heteroskedasticity, leverage effect, and conditional leptokurtosis. Our analysis covers both a continuous and discrete time framework. The results suggest that a non-zero autoregression coefficient tends to increase the deviation of option prices from Black and Scholes prices caused by stochastic volatility.     

Option pricing and the martingale restriction

In the absence of frictions, the value of the under-lying assetimplied by option prices must equal its actual market value.With frictions, however, this requirement need not hold. UsingS&P 100 index options data, I find that the implied costof the index is significantly higher in the options market thanin the stock market, and is directly related to measures oftransaction costs and liquidity. I show that the Black-Scholesmodel has strong bid-ask spread, trading volume, and open interestbiases. Option pricing models that relax the martingale restrictionperform significantly better.     

Option pricing with random risk aversion

Review of Quantitative Finance and Accounting - Based on a standard general equilibrium economy, we develop a framework for pricing European options where the risk aversion parameter is state...     

Option pricing in a Garch model with tempered stable innovations

The key problem for option pricing in Garch models is that the risk-neutral distribution of the underlying at maturity is unknown. Heston and Nandi solved this problem by computing the characteristic function of the underlying by a recursive procedure. Following the same idea, Christoffersen, Heston and Jacobs proposed a Garch-like model with inverse Gaussian innovations and recently Bellini and Mercuri obtained a similar procedure in a model with Gamma innovations. We present a model with tempered stable innovations that encompasses both the CHJ and the BM models as special cases. The proposed model is calibrated on S&P500 closing option prices and its performance is compared with the CHJ, the BM and the Heston–Nandi models.     

Option pricing in a lognormal securities market with discrete trading

This paper derives a call option valuation equation assuming discrete trading in securities markets where the underlying asset and market returns are bivariate lognormally distributed and investors have increasing, concave utility functions exhibiting skewness preference. Since the valuation does not require the continouus time riskfree hedging of Black and Scholes, nor the discrete time riskfree hedging of Cox, Ross and Rubinstein, market effects are introduced into the option valuation relation. The new option valuation seems to correct for the systematic mispricing of well-in and well-out of the money options by the Black and Scholes option pricing formula.     

Option pricing in a lognormal securities market with discrete trading: A comment

In a recent paper Lee et al. derive a pricing formula which is significantly different from that of Black and Scholes. Their derivation is inconsistent due to their failure to recognize that the rate of return of an option written on an asset whose rate of return is lognormally distributed will not be lognormally distributed.