When are Options Overpriced? The Black--Scholes Model and Alternative Characterisations of the Pricing Kernel

An important determinant of option prices is the elasticityof the pricing kernel used to price all claims in the economy.In this paper, we first show that for a given forward priceof the underlying asset, option prices are higher when the elasticityof the pricing kernel is declining than when it is constant.We then investigate the implications of the elasticity of thepricing kernel for the stochastic process followed by the underlyingasset. Given that the underlying information process followsa geometric Brownian motion, we demonstrate that constant elasticityof the pricing kernel is equivalent to a Brownian motion forthe forward price of the underlying asset, so that the Black–Scholesformula correctly prices options on the asset. In contrast,declining elasticity implies that the forward price processis no longer a Brownian motion: it has higher volatility andexhibits autocorrelation. In this case, the Black–Scholesformula underprices all options.     

Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.     

Bayesian range-based estimation of stochastic volatility models

Alizadeh, Brandt, and Diebold [2002. Journal of Finance 57, 1047–1091] propose estimating stochastic volatility models by quasi-maximum likelihood using data on the daily range of the log asset price process. We suggest a related Bayesian procedure that delivers exact likelihood based inferences. Our approach also incorporates data on the daily return and accommodates a nonzero drift. We illustrate through a Monte Carlo experiment that quasi-maximum likelihood using range data alone is remarkably close to exact likelihood based inferences using both range and return data.     

Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.     

Information and option pricings

How can one relate stock fluctuations and information-based human activities? We present a model of an incomplete market by adjoining the Black-Scholes exponential Brownian motion model for stock fluctuations with a hidden Markov process, which represents the state of information in the investors' community. The drift and volatility parameters take different values depending on the state of this hidden Markov process. Standard option pricing procedure under this model becomes problematic. Yet, with an additional economic assumption, we provide an explicit closed-form formula for the arbitrage-free price of the European call option. Our model can be discretized via a Skorohod embedding technique. We conclude with an example of a simulation of IBM stock, which shows that, not surprisingly, information does affect the market.     

Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment

By utilizing information about prices and trading volumes, we discuss the pricing of European contingent claims in a continuous-time hidden regime-switching environment. Hidden market sentiments described by the states of a continuous-time, finite-state, hidden Markov chain represent a common factor for an asset’s drift and volatility, as well as its trading volumes. Using observations about trading volumes, we present a filtered estimate of the hidden common factor. The asset pricing problem is then considered in a filtered market, where the hidden drift and volatility are replaced by their filtered estimates. We adopt the Esscher transform to select an equivalent martingale measure for pricing and derive a partial-differential integral equation for the option price.     

Drift Estimation of Generalized Security Price Processes from High Frequency Derivative Prices

This paper presents a framework for using high frequency derivative prices to estimate the drift of generalized security price processes. This work may be seen more generally as a quasi-likelihood approach to estimating continuous-time parameters of derivative pricing models using discrete option data. We develop a generalized derivative-based estimator for the drift where the underlying security price process follows any arbitrary state-time separable diffusion process (including arithmetic and geometric Brownian motion as special cases). The framework provides a method to measure premia in derivative prices, test for risk-neutral pricing and leads to a new empirical approach to pricing derivative contingent claims. A sufficient condition for the asymptotic consistency of the generalized estimator is also obtained. A study based on generating the S&P500 index and calls shows that the estimator can correctly estimate the drift parameter. This revised version was published online in November 2006 with corrections to the Cover Date.     

Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift

Abstract

We extend the work of Browne (1995) and Schmidli (2001), in which they minimize the probability of ruin of an insurer facing a claim process modeled by a Brownian motion with drift. We consider two controls to minimize the probability of ruin: (1) investing in a risky asset and (2) purchasing quota-share reinsurance. We obtain an analytic expression for the minimum probability of ruin and the corresponding optimal controls, and we demonstrate our results with numerical examples.     

Pricing methods and hedging strategies for volatility derivatives

We explore the valuation and hedging of discretely observed volatility derivatives using three different models for the price of the underlying asset: Geometric Brownian motion with constant volatility, a local volatility surface, and jump-diffusion. We begin by comparing the effects on valuation of variations in contract design, such as the differences between specifying log returns or actual returns and incorporating caps on the level of realized volatility. We then focus on the difficulties associated with hedging these products. Delta hedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing. Moreover, they provide limited protection in the jump-diffusion context. We study the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in vanilla options at each volatility observation.     

In the insurance business risky investments are dangerous: the case of negative risk sums

We investigate models with negative risk sums when the company invests its reserve into a risky asset whose price follows a geometric Brownian motion. Our main result is an exact asymptotic of the ruin probabilities for the case of exponentially distributed benefits. As in the case of non-life insurance with exponential claims, the ruin probabilities are either decreasing with a rate given by a power function (the case of small volatility) or equal to one identically (the case of large volatility). The result allows us to quantify the share of reserve to invest into such a risky asset to avoid a catastrophic outcome, namely the ruin with probability one. We address also the question of smoothness of the ruin probabilities as a function of the initial reserve for generally distributed jumps.     

An analytical approximation for pricing VWAP options

This paper proposes a unified approximation method for various options whose pay-offs depend on the volume weighted average price (VWAP). Despite their popularity in practice, very few pricing models have been developed in the literature. Also, in previous works, the underlying asset process has been restricted to a geometric Brownian motion. In contrast, our method is applicable to the general class of continuous Markov processes such as local volatility models. Moreover, our method can be used for any type of VWAP options with fixed-strike, floating-strike, continuously sampled, discretely sampled, forward-start and in-progress transactions.     

A new approach for option pricing under stochastic volatility

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives.     

Forecasting oil price realized volatility using information channels from other asset classes

Motivated from Ross (1989) who maintains that asset volatilities are synonymous to the information flow, we claim that cross-market volatility transmission effects are synonymous to cross-market information flows or “information channels” from one market to another. Based on this assertion we assess whether cross-market volatility flows contain important information that can improve the accuracy of oil price realized volatility forecasting. We concentrate on realized volatilities derived from the intra-day prices of the Brent crude oil and four different asset classes (Stocks, Forex, Commodities and Macro), which represent the different “information channels” by which oil price volatility is impacted from. We employ a HAR framework and estimate forecasts for 1-day to 66-days ahead. Our findings provide strong evidence that the use of the different “information channels” enhances the predictive accuracy of oil price realized volatility at all forecasting horizons. Numerous forecasting evaluation tests and alternative model specifications confirm the robustness of our results.     

Debt rollover-induced local volatility model

This paper introduces a structural scenario-based model with debt rollover risk and a higher-fidelity treatment of the bankruptcy procedure. The emerging stock price process is a generalized Brownian motion with state-dependent local volatility, and the resultant implied volatility smile is due exclusively to structural features (debt rollover and credit risks). Therefore, the model reinforces structural foundations of local volatility option pricing models. The paper advocates a joint modeling and calibration framework for multiple classes of derivatives on the firm’s asset value. In particular, an empirical application to Solar City equity and stock option valuation demonstrates the versatility and efficiency gains of the suggested model.

     

Skew Brownian Motion and Pricing European Options

Abstract

The volatility smile and systematic mispricing of the Black–Scholes option pricing model are the typical motivation for examining stochastic processes other than geometric Brownian motion to describe the underlying stock price. In this paper a new stochastic process is presented, which is a special case of the skew-Brownian motion of Itô and McKean. The process in question is the sum of a standard Brownian motion and an independent reflecting Brownian motion that is similar in construction to the stochastic representation of a skew-normal random variable. This stochastic process is taken in its exponential form to price European options. The derived option price nests the Black–Scholes equation as a special case and is flexible enough to accommodate stochastic volatility as well as stochastic skewness.     

Extracting Information from Options Markets: Smiles, State–Price Densities and Risk Aversion

In this paper, recent techniques of estimating implied information from derivatives markets are presented and applied empirically to the French derivatives market. We determine nonparametric implied volatility functions, state–price densities and historical densities from a high–frequency CAC 40 stock index option dataset. Moreover, we construct an estimator of the risk aversion function implied by the joint observation of the cross–section of option prices and time–series of underlying asset value. We report a decreasing implied volatility curve with the moneyness of the option. The estimated relative risk aversion functions are positive and globally consistent with the decreasing relative risk aversion assumption.     

The SIML Estimation of Integrated Covariance and Hedging Coefficient Under Round-off Errors,Micro-market Price Adjustments and Random Sampling

For estimating the integrated volatility and covariance by using high frequency data, Kunitomo and Sato (Math Comput Simul 81:1272–1289, 2011; N Am J Econ Finance 26:289–309, 2013) have proposed the separating information maximum likelihood (SIML) method when there are micro-market noises. The SIML estimator has reasonable finite sample properties and asymptotic properties when the sample size is large when the hidden efficient price process follows a Brownian semi-martingale. We shall show that the SIML estimation is useful for estimating the integrated covariance and hedging coefficient when we have round-off errors, micro-market price adjustments and noises, and when the high-frequency data are randomly sampled. The SIML estimation is consistent, asymptotically normal in the stable convergence sense under a set of reasonable assumptions and it has reasonable finite sample properties with these effects.