The performance of VIX option pricing models: Empirical evidence beyond simulation

We examine the pricing performance of VIX option models. Such models possess a wide‐range of underlying characteristics regarding the behavior of both the S&P500 index and the underlying VIX. Our tests employ three representative models for VIX options: Whaley ( 1993 ), Grunbichler and Longstaff ( 1996 ), Carr and Lee ( 2007 ), Lin and Chang ( 2009 ), who test four stochastic volatility models, as well as to previous simulation results of VIX option models. We find that no model has small pricing errors over the entire range of strike prices and times to expiration. In particular, out‐of‐the‐money VIX options are difficult to price, with Grunbichler and Longstaff's mean‐reverting model producing the smallest dollar errors in this category. Whaley's Black‐like option model produces the best results for in‐the‐money VIX options. However, the Whaley model does under/overprice out‐of‐the‐money call/put VIX options, which is opposite the behavior of stock index option pricing models. VIX options exhibit a volatility skew opposite the skew of index options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark31:251–281, 2011     

A generalization of the Barone‐Adesi and Whaley approach for the analytic approximation of American options

This article introduces a general quadratic approximation scheme for pricing American options based on stochastic volatility and double jump processes. This quadratic approximation scheme is a generalization of the Barone‐Adesi and Whaley approach and nests several option models. Numerical results show that this quadratic approximation scheme is efficient and useful in pricing American options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:478–493, 2009     

VIX option pricing

Substantial progress has been made in developing more realistic option pricing models for S&P 500 index (SPX) options. Empirically, however, it is not known whether and by how much each generalization of SPX price dynamics improves VIX option pricing. This article fills this gap by first deriving a VIX option model that reconciles the most general price processes of the SPX in the literature. The relative empirical performance of several models of distinct interest is examined. Our results show that state‐dependent price jumps and volatility jumps are important for pricing VIX options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:523–543, 2009     

Regime‐switching in stock index and Treasury futures returns and measures of stock market stress

We investigate bivariate regime‐switching in daily futures‐contract returns for the US stock index and ten‐year Treasury notes over the crisis‐rich 1997–2005 period. We allow the return means, volatilities, and correlation to all vary across regimes. We document a striking contrast between regimes, with a high‐stress regime that exhibits a much higher stock volatility, a much lower stock–bond correlation, and a higher mean bond return. The high‐stress regime is associated with higher average values of stock‐implied volatility, stock illiquidity, and stock and bond futures trading volume. The lagged implied volatility from equity‐index options is useful in modeling the time‐varying transition probabilities of the regime‐switching process. Our findings support the notions that: (1) stock market stress can have a material influence on Treasury bond pricing, and (2) the diversification benefits of combined stock–bond holdings tend to be greater during times with relatively high stock market stress. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:753–779, 2010     

Option pricing with a non‐zero lower bound on stock price

Black, F. and Scholes, M. (1973) assume a geometric Brownian motion for stock prices and therefore a normal distribution for stock returns. In this article a simple alternative model to Black and Scholes (1973) is presented by assuming a non‐zero lower bound on stock prices. The proposed stock price dynamics simultaneously accommodate skewness and excess kurtosis in stock returns. The feasibility of the proposed model is assessed by simulation and maximum likelihood estimation of the return probability density. The proposed model is easily applicable to existing option pricing models and may provide improved precision in option pricing. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:775–794, 2005     

Lévy betas: Static hedging with index futures

This study considers calibration to forward‐looking betas by extracting information on equity and index options from prices using Lévy models. The resulting calibrated betas are called Lévy betas. The objective of the proposed approach is to capture market expectations for future betas through option prices, as betas estimated from historical data may fail to reflect structural change in the market. By assuming a continuous‐time capital asset pricing model (CAPM) with Lévy processes, we derive an analytical solution to index and stock options, thus permitting the betas to be implied from observed option prices. One application of Lévy betas is to construct a static hedging strategy using index futures. Employing Hong Kong equity and index option data from September 16, 2008 to October 15, 2009, we show empirically that the Lévy betas during the sub‐prime mortgage crisis period were much more volatile than those during the recovery period. We also find evidence to suggest that the Lévy betas improve static hedging performance relative to historical betas and the forward‐looking betas implied by a stochastic volatility model.     

QUANTO LOOKBACK OPTIONS

The lookback feature in a quanto option refers to the payoff structure where the terminal payoff of the quanto option depends on the realized extreme value of either the stock price or the exchange rate. In this paper, we study the pricing models of European and American lookback options with the quanto feature. The analytic price formulas for two types of European-style quanto lookback options are derived. The success of the analytic tractability of these quanto lookback options depends on the availability of a succinct analytic representation of the joint density function of the extreme value and terminal value of the stock price and exchange rate. We also analyze the early exercise policies and pricing behaviors of the quanto lookback options with the American feature. The early exercise boundaries of these American quanto lookback options exhibit properties that are distinctive from other two-state American option 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     

PRICING OPTIONS ON VARIANCE IN AFFINE STOCHASTIC VOLATILITY MODELS

We consider the pricing of options written on the quadratic variation of a given stock price process. Using the Laplace transform approach, we determine semi‐explicit formulas in general affine models allowing for jumps, stochastic volatility, and the leverage effect. Moreover, we show that the joint dynamics of the underlying stock and a corresponding variance swap again are of affine form. Finally, we present a numerical example for the Barndorff‐Nielsen and Shephard model with leverage. In particular, we study the effect of approximating the quadratic variation with its predictable compensator.     

Knock‐in American options

A knock‐in American option under a trigger clause is an option contract in which the option holder receives an American option conditional on the underlying stock price breaching a certain trigger level (also called barrier level). We present analytic valuation formulas for knock‐in American options under the Black‐Scholes pricing framework. The price formulas possess different analytic representations, depending on the relation between the trigger stock price level and the critical stock price of the underlying American option. We also performed numerical valuation of several knock‐in American options to illustrate the efficacy of the price formulas. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:179–192, 2004     

Give everyone a prize? Employee stock options in private venture-backed firms

This study examines employee stock options in private entrepreneurial companies. I focus on private U.S. venture-backed firms because they are renowned for the intensity and organizational depth of their stock option grants. Contrary to simple stereotype, however, I show that 27% of U.S. venture-backed firms do not grant stock options to all employees. I seek to explain this by theorizing that the economic and legal settings in which venture-backed companies exist lead to both costs and benefits from the use of stock options to attract, compensate, incent, monitor, and retain certain employees, and that sometimes the costs exceed the benefits. I test the theory by determining whether variation in the organizational depth to which venture-backed firms grant employee stock options can be explained by proxies for these economic and legal costs. Such proxies include the fraction of employees who are in technical positions; the degree of flatness in the firm's organizational structure; its proximity to other venture-backed companies; the number of patents it has been granted; and the fraction of equity held by venture investors. The results support the theory and thereby imply that venture-backed firms grant employee stock options in an economically sophisticated manner.     

Pricing and hedging of quanto range accrual notes under Gaussian HJM with cross‐currency Levy processes

This study analyzes the pricing and hedging problems for quanto range accrual notes (RANs) under the Heath‐Jarrow‐Morton (HJM) framework with Levy processes for instantaneous domestic and foreign forward interest rates. We consider the effects of jump risk on both interest rates and exchange rates in the pricing of the notes. We first derive the pricing formula for quanto double interest rate digital options and quanto contingent payoff options; then we apply the method proposed by Turnbull (Journal of Derivatives, 1995, 3, 92–101) to replicate the quanto RAN by a combination of the quanto double interest rate digital options and the quanto contingent payoff options. Using the pricing formulas derived in this study, we obtain the hedging position for each issue of quanto RANs. In addition, by simulation and assuming the jump risk to follow a compound Poisson process, we further analyze the effects of jump risk and exchange rate risk on the coupons receivable in holding a RAN. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:973–998, 2009     

The valuation of reset options with multiple strike resets and reset dates

This article makes two contributions to the literature. The first contribution is to provide the closed‐form pricing formulas of reset options with strike resets and predecided reset dates. The exact closed‐form pricing formulas of reset options with strike resets and continuous reset period are also derived. The second contribution is the finding that the reset options not only have the phenomena of Delta jump and Gamma jump across reset dates, but also have the properties of Delta waviness and Gamma waviness, especially near the time before reset dates. Furthermore, Delta and Gamma can be negative when the stock price is near the strike resets at times close to the reset dates. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:87–107,2003     

Empirical performance of alternative pricing models of currency options

This article examines the out‐of‐sample pricing performance and biases of the Heston’s stochastic volatility and modified Black‐Scholes option pricing models in valuing European currency call options written on British pound. The modified Black‐Scholes model with daily‐revised implied volatilities performs as well as the stochastic volatility model in the aggregate sample. Both models provide close and similar correspondence to actual prices for options trading near‐ or at‐the‐money. The prices generated from the stochastic volatility model are subject to fewer and weaker aggregate pricing biases than are the prices from the modified Black‐Scholes model. Thus, the stochastic volatility model may provide improved estimates of the measures of option price sensitivities to key option parameters that may lead to more effective hedging and speculative strategies using currency options. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:265–291, 2000     

Empirical tests of canonical nonparametric American option‐pricing methods

Alcock and Carmichael (2008, The Journal of Futures Markets, 28, 717–748) introduce a nonparametric method for pricing American‐style options, that is derived from the canonical valuation developed by Stutzer (1996, The Journal of Finance, 51, 1633–1652). Although the statistical properties of this nonparametric pricing methodology have been studied in a controlled simulation environment, no study has yet examined the empirical validity of this method. We introduce an extension to this method that incorporates information contained in a small number of observed option prices. We explore the applicability of both the original method and our extension using a large sample of OEX American index options traded on the S&P100 index. Although the Alcock and Carmichael method fails to outperform a traditional implied‐volatility‐based Black–Scholes valuation or a binomial tree approach, our extension generates significantly lower pricing errors and performs comparably well to the implied‐volatility Black–Scholes pricing, in particular for out‐of‐the‐money American put options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:509–532, 2010     

Pricing of moving‐average‐type options with applications

Moving‐average‐type options are complex path‐dependent derivatives whose payoff depends on the moving average of stock prices. This article concentrates on two such options traded in practice: the moving‐average‐lookback option and the moving‐average‐reset option. Both options were issued in Taiwan in 1999, for example. The moving‐average‐lookback option is an option struck at the minimum moving average of the underlying asset's prices. This article presents efficient algorithms for pricing geometric and arithmetic moving‐average‐lookback options. Monte Carlo simulation confirmed that our algorithms converge quickly to the option value. The price difference between geometric averaging and arithmetic averaging is small. Because it takes much less time to price the geometric‐moving‐average version, it serves as a practical approximation to the arithmetic moving‐average version. When applied to the moving‐average‐lookback options traded on Taiwan's stock exchange, our algorithm gave almost the exact issue prices. The numerical delta and gamma of the options revealed subtle behavior and had implications for hedging. The moving‐average‐reset option was struck at a series of decreasing contract‐specified prices on the basis of moving averages. Similar results were obtained for such options with the same methodology. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:415–440, 2003     

Trinomial or binomial: Accelerating American put option price on trees

We investigate the pricing performance of eight trinomial trees and one binomial tree, which was found to be most effective in an earlier study, under 20 different implementation methodologies for pricing American put options. We conclude that the binomial tree, the Tian third‐order moment‐matching tree with truncation, Richardson extrapolation, and smoothing, performs better than the trinomial trees. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:826–839, 2009     

Stock option taxation: a missing piece in European innovation policy?

Europe continues to lag behind the USA in venture capital (VC) activity and in the creation of successful startups, and has recently been surpassed by China. This is despite the fact that many European countries have deep financial markets, strong legal institutions, and high R&D spending. We point to the tax treatment of employee stock options as an explanation for the stronger growth of the US VC sector. As a response to high uncertainty and transaction costs, VC financiers have developed a model in which founders and key recruitments are compensated with stock options under complex contracts. Low tax rates on employee stock options further raise the relative returns of working and investing in innovative entrepreneurial firms, and shift financial capital and talent to that sector. We measure the effective tax on stock options in VC-backed entrepreneurial firms in a number of developed economies. Countries with lower stock option taxation have higher VC activity and more high-growth expectation entrepreneurial activity. Based on these associations and the theoretical and empirical literature, we argue that more lenient taxation of gains on employee stock options can be a strategy for European countries to catch up in entrepreneurial finance. This tax policy would narrowly target entrepreneurial startups without requiring broad tax cuts. The favorable tax treatment of stock options allows the state to promote firms that rely on entrepreneurial finance and make use of these types of contracts without lowering taxes for other sectors of the economy.     

Lower‐boundary violations and market efficiency: Evidence from the German DAX‐index options market

The informational efficiency of the market for options on the German stock index DAX is examined using intraday transactions data. Problems of previous studies on options‐market efficiency, arising from dividend estimation and the early‐exercise effect, are avoided, because the DAX is a performance index and DAX options are European options. Ex‐post and ex‐ante tests are carried out to simulate trading strategies that exploit irrational lower‐boundary violations of observed option prices. Because the lower‐boundary conditions are solely based on arbitrage considerations, the test results do not depend on the assumption that investors use a particular option‐pricing model. The investigation shows that ex‐post profits are, in general, dramatically reduced when the execution of arbitrage strategies is delayed and/or transaction costs are accounted for. However, arbitrage restrictions, which rely on short selling of the component stocks of the index, tend to be violated more often and with higher persistence. An analysis of consecutive subsamples suggests that, over time, traders have been subjected to a learning process when pricing this relatively new instrument. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20: 405–424, 2000     

Pricing American options on foreign currency with stochastic volatility,jumps, and stochastic interest rates

By applying the Heath–Jarrow–Morton (HJM) framework, an analytical approximation for pricing American options on foreign currency under stochastic volatility and double jump is derived. This approximation is also applied to other existing models for the purpose of comparison. There is evidence that such types of jumps can have a critical impact on earlyexercise premiums that will be significant for deep out‐of‐the‐money options with short maturities. Moreover, the importance of the term structure of interest rates to early‐exercise premiums is demonstrated as is the sensitivity of these premiums to correlation‐related parameters. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:867–891, 2007