A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES

We propose a model which can be jointly calibrated to the corporate bond term structure and equity option volatility surface of the same company. Our purpose is to obtain explicit bond and equity option pricing formulas that can be calibrated to find a risk neutral model that matches a set of observed market prices. This risk neutral model can then be used to price more exotic, illiquid, or over‐the‐counter derivatives. We observe that our model matches the equity option implied volatility surface well since we properly account for the default risk in the implied volatility surface. We demonstrate the importance of accounting for the default risk and stochastic interest rate in equity option pricing by comparing our results to Fouque et al., which only accounts for stochastic volatility.     

PORTFOLIO OPTIMIZATION AND STOCHASTIC VOLATILITY ASYMPTOTICS

We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean‐reverting, this is a singular perturbation problem for a nonlinear Hamilton–Jacobi–Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a “practical” strategy that does not require tracking the fast‐moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single‐factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution.     

Improving volatility prediction and option valuation using VIX information: A volatility spillover GARCH model

We develop a new generalized autoregressive conditional heteroskedasticity (GARCH) model that accounts for the information spillover between two markets. This model is used to detect the usefulness of the CBOE volatility index (VIX) for improving the performance of volatility forecasting and option pricing. We find the significant ability of VIX to predict stock volatility both in-sample and out-of-sample. VIX information also helps to greatly reduce the option pricing error. The proposed volatility spillover GARCH model performs better than the related approaches proposed by Kanniainen et al. (2014, J Bank Finance, 43, pp. 200-211) and P. Christoffersen et al. (2014, J Financ Quant Anal, 49, pp. 663–697).     

A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.     

PRICING DERIVATIVES ON MULTISCALE DIFFUSIONS: AN EIGENFUNCTION EXPANSION APPROACH

Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a large class of derivative‐assets. The payoff of the derivative‐assets may be path‐dependent. In addition, the process underlying the derivatives may exhibit killing (i.e., jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility may be multiscale, in the sense that it may be driven by one fast‐varying and one slow‐varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of three derivative‐assets: a vanilla option on a defaultable stock, a path‐dependent option on a nondefaultable stock, and a bond in a short‐rate model.     

Option pricing under fast‐varying long‐memory stochastic volatility

Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long‐range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the option price in the regime where the volatility process is fast mean reverting. The volatility process is modeled as a smooth and bounded function of a fractional Ornstein–Uhlenbeck process. We give the expression for the implied volatility, which has a fractional term structure.     

A Note on Hedging in ARCH and Stochastic Volatility Option Pricing Models

Recently, Duan (1995) proposed a GARCH option pricing formula and a corresponding hedging formula. In a similar ARCH-type model for the underlying asset, Kallsen and Taqqu (1994) arrived at a hedging formula different from Duan's although they concur on the pricing formula. In this note, we explain this difference by pointing out that the formula developed by Kallsen and Taqqu corresponds to the usual concept of hedging in the context of ARCH-type models. We argue, however, that Duan's formula has some appeal and we propose a stochastic volatility model that ensures its validity. We conclude by a comparison of ARCH-type and stochastic volatility option pricing models.     

IMPLIED VOLATILITY IN THE HULL–WHITE MODEL

We study the implied volatility K ↦ I ( K ) in the Hull–White model of option pricing, and obtain asymptotic formulas for this function as the strike price K tends to infinity or zero. We also prove that the function I is convex near zero and concave near infinity, and characterize the behavior of the first two derivatives of this function.     

Volatility forecasts embedded in the prices of crude-oil options

This paper evaluates the ability of alternative option-implied volatility measures to forecast crude-oil return volatility. We find that a corridor implied volatility measure that aggregates information from a narrow range of option contracts consistently outperforms forecasts obtained by the popular Black–Scholes and model-free volatility expectations, as well as those generated by a realized volatility model. This measure ranks favorably in regression-based tests, delivers the lowest forecast errors under different loss functions, and generates economically significant gains in volatility timing exercises. Our results also show that the Chicago Board Options Exchange's “oil-VIX” index performs poorly, as it routinely produces the least accurate forecasts.     

Indian equity options: Smile,risk premiums,and efficiency

We study the pricing of equity options in India which is one of the world's largest options markets. Our findings are supportive of market efficiency: A parsimonious smile-adjusted Black model fits option prices well, and the implied volatility (IV) has incremental predictive power for future volatility. However, the risk premium embedded in IV for Single Stock Options appears to be higher than in other markets. The study suggests that even a very liquid market with substantial participation of global institutional investors can have structural features that lead to systematic departures from the behavior of a fully rational market while being “microefficient.”     

ANALYTICAL COMPARISONS OF OPTION PRICES IN STOCHASTIC VOLATILITY MODELS

This paper gives an ordering on option prices under various well-known martingale measures in an incomplete stochastic volatility model. Our central result is a comparison theorem that proves convex option prices are decreasing in the market price of volatility risk, the parameter governing the choice of pricing measure. The theorem is applied to order option prices under q -optimal pricing measures. In doing so, we correct orderings demonstrated numerically in Heath, Platen, and Schweizer ( Mathematical Finance , 11(4), 2001) in the special case of the Heston model.     

VIX futures and its closed‐form pricing through an affine GARCH model with realized variance

This paper studies the forecasting of volatility index (VIX) and the pricing of its futures by a generalized affine realized volatility model proposed by Christoffersen et al. This model is a weighted average of a GARCH and a pure realized variance (RV) model that incorporates each volatility component into the new dynamics. We rewrite the VIX in terms of both volatility components and then derive closed‐form formulas for the VIX forecasting and its futures pricing. Our empirical studies find that a unification of the GARCH and the RV in the modeling substantially improves the forecasting of this index and the pricing of its futures.     

Bounds on European Option Prices under Stochastic Volatility

In this paper we consider the range of prices consistent with no arbitrage for European options in a general stochastic volatility model. We give conditions under which the infimum and the supremum of the possible option prices are equal to the intrinsic value of the option and to the current price of the stock, respectively, and show that these conditions are satisfied in most of the stochastic volatility models from the financial literature. We also discuss properties of Black–Scholes hedging strategies in stochastic volatility models where the volatility is bounded.     

FAST SWAPTION PRICING IN GAUSSIAN TERM STRUCTURE MODELS

We propose a fast and accurate numerical method for pricing European swaptions in multifactor Gaussian term structure models. Our method can be used to accelerate the calibration of such models to the volatility surface. The pricing of an interest rate option in such a model involves evaluating a multidimensional integral of the payoff of the claim on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state space by a hyperplane tangent to the maximum probability point on the boundary and simplify the multidimensional integration into an analytical form. The maximum probability point can be determined using the gradient descent method. We demonstrate that our method is superior to previous methods by comparing the results to the price obtained by numerical integration.     

EXPLICIT IMPLIED VOLATILITIES FOR MULTIFACTOR LOCAL‐STOCHASTIC VOLATILITY MODELS

We consider an asset whose risk‐neutral dynamics are described by a general class of local‐stochastic volatility models and derive a family of asymptotic expansions for European‐style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three‐halves stochastic volatility, and SABR local‐stochastic volatility.     

CLOSED FORM PRICING FORMULAS FOR DISCRETELY SAMPLED GENERALIZED VARIANCE SWAPS

Most of the existing pricing models of variance derivative products assume continuous sampling of the realized variance processes, though actual contractual specifications compute the realized variance based on sampling at discrete times. We present a general analytic approach for pricing discretely sampled generalized variance swaps under the stochastic volatility models with simultaneous jumps in the asset price and variance processes. The resulting pricing formula of the gamma swap is in closed form while those of the corridor variance swaps and conditional variance swaps take the form of one‐dimensional Fourier integrals. We also verify through analytic calculations the convergence of the asymptotic limit of the pricing formulas of the discretely sampled generalized variance swaps under vanishing sampling interval to the analytic pricing formulas of the continuously sampled counterparts. The proposed methodology can be applied to any affine model and other higher moments swaps as well. We examine the exposure to convexity (volatility of variance) and skew (correlation between the equity returns and variance process) of these discretely sampled generalized variance swaps. We explore the impact on the fair strike prices of these exotic variance swaps with respect to different sets of parameter values, like varying sampling frequencies, jump intensity, and width of the monitoring corridor.     

EXACT SOLUTION OF A MARTINGALE STOCHASTIC VOLATILITY OPTION PROBLEM AND ITS EMPIRICAL EVALUATION

Exact explicit solution of the log-normal stochastic volatility (SV) option model has remained an open problem for two decades. In this paper, I consider the case where the risk-neutral measure induces a martingale volatility process, and derive an exact explicit solution to this unsolved problem which is also free from any inverse transforms. A representation of the asset price shows that its distribution depends on that of two random variables, the terminal SV as well as the time average of future stochastic variances. Probabilistic methods, using the author's previous results on stochastic time changes, and a Laplace–Girsanov Transform technique are applied to produce exact explicit probability distributions and option price formula. The formulae reveal interesting interplay of forces between the two random variables through the correlation coefficient. When the correlation is set to zero, the first random variable is eliminated and the option formula gives the exact formula for the limit of the Taylor series in Hull and White's (1987) approximation. The SV futures option model, comparative statics, price comparisons, the Greeks and practical and empirical implementation and evaluation results are also presented. A PC application was developed to fit the SV models to current market prices, and calculate other option prices, and their Greeks and implied volatilities (IVs) based on the results of this paper. This paper also provides a solution to the option implied volatility problem, as the empirical studies show that, the SV model can reproduce market prices, better than Black–Scholes and Black-76 by up to 2918%, and its IV curve can reproduce that of market prices very closely, by up to within its 0.37%.     

Switching asymmetric GARCH and options on a volatility index

Few proposed types of derivative securities have attracted as much attention and interest as option contracts on volatility. Grunbichler and Longstaff (1996) is the only study that proposes a model to value options written on a volatility index. Their model, which is based on modeling volatility as a GARCH process, does not take into account the switching regime and asymmetry properties of volatility. We show that the Grunbichler and Longstaff (1996) model underprices a three‐month option by about 10%. A Switching Regime Asymmetric GARCH is used to model the generating process of security returns. The comparison between the switching regime model and the traditional uni‐regime model among GARCH, EGARCH, and GJR‐GARCH demonstrates that a switching regime EGARCH model fits the data best. Next, the values of European call options written on a volatility index are computed using Monte Carlo integration. When comparing the values of the option based on the Switching Regime Asymmetric GARCH model and the traditional GARCH specification, it is found that the option values obtained from the different processes are very different. This clearly shows that the Grunbichler‐Longstaff model is too stylized to be used in pricing derivatives on a volatility index. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:251–282, 2004     

Volatility of volatility is (also) rough

Using high-frequency data for major volatility indexes, we compute the volatility of volatility and show that its logarithm follows a fractional Brownian motion with Hurst parameter smaller than 1/2 thereby extending to the volatility asset class the recent findings obtained for the equity index markets. The results confirm that the volatility of volatility is a rough process and it possesses the long memory property. We also show that the correlation between the volatility and the volatility of volatility is positive, consistent with observations in the volatility option market. Lastly, a robustness check using volatility futures confirms the findings.     

THE 4/2 STOCHASTIC VOLATILITY MODEL: A UNIFIED APPROACH FOR THE HESTON AND THE 3/2 MODEL

We introduce a new stochastic volatility model that includes, as special instances, the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). Our model exhibits important features: first, instantaneous volatility can be uniformly bounded away from zero, and second, our model is mathematically and computationally tractable, thereby enabling an efficient pricing procedure. This called for using the Lie symmetries theory for partial differential equations; doing so allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation scheme for the model, which is useful for numerical applications.