Option Prices, Implied Price Processes, and Stochastic Volatility

This paper characterizes all continuous price processes that are consistent with current option prices. This extends Derman and Kani (1994), Dupire (1994, 1997), and Rubinstein (1994), who only consider processes with deterministic volatility. Our characterization implies a volatility forecast that does not require a specific model, only current option prices. We show how arbitrary volatility processes can be adjusted to fit current option prices exactly, just as interest rate processes can be adjusted to fit bond prices exactly. The procedure works with many volatility models, is fast to calibrate, and can price exotic options efficiently using familiar lattice techniques.     

A closed-form GARCH option valuation model

This paper develops a closed-form option valuation formula fora spot asset whose variance follows a GARCH(p, q) process thatcan be correlated with the returns of the spot asset. It providesthe first readily computed option formula for a random volatilitymodel that can be estimated and implemented solely on the basisof observables. The single lag version of this model containsHeston's (1993) stochastic volatility model as a continuous-timelimit. Empirical analysis on S&P500 index options showsthat the out-of-sample valuation errors from the single lagversion of the GARCH model are substantially lower than thead hoc Black-Scholes model of Dumas, Fleming and Whaley (1998)that uses a separate implied volatility for each option to fitto the smirk/smile in implied volatilities. The GARCH modelremains superior even though the parameters of the GARCH modelare held constant and volatility is filtered from the historyof asset prices while the ad hoc Black-Scholes model is updatedevery period. The improvement is largely due to the abilityof the GARCH model to simultaneously capture the correlationof volatility, with spot returns and the path dependence involatility.     

Generalized parameter functions for option pricing

We extend the benchmark nonlinear deterministic volatility regression functions of Dumas et al. (1998) to provide a semi-parametric method where an enhancement of the implied parameter values is used in the parametric option pricing models. Besides volatility, skewness and kurtosis of the asset return distribution can also be enhanced. Empirical results, using closing prices of the S&P 500 index call options (in one day ahead out-of-sample pricing tests), strongly support our method that compares favorably with a model that admits stochastic volatility and random jumps. Moreover, it is found to be superior in various robustness tests. Our semi-parametric approach is an effective remedy to the curse of dimensionality presented in nonparametric estimation and its main advantage is that it delivers theoretically consistent option prices and hedging parameters. The economic significance of the approach is tested in terms of hedging, where the evaluation and estimation loss functions are aligned.     

Conditions on option prices for absence of arbitrage and exact calibration

Under the assumption of absence of arbitrage, European option quotes on a given asset must satisfy well-known inequalities, which have been described in the landmark paper of Merton [Merton, R., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4 (1), 141–183]. If we further assume that there is no interest rate volatility and that the underlying asset continuously pays deterministic dividends, cross-maturity inequalities must also be satisfied by the bid and ask option prices.     

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.     

Pricing double barrier options under a volatility regime-switching model with psychological barriers

The prices of lots of assets have been proved in literature to exhibit special behaviors around psychological barriers, which is an important fact needed to be considered when pricing derivatives. In this paper, we discuss the valuation problem of double barrier options under a volatility regime-switching model where there exist psychological barriers in the prices of underlying assets. The volatility can shift between two regimes, that is to say, when the asset price rises up or falls down through the psychological barrier, the volatility takes two different values. Using the Laplace transform approach, we obtain the price of the double barrier knock-out call option as well as its delta. We also provide the eigenfunction expansion pricing formula and examine the effect of the psychological barrier on the option price and delta, finding that the gamma of the option is discontinuous at such barriers.     

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.     

Double leverage cycle,interest rate,and financial crisis

We view mortgage as a risky derivative of its underlying house collateral and combine no-arbitrage valuation with equilibrium valuation approaches to develop a dynamic model of leverage cycle and interest rate. This model provides a unified explanation to pro-cyclical optimism, asset prices, and leverage, and counter-cyclical volatility and interest rate. In addition, the model shows that tightening funding margin in the mortgage securities market dampens optimism, asset prices, and leverage, whereas it raises volatility and interest rate in the housing market. A double leverage cycle leads to more volatile markets and a severe leverage cycle, thus resulting in worse financial crises.     

Cross-sectional tests of deterministic volatility functions

We study the cross-sectional performance of option pricing models in which the volatility of the underlying stock is a deterministic function of the stock price and time. For each date in our sample of FTSE 100 index option prices, we fit an implied binomial tree to the panel of all European style options with different strike prices and maturities and then examine how well this model prices a corresponding panel of American style options. We find that the implied binomial tree model performs no better than an ad-hoc procedure of smoothing Black–Scholes implied volatilities across strike prices and maturities. Our cross-sectional results complement the time-series findings of Dumas et al. [J. Finance 53 (1998) 2059].     

Bivariate normal mixture spread option valuation

We discuss the pricing and hedging of European spread options on correlated assets when the marginal distribution of each asset return is assumed to be a mixture of normal distributions. Being a straightforward two-dimensional generalization of a normal mixture diffusion model, the prices and hedge ratios have a firm behavioural and theoretical foundation. In this ‘bivariate normal mixture’ (BNM) model no-arbitrage option values are just weighted sums of different ‘2GBM’ option values that are based on the assumption of two correlated lognormal diffusions, and likewise for their sensitivities. The main advantage of this approach is that BNM option values are consistent with both volatility smiles and with the implied correlation ‘frown’. No other ‘frown consistent’ spread option valuation model has such straightforward implementation. We apply analytic approximations to compare BNM valuations of European spread options with those based on the 2GBM assumption and explain the differences between the two as a weighted sum of six second-order 2GBM sensitivities. We also examine BNM option sensitivities, finding that these, like the option values, can sometimes differ substantially from those obtained under the 2GBM model. Finally, we show how the correlation frown that is implied by the BNM model is affected as we change (a) the correlation structure and (b) the tail probabilities in the joint density of the asset returns.     

Return predictability and shareholders’ real options

This study re-interprets the properties of the residual income model by highlighting the shareholders’ abandonment (liquidation or adaptation) option. We estimate the value of this real option as an explicit component of abnormal earnings in the residual income model and test the improvement in valuation after incorporating it into the model. Relative to the traditional specification of the residual income model, this real options model has a stronger predictive power for future abnormal stock returns. We also find that the superior return predictability of the real options model is pronounced in the set of firms with a high probability of exercising liquidation options (for example, those with low profitability, low growth opportunities, high underlying asset volatility, and low intangible assets), which is consistent with the importance of shareholders’ abandonment option in equity valuation. The results are robust to extensive sensitivity checks.     

Index Option Pricing Models with Stochastic Volatility and Stochastic Interest Rates

This paper specifies a multivariate stochasticvolatility (SV) model for the S & P500 index and spot interest rateprocesses. We first estimate the multivariate SV model via theefficient method of moments (EMM) technique based on observations ofunderlying state variables, and then investigate the respective effects of stochastic interest rates, stochastic volatility, and asymmetric S & P500 index returns on option prices. We compute option prices using both reprojected underlying historical volatilities and the implied risk premiumof stochastic volatility to gauge each model's performance through direct comparison with observed market option prices on the index. Our major empirical findings are summarized as follows. First, 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. Second, similar to Melino and Turnbull (1990), our empirical findings strongly suggest the existence of a non-zero risk premium for stochastic volatility of asset returns. Based on the implied volatility risk premium, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information from 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 sufficientin modeling short-term kurtosis of asset returns, an SV model withfatter-tailed noise or jump component may have better explanatory power.     

Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing

This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly in the dynamics of the implied volatility surface. The paper derives a forward PIDE (PartialIntegro-Differential Equation) and demonstrates how this equationcan be used to fit the model to European option prices. For numerical pricing of general contingent claims, we develop an ADI finite difference method that is shown to be unconditionally stable and, if combined with Fast Fourier Transform methods, computationally efficient. The paper contains several detailed examples fromthe S&P500 market. This revised version was published online in November 2006 with corrections to the Cover Date.     

A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.As a concrete example, we specialize to a variant of the Hull-White or Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q.JEL Classification: D52, G13     

Pricing American interest rate claims with humped volatility models

Some of the most recent empirical studies on interest rate derivatives have found humped shapes in the volatility structure of interest rates. In this paper, we propose a simple model that allows for humped volatility structures, and that can be described by one state variable. With the model, American style claims can be priced very efficiently which is very important if the model has to be calibrated daily to market prices of standard American options. Furthermore, the model allows for explicit formulas for European style options. Finally, the computational efficiency of our model in the Li et al. (1995) framework is compared with the efficiency in a typical Hull and White (1993a, 1994, 1996) framework. In fact, we can use both procedures for our model, since we prove that if a deterministic volatility model can be embedded in either of these algorithms, then so it does in the other one. Empirical evidence from option data supporting our model is provided as well.     

An analytical approximation to the option formula for the GARCH model

This article derives an analytical approximation to the option formula for a spot asset price whose conditional variance equation follows a nonlinear asymmetric GARCH (NGARCH) process. The approximate option formula, which is just a volatility adjustment in comparison to the Black-Scholes (BS) formula, is very simple and provides the volatility term structure of spot asset prices. Also, the formula shows that the most characteristic feature of an NGARCH model appears in the vega of a European option, which depends on both the spread between the long-run variance and the current one and a parameter reproduced from the stationary property of the conditional variance. This methodology can be easily extended to an option formula for the generalized GARCH process.     

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.

     

Asset pricing under information with stochastic volatility

Based on a general specification of the asset specific pricing kernel, we develop a pricing model using an information process with stochastic volatility. We derive analytical asset and option pricing formulas. The asset prices in this rational expectations model exhibit crash-like, strong downward movements. The resulting option pricing formula is consistent with the strong negative skewness and high levels of kurtosis observed in empirical studies. Furthermore, we determine credit spreads in a simple structural model.      

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.     

American GARCH employee stock option valuation

We implement a flexible simulation-based approach for the fair value of employee stock option (ESO) that accounts for the vesting period, departure risk and voluntary suboptimal early exercise. We introduce GARCH effects on the underlying asset and we analyze the price bias with respect to the constant volatility case. We also perform a sensitivity analysis with respect to changes in several ESO characteristics. We compare this valuation with FAS 123 method revealing a FAS overvaluation. Finally, we value a real ESO plan providing the confidence intervals for the estimated ESO prices.