What is the correct meaning of implied volatility?

This paper presents a closed-form solution for the valuation of European options under the assumption that the excess returns of an underlying asset follow a diffusion process. In light of our model, the implied volatility computed from the Black–Scholes formula should be viewed as the volatility of excess returns rather than as the volatility of gross returns. Using the SPX and the OMX options data, we test whether implied volatility obtained from Black-Scholes option price explains the volatilities of excess returns better than gross returns, even though the result is not statistically significant.     

Dynamic, nonparametric hedging of European style contingent claims using canonical valuation

The canonical valuation, proposed by Stutzer [1996. Journal of Finance 51, 1633–1652], is a nonparametric option pricing approach for valuing European-style contingent claims. This paper derives risk-neutral dynamic hedge formulae for European call and put options under canonical valuation that obey put–call parity. Further, the paper documents the error-metrics of the canonical hedge ratio and analyzes the effectiveness of discrete dynamic hedging in a stochastic volatility environment. The results suggest that the nonparametric hedge formula generates hedges that are substantially unbiased and is capable of producing hedging outcomes that are superior to those produced by Black and Scholes [1973. Journal of Political Economy 81, 637–654] delta hedging.     

International stock markets interactions and conditional correlations

This paper investigates the transmission of price and volatility spillovers across the US and European stock markets in bivariate combinations. The framework used encompasses the most popular multivariate GARCH models, with News Impact Surfaces employed for interpretation. By using synchronous data the dynamic conditional correlation model (Engle, R., 2002. Dynamic conditional correlation: a simple class of multivariate GARCH models. Journal of Business and Economic Statistics 20, 339–350) is found to best capture the relationships for over half of the bivariate combinations of markets. Other findings include volatility spillovers from the US to European markets, and a reverse spillover. In addition, the magnitude of the correlation between markets is higher not only for negative shocks in both markets, but also when a combination of shocks of opposite signs occurs.     

The Valuation of Volatility Options

This paper examines the valuation of European- and American-style volatilityoptions based on a general equilibrium stochastic volatility framework.Properties of the optimal exercise region and of the option price areprovided when volatility follows a general diffusion process. Explicitvaluation formulas are derived in four particular cases. Emphasis is placedon the MRLP (mean-reverting in the log) volatility model which has receivedconsiderable empirical support. In this context we examine the propertiesand hedging behavior of volatility options. Unlike American options,European call options on volatility are found to display concavity at highlevels of volatility.     

Long memory options: LM evidence and simulations

This paper demonstrates the impact of the observed financial market persistence or long term memory on European option valuation by simple simulation. Many empirical researchers have observed the non-Fickian degrees of persistence in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black–Scholes’ geometric Brownian motion assumption. Moreover, Elliott and van der Hoek [Elliott, R.J., van der Hoek, J., 2003. A general fractional white noise theory and applications to finance. Math. Finance 13, 301–330] provide a theoretical framework for incorporating these findings into the Black–Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. By using a simple mono-fractal fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the Black–Scholes pricing formula. Long memory options are of considerable importance in corporate remuneration packages, since stock options are written on a company’s own shares for long expiration periods. It makes a significant difference in the valuation when an option is “blue” or when it is “red.” For a proper valuation of such stock options, the degrees of persistence of the companies’ share markets must be precisely measured and properly incorporated in the warrant valuation, otherwise substantial pricing errors may result.     

International linkage of the Russian market and the Russian financial crisis: A multivariate GARCH analysis

This study considers the linkage of the Russian equity market to the world market, examining the international transmission of the Russia's 1998 financial crisis utilizing the GARCH–BEKK model proposed by Engle and Kroner [Engle, R.F., Kroner, K.F., 1995. Multivariate simultaneous generalized ARCH. Economet. Theor. 11, 122–150]. We find evidence of direct linkage between the Russian equity market with regards to returns and volatility, while the weakness of the linkage suggests that the Russian equity market was only partially integrated into the world market. At the time of the crisis, evidence of contagion is clear.     

Pricing and hedging volatility smile under multifactor interest rate models

The paper extends Amin and Morton (1994), Zeto (2002), and Kuo and Paxson (2006) by considering jump-diffusion model of Das (1999) with various volatility functions in pricing and hedging Euribor options across strikes and maturities. Adding the jump element into a diffusion model helps capturing volatility smiles in the interest rate options markets, but specifying the mean-reversion volatility function improves the most. A humped volatility function with the additional jump component yields better in-sample and out-of-sample valuation, but level-dependent volatility becomes more crucial for hedging. The specification of volatility function is more crucial than merely adding jumps into any model and the effect of jumps declines as the maturity of options is longer.     

A jump diffusion model for VIX volatility options and futures

Volatility indices are becoming increasingly popular as a measure of market uncertainty and as a new asset class for developing derivative instruments. Although jumps are widely considered as a salient feature of volatility, their implications for pricing volatility options and futures are not yet fully understood. This paper provides evidence indicating that the time series behaviour of the VIX index is well approximated by a mean reverting logarithmic diffusion with jumps. This process is capable of capturing stylized facts of VIX dynamics such as fast mean-reversion at higher levels, level effects of volatility and large upward movements during times of market stress. Based on the empirical results, we provide closed-form valuation models for European options written on the spot and forward VIX, respectively.     

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.     

The determinants of bank interest rate margins: an international study

This paper studies the determinants of bank net interest margins (NIMs) in six selected European countries and the US during the period 1988–1995 for a sample of 614 banks. We apply the Ho and Saunders model (Ho, T., Saunders, A., 1981. The determinants of bank interest margins: theory and empirical evidence. Journal of Financial and Quantitative Analyses 16, 581–600) to a multicountry setting and decompose bank margins into a regulatory component, a market structure component and a risk premium component. The regulatory components in the form of interest-rate restrictions on deposits, reserve requirements and capital-to-asset ratios have a significant impact on banks NIMs. The empirical results suggest an important policy trade-off between assuring bank solvency—high capital-to-asset ratios—and lowering the cost of financial services to consumers—low NIMs. The more segmented or restricted the banking system—both geographically and by activity—the larger appears to be the monopoly power of existing banks, and the higher their spreads. Macro interest-rate volatility was found to have a significant impact on bank NIMs; this suggests that macro policies consistent with reduced interest-rate volatility could have a positive effect in reducing bank margins.     

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.     

Pricing volatility options under stochastic skew with application to the VIX index

In recent years, there has been a remarkable growth of volatility options. In particular, VIX options are among the most actively trading contracts at Chicago Board Options Exchange. These options exhibit upward sloping volatility skew and the shape of the skew is largely independent of the volatility level. To take into account these stylized facts, this article introduces a novel two-factor stochastic volatility model with mean reversion that accounts for stochastic skew consistent with empirical evidence. Importantly, the model is analytically tractable. In this sense, I solve the pricing problem corresponding to standard-start, as well as to forward-start European options through the Fast Fourier Transform. To illustrate the practical performance of the model, I calibrate the model parameters to the quoted prices of European options on the VIX index. The calibration results are fairly good indicating the ability of the model to capture the shape of the implied volatility skew associated with VIX options.     

The valuation of multivariate contingent claims under transformed trinomial approaches

This study develops a transformed-trinomial approach for the valuation of contingent claims written on multiple underlying assets. Our model is characterized by an extension of the Camara and Chung (J Futur Mark 26: 759–787, 2006) transformed-binomial model for pricing options with one underlying asset, and a discrete-time version of the Schroder (J Finance 59(5): 2375–2401, 2004) model. However, unlike the Schroder model, our model can facilitate straightforward valuation of American-style multivariate contingent claims. The major advantage of our transformed-trinomial approach is that it can easily tackle the volatility skew observed within the markets. We go on to use numerical examples to demonstrate the way in which our transformed-trinomial approach can be utilized for the valuation of multivariate contingent claims, such as binary options.     

An exact and explicit solution for the valuation of American put options

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.     

Analytical pricing of single barrier options under local volatility models

This paper considers a single barrier option under a local volatility model and shows that any down-and-in option can be priced by a combination of three standard European options whose volatility functions are connected through symmetrization. The symmetrized volatility function is approximated by a sequence of smooth functions that converges to the original one. An approximation formula is developed to price the standard European options with the approximated volatility functions. Finally, we apply the Aitken convergence accelerator to obtain an approximate price of the down-and-in option. Other single barrier options are priced in a similar fashion.     

Option Pricing Under Autoregressive Random Variance Models

Abstract

The autoregressive random variance (ARV) model introduced by Taylor (1980, 1982, 1986) is a popular version of stochastic volatility (SV) models and a discrete-time simplification of the continuous-time diffusion SV models. This paper introduces a valuation model for options under a discrete-time ARV model with general stock and volatility innovations. It employs the discretetime version of the Esscher transform to determine an equivalent martingale measure under an incomplete market. Various parametric cases of the ARV models, are considered, namely, the log-normal ARV models, the jump-type Poisson ARV models, and the gamma ARV models, and more explicit pricing formulas of a European call option under these parametric cases are provided. A Monte Carlo experiment for some parametric cases is also conducted.     

The Valuation of Options on Futures Contracts

Rational restrictions are derived for the values of American options on futures contracts. For these options, the optimal policy, in general, involves premature exercise. A model is developed for valuing options on futures contracts in a constant interest rate setting. Despite the fact that premature exercise may be optimal, the value of this American feature appears to be small and a European formula due to Black serves as a useful approximation. Finally, a model is developed to value these options in a world with stochastic interest rates. It is shown that the pricing errors caused by ignoring the location of the interest rate (relative to its long-run mean) range from ?5% to 7%, when the current rate is ±200 basis points from its long-run value. The role of interest rate expectations is, therefore, crucial to the valuation. Optimal exercise policies are found from numerical methods for both models.     

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].     

Valuation of FX barrier options under stochastic volatility

This paper describes European-style valuation and hedging procedures for a class of knockout barrier options under stochastic volatility. A pricing framework is established by applying mean self-financing arguments and the minimal equivalent martingale measure. Using appropriate combinations of stochastic numerical and variance reduction procedures we demonstrate that fast and accurate valuations can be obtained for down-and-out call options for the Heston model.