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.     

OPTIMAL STATIC–DYNAMIC HEDGES FOR BARRIER OPTIONS

We study optimal hedging of barrier options, using a combination of a static position in vanilla options and dynamic trading of the underlying asset. The problem reduces to computing the Fenchel–Legendre transform of the utility-indifference price as a function of the number of vanilla options used to hedge. Using the well-known duality between exponential utility and relative entropy, we provide a new characterization of the indifference price in terms of the minimal entropy measure, and give conditions guaranteeing differentiability and strict convexity in the hedging quantity, and hence a unique solution to the hedging problem. We discuss computational approaches within the context of Markovian stochastic volatility models.     

Volatility options: Hedging effectiveness,pricing, and model error

Motivated by the growing literature on volatility options and their imminent introduction in major exchanges, this article addresses two issues. First, the question of whether volatility options are superior to standard options in terms of hedging volatility risk is examined. Second, the comparative pricing and hedging performance of various volatility option pricing models in the presence of model error is investigated. Monte Carlo simulations within a stochastic volatility setup are employed to address these questions. Alternative dynamic hedging schemes are compared, and various option‐pricing models are considered. It is found that volatility options are not better hedging instruments than plain‐vanilla options. Furthermore, the most naïve volatility option‐pricing model can be reliably used for pricing and hedging purposes. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:1–31, 2006     

A new scheme for static hedging of European derivatives under stochastic volatility models

This study proposes a new scheme for static hedging of European path‐independent derivatives under stochastic volatility models. First, we show that pricing European path‐independent derivatives under stochastic volatility models is transformed to pricing those under one‐factor local volatility models. Next, applying an efficient static replication method for one‐dimensional price processes developed by Takahashi and Yamazaki (2008), we present a static hedging scheme for European path‐independent derivatives. Finally, a numerical example comparing our method with a dynamic hedging method under Heston's (1993) stochastic volatility model is used to demonstrate that our hedging scheme is effective in practice. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:397–413, 2009     

Market Volatility and Feedback Effects from Dynamic Hedging

In this paper we analyze the manner in which the demand generated by dynamic hedging strategies affects the equilibrium price of the underlying asset. We derive an explicit expression for the transformation of market volatility under the impact of such strategies. It turns out that volatility increases and becomes time and price dependent. The strength of these effects however depends not only on the share of total demand that is due to hedging, but also significantly on the heterogeneity of the distribution of hedged payoffs. We finally discuss in what sense hedging strategies derived from the assumption of constant volatility may still be appropriate even though their implementation obviously violates this assumption.     

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.     

HEDGING WITH ENERGY

In the setting of diffusion models for price evolution, we suggest an easily implementable approximate evaluation formula for measuring the errors in option pricing and hedging due to volatility misspecification. The main tool we use in this paper is a (suitably modified) classical inequality for the L ² norm of the solution, and the derivatives of the solution, of a partial differential equation (the so-called "energy" inequality). This result allows us to give bounds on the errors implied by the use of approximate models for option valuation and hedging and can be used to justify formally some "folk" belief about the robustness of the Black and Scholes model. Surprisingly enough, the result can also be applied to improve pricing and hedging with an approximate model. When statistical or a priori information is available on the "true" volatility, the error measure given by the energy inequality can be minimized w.r.t. the parameters of the approximating model. The method suggested in this paper can help in conjugating statistical estimation of the volatility function derived from flexible but computationally cumbersome statistical models, with the use of analytically tractable approximate models calibrated using error estimates.     

A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets

This paper provides comparative theoretical and numerical results on risks, values, and hedging strategies for local risk-minimization versus mean-variance hedging in a class of stochastic volatility models. We explain the theory for both hedging approaches in a general framework, specialize to a Markovian situation, and analyze in detail variants of the well-known Heston (1993) and Stein and Stein (1991) stochastic volatility models. Numerical results are obtained mainly by PDE and simulation methods. In addition, we take special care to check that all of our examples do satisfy the conditions required by the general theory.     

The compatibility of one‐factor market models in caps and swaptions markets: Evidence from their dynamic hedging performance

This study examines the dynamic hedging performance of the one‐factor LIBOR and swap market models in both caps and swaptions markets, using a procedure similar to the way that these models are used in practice. The effects of different calibration methods on model performance are investigated as well. The LIBOR market models and the swap market models are calibrated to the cross‐sectional Black implied volatilities for caps and swaptions respectively; the test is based on their effectiveness in hedging floors and swaptions that are not used in the calibration. We find that the LIBOR market models outperform the swap market models in hedging floors and perform as well as the swap market models in hedging swaptions. Our results also show that incorporating a humped volatility structure into these models does not significantly improve their hedging performance. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:109–130, 2008     

Local Vega Index and Variance Reduction Methods

In this article we discuss a generalization of the Greek called vega which is used to study the stability of option prices and hedging portfolios with respect to the volatility in various models. We call this generalization the local vega index. We compute through Monte Carlo simulations this index in the cases of Asian options under the classical Black-Scholes setup. Simulation methods using Malliavin calculus and kernel density estimation are compared. Variance reduction methods are discussed.     

PARTIAL HEDGING IN A STOCHASTIC VOLATILITY ENVIRONMENT

We consider the problem of partial hedging of derivative risk in a stochastic volatility environment. It is related to state-dependent utility maximization problems in classical economics. We derive the dual problem from the Legendre transform of the associated Bellman equation and interpret the optimal strategy as the perfect hedging strategy for a modified claim. Under the assumption that volatility is fast mean-reverting and using a singular perturbation analysis, we derive approximate value functions and strategies that are easy to implement and study. The analysis identifies the usual mean historical volatility and the harmonically averaged long-run volatility as important statistics for such optimization problems without further specification of a stochastic volatility model. The approximation can be improved by specifying a model and can be calibrated for the leverage effect from the implied volatility skew. We study the effectiveness of these strategies using simulated stock paths.     

Derivative pricing model and time‐series approaches to hedging: A comparison

This research compares derivative pricing model and statistical time‐series approaches to hedging. The finance literature stresses the former approach, while the applied economics literature has focused on the latter. We compare the out‐of‐sample hedging effectiveness of the two approaches when hedging commodity price risk using futures contracts. For various methods of parameter estimation and inference, we find that the derivative pricing models cannot out‐perform a vector error‐correction model with a GARCH error structure. The derivative pricing models' unpalatable assumption of deterministically evolving futures volatility seems to impede their hedging effectiveness, even when potentially foresighted optionimplied volatility term structures are employed. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:613–641, 2005     

Optimal futures hedging for energy commodities: An application of the GAS model

This paper applies generalized autoregressive score-driven (GAS) models to futures hedging of crude oil and natural gas. For both commodities, the GAS framework captures the marginal distributions of spot and futures returns and corresponding dynamic copula correlations. We compare within-sample and out-of-sample hedging effectiveness of GAS models against constant ordinary least square (OLS) strategy and time-varying copula-based GARCH models in terms of volatility reduction and Value at Risk reduction. We show that the constant OLS hedge ratio is not inherently inferior to the time-varying alternatives. Nonetheless, GAS models tend to exhibit better hedging effectiveness than other strategies, particularly for natural gas.     

Hedging under model misspecification: All risk factors are equal,but some are more equal than others …

It is often difficult to distinguish among different option pricing models that consider stochastic volatility and/or jumps based on a cross‐section of European option prices. This can result in model misspecification. We analyze the hedging error induced by model misspecification and show that it can be economically significant in the cases of a delta hedge, a minimum‐variance hedge, and a delta‐vega hedge. Furthermore, we explain the surprisingly good performance of a simple ad‐hoc Black‐Scholes hedge. We compare realized hedging errors (an incorrect hedge model is applied) and anticipated hedging errors (the hedge model is the true one) and find that there are substantial differences between the two distributions, particularly depending on whether stochastic volatility is included in the hedge model. Therefore, hedging errors can be useful for identifying model misspecification. Furthermore, model risk has severe implications for risk measurement and can lead to a significant misestimation, specifically underestimation, of the risk to which a hedged position is exposed. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Implied deterministic volatility functions: An empirical test for Euribor options

This study proposes the implied deterministic volatility function (IDVF) for the volatility as the function of moneyness and time in the Heath, Jarrow, and Morton (1992) model to price and hedge Euribor options across moneyness and maturities from 1 January 2003 to 31 December 2005. The IDVF models are extended to two‐ and three‐factor models, indicating that they are potential candidates for interest rate risk management. Based on the criteria of in‐sample fitting, prediction, and hedging, it is found that two‐factor IDVF models provide the best in‐sample and prediction performance, whereas three‐factor IDVF models yield the best results for hedging. Correctly specified multifactor models with the volatility as the function of moneyness and time can replace inappropriate onefactor models. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:319–347, 2009     

MEAN–VARIANCE HEDGING AND OPTIMAL INVESTMENT IN HESTON'S MODEL WITH CORRELATION

This paper solves the mean–variance hedging problem in Heston's model with a stochastic opportunity set moving systematically with the volatility of stock returns. We allow for correlation between stock returns and their volatility (so-called leverage effect). Our contribution is threefold: using a new concept of opportunity-neutral measure we present a simplified strategy for computing a candidate solution in the correlated case. We then go on to show that this candidate generates the true variance-optimal martingale measure; this step seems to be partially missing in the literature. Finally, we derive formulas for the hedging strategy and the hedging error.     

The performance of traders' rules in options market

This study focuses on the usefulness of the traders' rules to predict future implied volatilities for pricing and hedging KOSPI 200 index options. There are two versions of this approach. In the “relative smile” approach, the implied volatility skew is treated as a fixed function of moneyness. In the “absolute smile” approach, the implied volatility skew is treated as a fixed function of the strike price. It is found that the “absolute smile” approach shows better performance than Black, F. and Scholes, L. ( 1973 ) model and the stochastic volatility model for both pricing and hedging options. Consistent with Jackwerth, J. C. and Rubinstein, M. (2001) and Li, M. and Pearson, N. D. (2007), the traders' rules dominate mathematically more sophisticated model, that is, the stochastic volatility model. The traders' rules can be an alternative to the sophisticated and complicated models for pricing and hedging options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:999–1020, 2009     

Pricing American options with stochastic volatility: Evidence from S&P 500 futures options

This article is the first attempt to test empirically a numerical solution to price American options under stochastic volatility. The model allows for a mean‐reverting stochastic‐volatility process with non‐zero risk premium for the volatility risk and correlation with the underlying process. A general solution of risk‐neutral probabilities and price movements is derived, which avoids the common negative‐probability problem in numerical‐option pricing with stochastic volatility. The empirical test shows clear evidence supporting the occurrence of stochastic volatility. The stochastic‐volatility model outperforms the constant‐volatility model by producing smaller bias and better goodness of fit in both the in‐sample and out‐of‐sample test. It not only eliminates systematic moneyness bias produced by the constant‐volatility model, but also has better prediction power. In addition, both models perform well in the dynamic intraday hedging test. However, the constant‐volatility model seems to have a slightly better hedging effectiveness. The profitability test shows that the stochastic volatility is able to capture statistically significant profits while the constant volatility model produces losses. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:625–659, 2000     

Regime‐dependent smile‐adjusted delta hedging

We introduce several regime‐dependent smile‐adjusted deltas and compare their efficiency with the smile‐adjusted deltas that are popular with option traders. Using years of daily option prices, out‐of‐sample hedging performance tests for options of all moneyness and maturities and daily, weekly, or fortnightly rebalancing show that even the simplest regime‐dependent smile‐adjustment consistently outperforms implied BSM delta hedging and local volatility and minimum variance smile‐adjustments. Markov‐switching deltas offer the best performance, with delta‐hedging errors often half the size of implied BSM hedging errors. During volatile markets risk reduction from regime‐dependent delta hedging is much greater than during tranquil periods.     

Volatility and trading demands in stock index futures

In this study we examine how volatility and the futures risk premium affect trading demands for hedging and speculation in the S&P 500 Stock Index futures contracts. To ascertain if different volatility measures matter in affecting the result, we employ three volatility estimates. Our empirical results show a positive relation between volatility and open interest for both hedgers and speculators, suggesting that an increase in volatility motivates both hedgers and speculators to engage in more trading in futures markets. However, the influence of volatility on futures trading, especially for hedging, is statistically significant only when spot volatility is used. We also find that the demand to trade by speculators is more sensitive to changes in the futures risk premium than is the demand to trade by hedgers. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:399–414, 2003