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 NONZERO‐SUM GAME APPROACH TO CONVERTIBLE BONDS: TAX BENEFIT,BANKRUPTCY COST,AND EARLY/LATE CALLS

Convertible bonds are hybrid securities that embody the characteristics of both straight bonds and equities. The conflicts of interest between bondholders and shareholders affect the security prices significantly. In this paper, we investigate how to use a nonzero‐sum game framework to model the interaction between bondholders and shareholders and to evaluate the bond accordingly. Mathematically, this problem can be reduced to a system of variational inequalities and we explicitly derive the Nash equilibrium to the game. Our model shows that credit risk and tax benefit have considerable impacts on the optimal strategies of both parties. The shareholder may issue a call when the debt is in‐the‐money or out‐of‐the‐money. This is consistent with the empirical findings of “late and early calls.” In addition, the optimal call policy under our model offers an explanation for certain stylized patterns related to the returns of company assets and stocks on call.     

STOCHASTIC VOLATILITY MODELS AND THE PRICING OF VIX OPTIONS

In this paper, we examine and compare the performance of a variety of continuous‐time volatility models in their ability to capture the behavior of the VIX. The “3/2‐ model” with a diffusion structure which allows the volatility of volatility changes to be highly sensitive to the actual level of volatility is found to outperform all other popular models tested. Analytic solutions for option prices on the VIX under the 3/2‐model are developed and then used to calibrate at‐the‐money market option prices.     

Effects of rollover strategies and information stability on the performance measures in options markets: An examination of the KOSPI 200 index options market

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 two rollover strategies (nearest‐to‐next strategy and next‐to‐next) used in the AHBS model to investigate their effect on pricing errors. We suggest a new rollover strategy, next‐to‐next strategy, and demonstrate that our rollover strategy produces more consistent estimates between in‐sample market and model option prices. Probably even more important is that our new rollover strategy makes more accurate out‐of‐sample forecasts for 1‐day or 1‐week ahead prices. Prior literature has documented some anomalies associated with the use of AHBS model, for example, an overfitting problem. A secondary contribution is that our new rollover strategy does not suffer from this overfitting critique. Third, this study uses the mean square error for out‐of‐sample pricing and price changes to determine how the options investors are influenced by moneyness. The results indicate that underpricing (or overpricing) by the AHBS model for the near‐the‐money category is more likely to be maintained for the next several trading days but that such a phenomenon is disappeared for the deep out‐of‐the‐money category. Finally, we suggest the ratio of the number of option contracts to differences in strike prices available for trading between the current day and the previous day(s) as a good categorizing factor for options, such as moneyness. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

AN EQUILIBRIUM GUIDE TO DESIGNING AFFINE PRICING MODELS

The paper examines equilibrium models based on Epstein–Zin preferences in a framework in which exogenous state variables follow affine jump diffusion processes. A main insight is that the equilibrium asset prices can be computed using a standard machinery of affine asset pricing theory by imposing parametric restrictions on market prices of risk, determined inside the model by preference and model parameters. An appealing characteristic of the general equilibrium setup is that the state variables have an intuitive and testable interpretation as driving the consumption and dividend dynamics. We present a detailed example where large shocks (jumps) in consumption volatility translate into negative jumps in equilibrium prices of the assets as agents demand a higher premium to compensate for higher risks. This endogenous “leverage effect,” which is purely an equilibrium outcome in the economy, leads to significant premiums for out‐of‐the‐money put options. Our model is thus able to produce an equilibrium “volatility smirk,” which realistically mimics that observed for index options.     

Size matters for OTC market makers: General results and dimensionality reduction techniques

In most over‐the‐counter (OTC) markets, a small number of market makers provide liquidity to other market participants. More precisely, for a list of assets, they set prices at which they agree to buy and sell. Market makers face therefore an interesting optimization problem: they need to choose bid and ask prices for making money while mitigating the risk associated with holding inventory in a volatile market. Many market‐making models have been proposed in the academic literature, most of them dealing with single‐asset market making whereas market makers are usually in charge of a long list of assets. The rare models tackling multiasset market making suffer however from the curse of dimensionality when it comes to the numerical approximation of the optimal quotes. The goal of this paper is to propose a dimensionality reduction technique to address multiasset market making by using a factor model. Moreover, we generalize existing market‐making models by the addition of an important feature: the existence of different transaction sizes and the possibility for the market makers in OTC markets to answer different prices to requests with different sizes.     

房地产价格与货币供应量的波动溢出效应

基于VAR模型和GARCH模型对中国房地产价格与货币供应量的波动溢出效应进行实证研究,发现货币供应量与房地产价格之间不存在显著的动态相关性,也不存在明显的波动溢出效应,中央银行没有必要用货币政策去直接调控房地产价格。     

A cointegrated commodity pricing model

We propose a commodity pricing model that extends the Gibson–Schwartz two‐factor model to incorporate the effect of linear relations among commodity spot prices, and provide a condition under which such linear relations represent cointegration. We derive futures and call option prices for the proposed model, and indicate that, unlike in Duan and Pliska (2004), the linear relations among commodity prices should affect commodity derivative prices, even when the volatilities of commodity returns are constant. Using crude oil and heating oil market data, we estimate the model and apply the results to the hedging of long‐term futures using short‐term ones.     

RE‐EXAMINING THE IMPLICATIONS OF THE NEW CONSENSUS: ENDOGENOUS MONEY AND TAYLOR RULES IN A SIMPLE NEOCLASSICAL MACRO MODEL

This paper re‐examines the impact of endogenous money in a neoclassical model with interest‐sensitive expenditures. It first outlines a benchmark model with exogenous money and the usual full employment and money growth‐determined inflation results. It then replaces exogenous money with endogenous money, which is shown to generate model indeterminacy. Two methods of resolving this indeterminacy are then explored: money illusion and a Taylor rule for monetary policy, a key feature of new consensus models. The paper concludes that endogenous money has negative implications for the behaviour and interpretation of neoclassical and new consensus models.     

A Study of Arbitrage Efficiency Between the FTSE‐100 Index Futures and Options Contracts

Despite the importance of the London markets and the significance of the relationship for market makers, little published research is available on arbitrage between the FTSE‐100 Index futures and the FTSE‐100 European index options contracts. This study uses the put–call–futures parity condition to throw light on the relationship between options and futures written against the FTSE Index. The arbitrage methodology adopted in this study avoids many of the problems that have affected prior research on the relationship between options or futures prices and the underlying index. The problems that arise from nonsynchroneity between options and futures prices are reduced by the matching of options and futures prices within narrow time intervals with time‐stamped transaction data. This study allows for realistic trading and market‐impact costs. The feasibility of strategies such as execute‐and‐hold and early unwinding is examined with both ex‐post and ex‐ante simulation tests that take into consideration possible execution time lags for the arbitrage trade. This study reveals that the occurrence of matched put–call–futures trios exhibits a U‐shaped intraday pattern with a concentration at both open and close, although the magnitude of observed mispricings has no discernible intraday pattern. Ex‐post arbitrage profits for traders facing transaction costs are concentrated in at‐the‐money options. As in other major markets, despite important microstructure differences, opportunities are generally rapidly extinguished in less than 3 min. The results suggest that arbitrage opportunities for traders facing transaction costs are small in number and confirm the efficiency of trading on the London International Financial Futures and Options Exchange. © 2002 John Wiley & Sons, Inc. Jrl Fut Mark 22:31–58, 2002     

Semi‐efficient valuations and put‐call parity

We propose an approach to the valuation of payoffs in general semimartingale models of financial markets where prices are nonnegative. Each asset price can hit 0; we only exclude that this ever happens simultaneously for all assets. We start from two simple, economically motivated axioms, namely, absence of arbitrage (in the sense of NUPBR) and absence of relative arbitrage among all buy‐and‐hold strategies (called static efficiency). A valuation process for a payoff is then called semi‐efficient consistent if the financial market enlarged by that process still satisfies this combination of properties. It turns out that this approach lies in the middle between the extremes of valuing by risk‐neutral expectation and valuing by absence of arbitrage alone. We show that this always yields put‐call parity, although put and call values themselves can be nonunique, even for complete markets. We provide general formulas for put and call values in complete markets and show that these are symmetric and that both contain three terms in general. We also show that our approach recovers all the put‐call parity respecting valuation formulas in the classic theory as special cases, and we explain when and how the different terms in the put and call valuation formulas disappear or simplify. Along the way, we also define and characterize completeness for general semimartingale financial markets and connect this to the classic theory.     

Variance risk premiums and predictive power of alternative forward variances in the corn market

We propose a fear index for corn using the variance swap rate synthesized from out‐of‐the‐money call and put options as a measure of implied variance. We find negative and time‐varying variance risk premiums (realized variance minus implied variance) in the corn market from 1987 to 2009. Our results contrast with Egelkraut, Garcia, and Sherrick (2007), but are in line with the findings of Simon (2002). We conclude that our synthesized model‐free implied variance estimation procedure contains superior information about future realized variance relative to traditional model‐dependent estimating procedures: the implied variance model by Black (1976) and the seasonal GARCH(1, 1) forecasted variance model. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 32:587–608, 2012     

Is there information in the volatility skew?

Since the 1987 crash, option prices have exhibited a strong negative skew, implying higher implied volatility for out‐of‐the‐money puts than at‐ and in‐the‐money puts. This has resulted in incorporating multiple jumps and stochastic volatility within the data generating process to improve the Black–Scholes model in an attempt to capture negative skewness and a highly leptokurtic distribution. The general conclusion is that there is a large jump premium in the short term, which best explains the significant negative skew for short maturity options. Alternative explanations for the negative skew are related to market liquidity driven by demand shocks and supply shortages. Regardless of the explanation for the negative skew, we assess the information content in the shape of the skew to infer if the option market can accurately forecast stock market crashes and/or spikes upward. We demonstrate, using all options on the S&P 100 from 1984–2006, that the shape of the skew can reveal with significant probability when the market will crash or spike. However, we find the magnitude of the spike prediction is not economically significant. Our findings are strongest for the short‐term out‐of‐the money puts, consistent with the notion of investors' aversion to large negative movements. We also find that the power of the crash/spike prediction decreases with an increase in the time to option maturity. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:921–959, 2007     

Option Pricing With V. G. Martingale Components1

European call options are priced when the uncertainty driving the stock price follows the V. G. stochastic process (Madan and Seneta 1990). the incomplete markets equilibrium change of measure is approximated and identified using the log return mean, variance, and kurtosis. an exact equilibrium interpretation is also provided, allowing inference about relative risk aversion coefficients from option prices. Relative to Black-Scholes, V. G. option values are higher, particularly so for out of the money options with long maturity on stocks with high means, low variances, and high kurtosis.     

International evidence on alternative models of the term structure of volatilities

The term structure of instantaneous volatilities (TSV) of forward rates for different monetary areas (euro, U.S. dollar and British pound) is examined using daily data from at‐the‐money cap markets. During the sample period (two and a half years), the TSV experienced severe changes both in level and shape. Two new functional forms of the instantaneous volatility of forward rates are proposed and tested within the LIBOR Market Model framework. Two other alternatives are calibrated and used as benchmarks to test the accuracy of the new models. The two new models provide more flexibility to adequately calibrate the observed cap prices, although this improved accuracy in replicating cap prices produces some instability in parameter estimates. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:653–683, 2009     

Intraday price formation and bid–ask spread components: A new approach using a cross‐market model

This study examines the intraday formation process of transaction prices and bid–ask spreads in the KOSPI 200 futures market. By extending the structural model of Madhavan, A., Richardson, M., and Roomans, M. ( 1997 ), we develop a unique cross‐market model that can decompose spread components and explain intraday price formation for the futures market by using the order flow information from the KOSPI 200 options market, which is a market that is closely related to the futures market as well as considered to be one of the most remarkable options markets in the world. The empirical results indicate that the model‐implied spread and the permanent component of the spread that results from informed trading tend to be underestimated without the inclusion of options market information. Further, the results imply that trades of in‐the‐money options, which have high delta values, generally incur a more adverse information cost component (the permanent spread component) of the futures market than those of out‐of‐the‐money options, which have relatively low delta values. Finally, we find that the adverse information cost component that is estimated from the cross‐market model exhibits a nearly U‐shaped intraday pattern; however, it sharply decreases at the end of the trading day. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

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     

A maximal affine stochastic volatility model of oil prices

This study develops and estimates a stochastic volatility model of commodity prices that nests many of the previous models in the literature. The model is an affine three‐factor model with one state variable driving the volatility and is maximal among all such models that are also identifiable. The model leads to quasi‐analytical formulas for futures and options prices. It allows for time‐varying correlation structures between the spot price and convenience yield, the spot price and its volatility, and the volatility and convenience yield. It allows for expected mean‐reversion in the short term and for an increasing expected long‐term price, and for time‐varying risk premia. Furthermore, the model allows for the situation in which options' prices depend on risk not fully spanned by futures prices. These properties are desirable and empirically important for modeling many commodities, especially crude oil. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:101–133, 2010     

The predictive failure of the Baba, Hendry and Starr model of M1

restored a stable specification for M1 demand using an error-correction model which allowed for learning about new assets and incorporated a volatility term in long-term interest rates. Our study replicated their in-sample results, but found that their model completely breaks down over longer sample periods. We argue that this predictive failure could have been anticipated by sensitivity analysis. Their specification appears to have underestimated the interest rate elasticity of money demand because of the learning-adjustment mechanism. Our results also call into question their basic use of volatility in narrow money demand models.     

Black‐Scholes‐Merton revisited under stochastic dividend yields

European options are priced in a framework à la Black‐Scholes‐Merton, which is extended to incorporate stochastic dividend yield under a stochastic mean–reverting market price of risk. Explicit formulas are obtained for call and put prices and their Greek parameters. Some well‐known properties of the Black‐Scholes‐Merton formula fail to hold in this setting. For example, the delta of the call can be negative and even greater than one in absolute terms. Moreover, call prices can be a decreasing function of the underlying volatility although the latter is constant. Finally, and most importantly, option prices highly depend on the features of the market price of risk, which does not need to be specified at all in the standard Black‐Scholes‐Merton setting. The results are simulated in order to assess the economic impact of assuming that the dividend yield is deterministic when it is actually stochastic, as well as to assess the economic importance of the features of the market price of risk. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:703–732, 2006