Futures-Style Options on Euro-deposit Futures: Nihil sub Sole Novi?

Euro-deposit futures play a relevant role among the derivative products traded in official markets. As opposed to most futures contracts, the underlying instrument is not represented by a traded asset but by a linear transformation of an interest rate, the Libor. The options written on Euro-deposit futures that are traded at the London International Financial Futures & Options Exchange (LIFFE) are subject to daily marking to market, as the underlying futures; thus, they are called futures-style options or pure futures options. These options are often priced with the Black (1976) formula, whose use entails several shortcomings. A more realistic alternative is represented by the univariate Cox, Ingersoll and Ross (1985) model. The closed-form solutions for the prices of Euro-deposit futures and futures-style options on Euro-deposit futures obtained in the CIR model are two major original contributions presented in this paper. Other original contributions involve the determination of the relation between futures rates and forward rates and the derivation of the equivalent portfolio for the hedging of futures-style options on Euro-deposit futures.     

Commodity derivatives valuation with autoregressive and moving average components in the price dynamics

In this paper, we develop a continuous time factor model of commodity prices that allows for higher-order autoregressive and moving average components. We document the need for these components by analyzing the convenience yield’s time series dynamics. The model we propose is analytically tractable and allows us to derive closed-form pricing formulas for futures and options. Empirically, we estimate a parsimonious version of the general model for the crude oil futures market and demonstrate the model’s superior performance in pricing nearby futures contracts in- and out-of-sample. Most notably, the model substantially improves the pricing of long-horizon contracts with information from the short end of the futures curve.     

Pricing U.S. Dollar Index Futures Options: An Empirical Investigation

This paper develops a pricing model and empirically tests the pricing efficiency of options on the U.S. Dollar Index (USDX) futures contract. Empirical tests of the model indicate that the market consistently overprices these options relative to the derived model. This overpricing is more pronounced for out‐of‐the‐money options than for in‐the‐money options and more pronounced for put options than for call options. To validate the above results, delta neutral portfolios are created for one‐ and two‐day holding periods and consistently generate positive arbitrage profits, indicating that on average the market overprices the options on the USDX futures contracts.     

An Empirical Investigation of the Market for Comex Gold Futures Options

Option-pricing models that assume a constant interest rate may misprice futures options if the interest rate fluctuates significantly or if the price of the underlying asset is correlated with the interest rate. The futures option-pricing model of Ramaswamy and Sundaresan allows for a stochastic interest rate and correlation of the underlying asset's price with the interest rate. Using a data set of daily closing prices for Comex gold futures options, this paper tests the Ramaswamy and Sundaresan model against a constant interest rate model. Results indicate that the stochastic interest rate model is a superior predictor of market prices.     

Futures and futures options with basis risk: theoretical and empirical perspectives

Under a no-arbitrage assumption, the futures price converges to the spot price at the maturity of the futures contract, where the basis equals zero. Assuming that the basis process follows a modified Brownian bridge process with a zero basis at maturity, we derive the closed-form solutions of futures and futures options with the basis risk under the stochastic interest rate. We make a comparison of the Black model under a stochastic interest rate and our model in an empirical test using the daily data of S&P 500 futures call options. The overall mean errors in terms of index points and percentage are ?4.771 and ?27.83%, respectively, for the Black model and 0.757 and 1.30%, respectively, for our model. This evidence supports the occurrence of basis risk in S&P 500 futures call options.     

PRICING CURRENCY FUTURES OPTIONS WITH LOGNORMALLY DISTRIBUTED JUMPS

We find that a mixed diffusion-jump process fits most daily currency futures price series better than a mixture of normal densities and, especially, an asymmetric stable Paretian model. We also find that Merton's (1976) mixed diffusion-jump option pricing model outperforms Black's (1 976) model for valuing currency futures options. Our results suggest that researchers should begin to consider the possibility of jump processes as time-independent models of other futures price series.     

A test of the beta model on Eurodollar futures options

The paper reports empirical tests of the beta model for pricing fixed-income options. The beta model resembles the Black–Scholes model with the lognormal probability distribution replaced by a beta probability distribution. The test is based on 32 817 daily prices of Eurodollar futures options and concludes that the beta model is more accurate than alternative option pricing models.     

Calibrating a market model with stochastic volatility to commodity and interest rate risk

Based on the multi-currency LIBOR Market Model, this paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function allowing the model to simultaneously fit the implied volatility surfaces of commodity and interest rate options. Since liquid market prices are only available for options on commodity futures, rather than forwards, a convexity correction formula for the model is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given correlation structure between forward interest rates and commodity prices (cross-correlation). When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), while in the case of futures a (rapidly converging) iterative fitting procedure is presented. The fitting of cross-correlation is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the ‘orthonormal Procrustes’ problem in linear algebra. The calibration approach is demonstrated in an application to market data for oil futures.     

Pricing a class of exotic commodity options in a multi-factor jump-diffusion model

In a recent paper, Crosby introduced a multi-factor jump-diffusion model which would allow futures (or forward) commodity prices to be modelled in a way which captured empirically observed features of the commodity and commodity options markets. However, the model focused on modelling a single individual underlying commodity. In this paper, we investigate an extension of this model which would allow the prices of multiple commodities to be modelled simultaneously in a simple but realistic fashion. We then price a class of simple exotic options whose payoff depends on the difference (or ratio) between the prices of two different commodities (for example, spread options), or between the prices of two different (i.e. with different tenors) futures contracts on the same underlying commodity, or between the prices of a single futures contract as observed at two different calendar times (for example, forward start or cliquet options). We show that it is possible, using a Fourier transform-based algorithm, to derive a single unifying form for the prices of all these aforementioned exotic options and some of their generalizations. Although we focus on pricing options within the model of Crosby, most of our results would be applicable to other models where the relevant ‘extended’ characteristic function is available in analytical form.     

Futures Options and the Volatility of Futures Prices

Assuming nonstochastic interest rates, European futures options are shown to be European options written on a particular asset referred to as a futures bond. Consequently, standard option pricing results may be invoked and standard option pricing techniques may be employed in the case of European futures options. Additional arbitrage restrictions on American futures options are derived. The efficiency of a number of futures option markets is examined. Assuming that at-the-money American futures options are priced accurately by Black's European futures option pricing model, the relationship between market participants' ex ante assessment of futures price volatility and the term to maturity of the underlying futures contract is also investigated empirically.     

Analytical bounds for Treasury bond futures prices

The pricing of delivery options, particularly timing options, in Treasury bond futures is prohibitively expensive. Recursive use of the lattice model is unavoidable for valuing such options, as Boyle in J Finance 14(1):101?C113, (1989) demonstrates. As a result, the main purpose of this study is to derive upper bounds and lower bounds for Treasury bond futures prices. This study first shows that the popular preference-free, closed form cost of carry model is an upper bound for the Treasury bond futures price. Then, the next step is to derive analytical lower bounds for the futures price under one and two-factor Cox-Ingersoll-Ross models of the term structure. The bound under the two-factor Cox-Ingersoll-Ross model is then tested empirically using weekly futures prices from January 1987 to December 2000.     

ESTIMATING THE VALUE OF DELIVERY OPTIONS IN FUTURES CONTRACTS

We analyze the effect various delivery options embedded in commodity futures contracts have on the futures price. The two embedded options considered are the timing and location options. We show that early delivery is always optimal when only a timing option is present, but not so when joint options are present. The estimates of the combined options are much smaller than the comparable estimates for the timing option alone. The average value of the joint option is about 5% of the average basis on the first day of the maturity month. This suggests that joint options can increase deliverable supplies while potentially having only a small effect on basis behavior.     

An Intraweek Seasonality in the Implied Volatilities of Individual and Index Options

This paper studies intraweek seasonalities in the implied volatilities of options on stock market indices. Oneway analysis of variance isolates the daily behavior of implied volatilities. The differential between call implied volatility and put implied volatility tends to drop on Friday and rise on Monday. Relying on a synthetic futures contract created from options, an explanatory model is proposed. The model complements previous research on the difference between the intraweek behavior of stock market indices and that of derivative instruments based on the indices.     

Noise, equity prices, and hedging: A new approach

The existence of noise trading in equity markets has possible economic implications for arbitrage, and asset pricing. In terms of pricing, noise trading can lead to excess volatility which has been shown to influence the value of options and futures. Furthermore, option research shows that modeling volatility leads to improved hedging performance. To this end, we derive a general hedging model for equity index futures in the presence of noise trading. Our analysis shows how the level and dynamics of noise trading should influence a hedger's behavior. Finally, we empirically test our model using the NASDAQ-100 index futures and FTSE 100 index futures over the period of January 1998 to May 2003.     

Interest Rate Futures Options and Interest Rate Options

This paper derives pricing models of interest rate options and interest rate futures options. The models utilize the arbitrage-free interest rate movements model of Ho and Lee. In their model, they take the initial term structure as given, and for the subsequent periods, they only require that the bond prices move relative to each other in an arbitrage-free manner. Viewing the interest rate options as contingent claims to the underlying bonds, we derive the closed-form solutions to the options. Since these models are sufficiently simple, they can be used to investigate empirically the pricing of bond options. We also empirically examine the pricing of Eurodollar futures options. The results show that the model has significant explanatory power and, on average, has smaller estimation errors than Black's model. The results suggest that the model can be used to price options relative to each other, even though they may have different expiration dates and strike prices.     

The Pricing of Futures and Options Contracts on the Value Line Index

This paper considers the problems peculiar to the Value Line Index, because of its use of geometric averaging, as regards the pricing of options and futures on that index. The Value Line Composite Index (VLCI) is an equally weighted geometric average index of nearly 1700 stocks. The VLCI futures market has existed since 1982 while the VLCI options market was established in 1985. This paper provides valuation formulas and analyzes the economic properties of these contracts. Because of the geometric averaging in the VLCI, its contingent claims have special properties. For example, the futures price may fall short of the spot price and the value of a VLCI call option may decline when the volatility of the index is increased. VLCI futures are shown to provide a direct means for duplicating an equally weighted portfolio of the underlying stocks.     

A four-factor stochastic volatility model of commodity prices

The number of factors driving the uncertain dynamics of commodity prices has been a central consideration in financial literature. While the majority of empirical studies relies on the assumption that up to three factors are sufficient to explain all relevant uncertainty inherent in commodity spot, futures, and option prices, evidence from Trolle and Schwartz (Rev Financ Stud 22(11):4423–4461, 2009b) and Hughen (J Futures Mark 30(2):101–133, 2010) indicates a need for additional risk factors. In this article, we propose a four-factor maximal affine stochastic volatility model that allows for three independent sources of risk in the futures term structure and an additional, potentially unspanned stochastic volatility process. The model principally integrates the insights from Hughen (2010) and Tang (Quant Finance 12(5):781–790, 2012) and nests many well-known models in the literature. It can account for several stylized facts associated with commodity dynamics such as mean reversion to a stochastic level, stochastic volatility in the convenience yield, a time-varying correlation structure, and time-varying risk-premia. In-sample and out-of-sample tests indicate a superior model fit to futures and options data as well as lower hedging errors compared to three-factor benchmark models. The results also indicate that three factors are not sufficient to model the joint dynamics of futures and option prices accurately.     

Scenarios on European energy security: Outcomes of natural gas strategy in 2020

This paper adopts a futuristic methodology and analyzes the role of natural gas in European energy security in order to transform economic and policy uncertainties into meaningful scenarios. It implements “trend analysis” to forecast the volume of gas needed until 2020 by elaborating the estimates of the EU Commission and “scenario building” to come up with alternative futures forging different regional implications. The economic analysis stems from four scenarios as introduced by the EU Commission: (1) baseline scenario with average oil price of 61$/bbl; (2) baseline scenario with average oil price of 100$/bbl; (3) New Energy Policy scenario with average oil price of 61$/bbl; (4) New Energy Policy scenario with average oil price of 100$/bbl. The policy analysis is derived from the options, restraints, priorities and strategies of the concerned actors which include the EU Commission, selected EU members, suppliers and transit countries. The analysis on actors results in four policy scenarios: (1) Russia first; (2) Russia everywhere; (3) security first; (4) each for itself. The matrix, which excludes the possibility of unprecedented developments such as a drastic increase in European shale gas production or continuing global recession, clusters 16 contingencies. The paper, within this context, gives an idea on how alternative policy options of European energy security may lead to different futures based on oil prices, environmental commitments and strategic initiatives of the concerned actors.     

On the realized volatility of the ECX CO2 emissions 2008 futures contract: distribution, dynamics and forecasting

This article documents the conditional and unconditional distributions of the realized volatility for the 2008 futures contract in the European climate exchange (ECX), which is valid under the EU emissions trading scheme (EU ETS). Realized volatility measures from naive, kernel-based and subsampling estimators are used to obtain inferences about the distributional and dynamic properties of the ECX emissions futures volatility. The distribution of the daily realized volatility in logarithmic form is shown to be close to normal. The mixture-of-normals hypothesis is strongly rejected, as the returns standardized using daily measures of volatility clearly departs from normality. A simplified HAR-RV model (Corsi in J Financ Econ 7:174–196, 2009) with only a weekly component, which reproduces long memory properties of the series, is then used to model the volatility dynamics. Finally, the predictive accuracy of the HAR-RV model is tested against GARCH specifications using one-step-ahead forecasts, which confirms the HAR-RV superior ability.     

The Valuation of Options on Futures Contracts

Rational restrictions are derived for the values of American options on futures contracts. For these options, the optimal policy, in general, involves premature exercise. A model is developed for valuing options on futures contracts in a constant interest rate setting. Despite the fact that premature exercise may be optimal, the value of this American feature appears to be small and a European formula due to Black serves as a useful approximation. Finally, a model is developed to value these options in a world with stochastic interest rates. It is shown that the pricing errors caused by ignoring the location of the interest rate (relative to its long-run mean) range from ?5% to 7%, when the current rate is ±200 basis points from its long-run value. The role of interest rate expectations is, therefore, crucial to the valuation. Optimal exercise policies are found from numerical methods for both models.