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     

Dividend‐Rollover Effect and the Ad Hoc Black‐Scholes Model

One of the most widely used option‐valuation models among practitioners is the ad hoc Black‐Scholes (AHBS) model. The main contribution of this study is methodological. We carefully consider three dividend strategies (No dividend, Implied‐forward dividend, and Actual dividend) for the AHBS model to investigate their effect on pricing errors. We suggest a new dividend strategy, implied‐forward dividend, which incorporates expectational information on dividends embedded in option prices. We demonstrate that our implied‐forward dividend strategy produces more consistent estimates between in‐sample market and model option prices. More importantly our new implied‐forward dividend strategy makes more accurate out‐of‐sample forecasts for one‐day or one‐week ahead prices. Second, we document that both a “Return‐volatility” Smile and a “Return‐pricing Error” Smile exist. From these return characteristics, we make two conclusions: (1) the return dependency of implied volatility is an important explanatory variable and should be controlled to reduce the pricing error of an AHBS model, and (2) it is important for the hedging horizon to be based on return size, that is, the larger the contemporaneous return, the more frequent an option issuer must rebalance the option's hedge. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 32:742‐772, 2012     

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     

Pricing Eurodollar futures options using the BDT term structure model: The effect of yield curve smoothing

This article focuses on pricing Eurodollar futures options using the single‐factor Black, Derman, and Toy (1990) term structure model with particular emphasis on yield curve smoothing. Of the various approaches, the maximum smoothness forward rate approach developed by Adams and van Deventer (1994), cubic yield spline, and linear interpolation are used to produce finely spaced binomial trees. We compare the pricing accuracy associated with the use of yield curve smoothing techniques within the BDT framework. The findings provide the first supporting evidence that using a forward rate curve with maximum smoothness together with a time‐varying volatility structure improves best the performance of the BDT model. The empirical results are found to be robust across factors affecting the option price such as time‐to‐expiration, moneyness, and trading volume. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:293–306, 2000     

Valuing real options using implied binomial trees and commodity futures options

A real option on a commodity is valued using an implied binomial tree (IBT) calibrated using commodity futures options prices. Estimating an IBT in the absence of spot options (the norm for commodities) allows real option models to be calibrated for the first time to market‐implied probability distributions for commodity prices. In addition, the existence of long‐dated futures options means that good volatility estimates may now be incorporated into capital budgeting evaluations of real options projects with long planning horizons. An example is given using gold futures options and a real option to extract gold from a mine. A detailed out‐of‐sample test is included that shows how IBT option pricing errors evolve on subtrees emanating from future levels of the underlying asset. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:203–226, 2007     

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 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     

Show me the money: Option moneyness concentration and future stock returns

Informed traders often use options that are not in-the-money due to higher potential gains for a smaller upfront cost. Thus, trading activity by option moneyness should be a gauge of informed option trading. We construct a dollar volume-weighted average moneyness measure to capture option trading activity at different moneyness levels. Stock returns increase with this measure, suggesting more trading activity in options with higher leverage predicts future stock returns. Our results hold cross-sectionally and at the portfolio level yielding a Fama–French five-factor α of 12% per year for all stocks and 33% per year for high implied volatility stocks.     

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     

Hedging under the influence of transaction costs: An empirical investigation on FTSE 100 index options

The Black–Scholes (BS; F. Black & M. Scholes, 1973) option pricing model, and modern parametric option pricing models in general, assume that a single unique price for the underlying instrument exists, and that it is the mid‐ (the average of the ask and the bid) price. In this article the authors consider the Financial Times and London Stock Exchange (FTSE) 100 Index Options for the time period 1992–1997. They estimate the ask and bid prices for the index, and show that, when substituted for the mid‐price in the BS formula, they provide superior option price predictors, for call and put options, respectively. This result is reinforced further when they .t a non‐parametric neural network model to market prices of liquid options. The empirical .ndings in this article suggest that the ask and bid prices of the underlying asset provide a superior fit to the mid/closing price because they include market maker's, compensation for providing liquidity in the market for constituent stocks of the FTSE 100 index. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:471–494, 2007     

Analytic approximation formulae for pricing forward‐starting Asian options

In this article we first identify a missing term in the Bouaziz, Briys, and Crouhy ( 1994 ) pricing formula for forward‐starting Asian options and derive the correct one. First, illustrate in certain cases that the missing term in their pricing formula could induce large pricing errors or unreasonable option prices. Second, we derive new analytic approximation formulae for valuing forward‐starting Asian options by adding the second‐order term in the Taylor series. We show that our formulae can accurately value forward‐starting Asian options with a large underlying asset's volatility or a longer time window for the average of the underlying asset prices, whereas the pricing errors for these options with the previously mentioned formula could be large. Third, we derive the hedge ratios for these options and compare their properties with those of plain vanilla options. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:487–516, 2003     

Pricing options using implied trees: Evidence from FTSE‐100 options

Previously, few, if any, comparative tests of performance of Jackwerth's ( 1997 ) generalized binomial tree (GBT) and Derman and Kani ( 1994 ) implied volatility tree (IVT) models were done. In this paper, we propose five different weight functions in GBT and test them empirically compared to both the Black‐Scholes model and IVT. We use the daily settlement prices of FTSE‐100 index options from January to November 1999. With both American and European options traded on the FTSE‐100 index, we construct both GBT and IVT from European options and examine their performance in both the hedging of European option and the pricing of its American counterpart. IVT is found to produce least hedging errors and best results for American call options with earlier maturity than the maturity span of the implied trees. GBT appears to produce better results for American ATM put pricing for any maturity, and better in‐sample fit for options with maturity equal to the maturity span of the implied trees. Deltas calculated from IVT are consistently lower (higher) than Black‐Scholes deltas for both European and American calls (puts) in absolute term. The reverse holds true for GBT deltas. These empirical findings about the relative performance of GBT, IVT, and Standard Black‐Scholes models are important to practitioners as they indicate that different methods should be used for different applications, and some cautions should be exercised. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:601–626, 2002     

A non‐lattice pricing model of American options under stochastic volatility

In this article, an analytical approach to American option pricing under stochastic volatility is provided. Under stochastic volatility, the American option value can be computed as the sum of a corresponding European option price and an early exercise premium. By considering the analytical property of the optimal exercise boundary, the formula allows for recursive computation of the American option value. Simulation results show that a nonlattice method performs better than the lattice‐based interpolation methods. The stochastic volatility model is also empirically tested using S&P 500 futures options intraday transactions data. Incorporating stochastic volatility is shown to improve pricing, hedging, and profitability in actual trading. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:417–448, 2006     

Pricing FTSE 100 index options under stochastic volatility

The autoregressive conditional heteroscedasticity/generalized autoregressive conditional heteroscedasticity (ARCH/GARCH) literature and studies of implied volatility clearly show that volatility changes over time. This article investigates the improvement in the pricing of Financial Times‐Stock Exchange (FTSE) 100 index options when stochastic volatility is taken into account. The major tool for this analysis is Heston’s (1993) stochastic volatility option pricing formula, which allows for systematic volatility risk and arbitrary correlation between underlying returns and volatility. The results reveal significant evidence of stochastic volatility implicit in option prices, suggesting that this phenomenon is essential to improving the performance of the Black–Scholes model (Black & Scholes, 1973) for FTSE 100 index options. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:197–211, 2001     

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     

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     

Box‐spread arbitrage efficiency of Nifty index options: The Indian evidence

This study examines the market efficiency for the European style Nifty index options using the box‐spread strategy. Time‐stamped transactions data are used to identify the mispricing and arbitrage opportunities for options with this modelfree approach. Profit opportunities, after accounting for the transaction costs, are quite frequent, but do not persist even for two minutes. The mispricing is higher for the contracts with higher liquidity (immediacy) risk captured by the moneyness (the difference between the strike prices and the spot price) and the volatility of the underlying. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:544–562, 2009     

Role of delivery options in basis convergence

The corn futures contract, traded on the Chicago Board of Trade, provides sellers with delivery options about the timing of delivery, the location of delivery, and the grade to be delivered. These options presumably have values that can vary from one delivery month to the next. The joint values of the timing and location options are estimated for each delivery month for the years 1989 through 1997. These estimates are then used in regression models to determine the degree to which they influence basis variability on the first day of the maturity month. Econometric models are also developed to see if the estimated implicit options values are useful in improving the forecasts of basis convergence over the 2‐month period prior to maturity. The results suggested that variation in the delivery options values in the corn futures contract does indeed help explain basis variability on the first day of maturity. An option‐value variable, based on estimated values two months prior to maturity, resulted in occasional, small improvements (from a statistical point of view) in the precision of forecasts. The existence of delivery options increases basis variability at maturity, but it is difficult to use this information to improve forecasts of basis convergence. One limitation of the analysis is that the Chicago cash market had few transactions per day during the sample period, and hence the reported spot prices may be inadequate for making high‐quality estimates of the options values. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:783–809, 2002     

Option happiness and liquidity: Is the dynamics of the volatility smirk affected by relative option liquidity?

This study investigates the dynamic relationship between option happiness (the steepness of the volatility smirk) and relative index option liquidity. We find that, on a daily basis, option happiness is significantly dependent on the relative liquidity between option series with different moneyness. In particular, deterioration (improvement) in liquidity of an out‐of‐the‐money put option relative to a concurrent at‐the‐money call option would lead to higher (lower) option happiness. This relationship is robust to relative option liquidity measures based on bid–ask spreads, option price impacts, and option order book imbalances. The results also show a significant maturity effect in option happiness, consistent with the notion that options are “dying smiling.” © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

The economic significance of conditional skewness in index option markets

This study examines whether conditional skewness forecasts of the underlying asset returns can be used to trade profitably in the index options market. The results indicate that a more general skewness‐based option‐pricing model can generate better trading performance for strip and strap trades. The results show that conditional skewness model forecasts, when combined with forward‐looking option implied volatilities, can significantly improve the performance of skewness‐based trades but trading costs considerably weaken the profitability of index option strategies. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:378–406, 2010