Can we forecast the implied volatility surface dynamics of equity options? Predictability and economic value tests

We examine whether the dynamics of the implied volatility surface of individual equity options contains exploitable predictability patterns. Predictability in implied volatilities is expected due to the learning behavior of agents in option markets. In particular, we explore the possibility that the dynamics of the implied volatility surface of individual stocks may be associated with movements in the volatility surface of S&P 500 index options. We present evidence of strong predictable features in the cross-section of equity options and of dynamic linkages between the volatility surfaces of equity and S&P 500 index options. Moreover, time-variation in stock option volatility surfaces is best predicted by incorporating information from the dynamics in the surface of S&P 500 options. We analyze the economic value of such dynamic patterns using strategies that trade straddle and delta-hedged portfolios, and find that before transaction costs such strategies produce abnormal risk-adjusted returns.     

Options Trading Based on the Forecasting of Volatility Direction with the Incorporation of Investor Sentiment

Using options price data on the Taiwanese stock market, we propose an options trading strategy based on the forecasting of volatility direction. The forecasting models are constructed with the incorporation of absolute returns, heterogeneous autoregressive-realized volatility (HAR-RV), and proxy of investor sentiment. After we take into consideration the margin-based transaction costs, the results of our simulated trading indicate that a straddle trading strategy that considers the forecasting of volatility direction with the incorporation of market turnover achieves the best Sharpe ratios. Our trading algorithm bridges the gap between options trading, market volatility, and the information content of investor overreaction.     

An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston [(1993) The Review of Financial studies, 6, 327–343], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.     

Geometric Asian option pricing in general affine stochastic volatility models with jumps

In this paper, we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain closed form solutions for some relevant affine model classes.     

The relationship between implied and realized volatility: evidence from the Australian stock index option market

This paper examines the relationship between the volatility implied in option prices and the subsequently realized volatility by using the S&P/ASX 200 index options (XJO) traded on the Australian Stock Exchange (ASX) during a period of 5 years. Unlike stock index options such as the S&P 100 index options in the US market, the S&P/ASX 200 index options are traded infrequently and in low volumes, and have a long maturity cycle. Thus an errors-in-variables problem for measurement of implied volatility is more likely to exist. After accounting for this problem by instrumental variable method, it is found that both call and put implied volatilities are superior to historical volatility in forecasting future realized volatility. Moreover, implied call volatility is nearly an unbiased forecast of future volatility.

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.     

Realizing smiles: Options pricing with realized volatility

We develop a discrete-time stochastic volatility option pricing model exploiting the information contained in the Realized Volatility (RV), which is used as a proxy of the unobservable log-return volatility. We model the RV dynamics by a simple and effective long-memory process, whose parameters can be easily estimated using historical data. Assuming an exponentially affine stochastic discount factor, we obtain a fully analytic change of measure. An empirical analysis of Standard and Poor's 500 index options illustrates that our model outperforms competing time-varying and stochastic volatility option pricing models.     

The relation between implied and realised volatility in the Danish option and equity markets

We show that the conclusions to be drawn concerning the informational efficiency of illiquid options markets depend critically on whether one carefully recognises and appropriately deals with the econometrics of the errors‐in‐variables problem. This paper examines the information content of options on the Danish KFX share index. We consider the relation between the volatility implied in an option's price and the subsequently realised index return volatility. Since these options are traded infrequently and in low volumes, the errors‐in‐variables problem is potentially large. We address the problem directly using instrumental variables techniques. We find that when measurement errors are controlled for, call option prices even in this very illiquid market contain information about future realised volatility over and above the information contained in historical volatility.     

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.     

An almost Markovian LIBOR market model calibrated to caps and swaptions

A new variant of the LIBOR market model is implemented and calibrated simultaneously to both at-the-money and out-of-the-money caps and swaptions. This model is a two-factor version of a new class of the almost Markovian LIBOR market models with properties long sought after: (i) the almost Markovian parameterization of the LIBOR market model volatility functions is unique and asymptotically exact in the limit of a short time horizon up to a few years, (ii) only minimum plausible assumptions are required to derive the implemented volatility parameterization, (iii) the calibration yields very good results, (iv) the calibration is almost immediate, (v) the implemented LIBOR market model has a related short-rate model. Numerical results for the two-factor case show that the volatility functions for the LIBOR market model can be imported into its short-rate model cousin without adjustment.     

Valuation of commodity derivatives in a new multi-factor model

This paper extends existing commodity valuation models to allow for stochastic volatility and simultaneous jumps in the spot price and spot volatility. Closed-form valuation formulas for forwards, futures, futures options, geometric Asian options and commodity-linked bonds are obtained using the Heston (1993) and Bakshi and Madan (2000) methodology. Stochastic volatility and jumps do not affect the futures price at a given point in time. However, numerical examples indicate that they play important roles in pricing options on futures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Trading volatility spreads: a test of index option market efficiency

If returns on two assets share common volatility components, the prices of options on the assets should be interdependent and the implied volatility spread should mean revert. We first demonstrate, using the canonical correlation method, that there is a common component in the volatilities of the returns on S&P 100 and S&P 500 indices. We then exploit this commonality by trading on the volatility spread between tick-by-tick OEX and SPX call options listed on the CBOE. Our vega-delta-neutral strategies generated significant profits, even after transaction costs are taken into account. The results suggest that the two options markets are not jointly efficient.     

Is Volatility Risk for the British Pound Priced in U.S. Options Markets?

This paper estimates the premium for volatility risk for European currency options written on British pounds. The average annualized premium for volatility risk is neither statistically different from zero nor invariant to the option's moneyness. However, the risk premium is positively and nonproportionaly related to the level of volatility, except for out‐of‐the‐money options. Finding a zero premium for volatility risk does not undermine the assumption of a zero‐price volatility risk in many extant stochastic‐volatility option pricing models and the option pricing formulas in those models.     

Dispersion trading: Empirical evidence from U.S. options markets

This paper develops empirical evidence on the viability of a form of volatility trading known as “dispersion trading.” The results shed light on the efficiency with which U.S. options markets price volatility.Using end-of-day implied volatilities extracted from equity option prices for the stocks that comprise the S&P 500, the implied volatility of the S&P 500 is computed using a modification of the Markowitz variance equation. This Markowitz-implied volatility is then compared to the implied volatility of the S&P 500 extracted directly from index options on the S&P 500. These contemporaneous measures of implied volatility are then examined for exploitable discrepancies both with and without transaction costs. The study covers the period October 31, 2005 through November 1, 2007.It is shown that, from a trader's perspective, index option implied volatility tended to be more often “rich” and component volatilities tended to be more often “cheap.” Nevertheless, there were times when the opposite was true; suggesting that potential dispersion trades can run in either direction.     

The volatility term structure in a lognormal process for the short rate

One-factor Markov models are widely used by practitioners for pricing financial options. Their simplicity facilitates their calibration to the intial conditions and permits fast computer Implementations. Nevertheless, the danger remains that such models behave unrealistically, if the calibration of the volatility is not properly done. Here, we study a lognormal process and investigate how to specify the volatility constraints in such a way that the term structure of volatility at future times, as implied by the short rate process, has a realistic and stable shape. However, the drifting down of the volatility term structure is unavoidable. As a result, there is a tendency to underestimate option prices.     

An Improved Approach to Computing Implied Volatility

A well‐known problem in finance is the absence of a closed form solution for volatility in common option pricing models. Several approaches have been developed to provide closed form approximations to volatility. This paper examines Chance's (1993, 1996) model, Corrado and Miller's (1996) model and Bharadia, Christofides and Salkin's (1996) model for approximating implied volatility. We develop a simplified extension of Chance's model that has greater accuracy than previous models. Our tests indicate dramatically improved results.     

Modelling the Implied Volatility of Options on Long Gilt Futures

This paper investigates the properties of implied volatility series calculated from options on Treasury bond futures, traded on LIFFE. We demonstrate that the use of near-maturity at the money options to calculate implied volatilities causes less mis-pricing and is therefore superior to, a weighted average measure encompassing all relevant options. We demonstrate that, whilst a set of macroeconomic variables has some predictive power for implied volatilities, we are not able to earn excess returns by trading on the basis of these predictions once we allow for typical investor transactions costs.     

Options on realized variance by transform methods: a non-affine stochastic volatility model

In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model by developing efficient transform-based pricing methods. This non-affine model gives prices of options on realized variance that allow upward-sloping implied volatility of variance smiles. Heston's model [Rev. Financial Stud., 1993, 6, 327–343], the benchmark affine stochastic volatility model, leads to downward-sloping volatility of variance smiles—in disagreement with variance markets in practice. Using control variates, we propose a robust method to express the Laplace transform of the variance call function in terms of the Laplace transform of the realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm that gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps.     

Pricing via recursive quantization in stochastic volatility models

We provide the first recursive quantization-based approach for pricing options in the presence of stochastic volatility. This method can be applied to any model for which an Euler scheme is available for the underlying price process and it allows one to price vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991, (4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108]. A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.     

Valuing executive stock options: performance hurdles, early exercise and stochastic volatility

Accounting standards require companies to assess the fair value of any stock options granted to executives and employees. We develop a model for accurately valuing executive and employee stock options, focusing on performance hurdles, early exercise and uncertain volatility. We apply the model in two case studies and show that properly computed fair values can be significantly lower than traditional Black–Scholes values. We then explore the implications for pay-for-performance sensitivity and the design of effective share-based incentive schemes. We find that performance hurdles can require a much greater fraction of total compensation to be a fixed salary, if pre-existing incentive levels are to be maintained.