The analytic valuation of American options

No analytic solution exists for the valuation of American optionswritten on futures contracts and foreign currencies for whichearly exercise may be optimal. This article formulates the Americanoption valuation problem in economically and mathematicallymeaningful ways. This enables us to derive valuation formulasfor American options. The properties associated with the optimalexercise boundary are examined, and a numerical technique toimplement the valuation formulas is presented.     

Maturity and exercise price of executive stock options

Using a simple three-period model in which a manager can gather information before making an investment decision, this paper studies optimal contracts with various stock options. In particular, we show how the exercise price of executive stock options is related to a base salary, the size of the option grant, leverage, and the riskiness of a desired investment policy. The optimal exercise price increases in the size of grant and the base salary and decreases in leverage and the riskiness of a desired investment policy. Other things equal, the optimal exercise price of European options with a longer maturity should increase more for an increase in the base salary and the size of grant and decrease more for an increase in leverage than the one with a shorter maturity. The optimal exercise price of American options is determined by the optimal exercise prices of European options with different maturities. Given the fixed exercise price, the size of the option grant does not decrease in the face value of debt.     

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.     

Combining the Effects of OLS and Spread on Futures Hedging: Evidence from the Taiwan Stock Index

Abstract:

This study proposes a dynamic hedge ratio, the combined ordinary least squares spread (COLSS), which combines the hedge ratio of ordinary least squares and the value of spread. Using this dynamic ratio for hedging with futures contracts, one can replace spot risk with spread risk. The COLSS captures not only the long-run equilibrium between spot and futures returns, but also the short-run deviation from equilibrium. The spread is forecast by one-period lagged stock market factors and high-order moments that are estimated by an options model. In the in-sample and out-of-sample tests, the COLSS strategy achieves significant risk reduction and outperforms the alternative models by a large utility improvement.     

A MULTIFACTOR MODEL OF THE QUALITY OPTION IN TREASURY FUTURES CONTRACTS

The values of quality options in Treasury futures contracts are set relative to the prices of all coupon bonds in their respective deliverable sets. As a result, any model used to value the quality option should set its price relative to the set of observed bond prices. This requirement rules out the use of most simple equilibrium models that represent all bond prices in terms of a finite number of state variables. We use the two-factor Heath-Jarrow-Morton model, which permits claims to be priced relative to observable bond prices, to investigate the potential value of the quality option in Treasury bond and note futures. We show that the quality option has significantly more value in a two-factor interest rate economy than in a single-factor economy, and that ignoring it could lead to significant mispricing.     

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.     

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.     

New No-arbitrage Conditions and the Term Structure of Interest Rate Futures

Interest rate futures are basic securities and at the same time highly liquid traded objects. Despite this observation, most models of the term structure of interest rate assume forward rates as primary elements. The processes of futures prices are therefore endogenously determined in these models. In addition, in these models hedging strategies are based on forward and/or spot contracts and only to a limited extent on futures contracts. Inspired by the market model approach of forward rates by Miltersen, Sandmann, and Sondermann (J Finance 52(1); 409–430, 1997), the starting point of this paper is a model of futures prices. Using, as the input to the model, the prices of futures on interest related assets new no-arbitrage restrictions on the volatility structure are derived. Moreover, these restrictions turn out to prevent an application of a market model based on futures prices.     

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.     

An equilibrium approach to pricing foreign currency options

The paper presents a modified version of the Garman-Kohlhagen formula for pricing European currency options. The equilibrium approach deviates from the no-arbitrage approach by allowing domestic and foreign interest rates and their dynamics to be determined endogenously in the model. By using the relations between exchange rate dynamics and the dynamics of interest rates, I provide a new characterisation of the relevant volatilities for European currency option pricing, which only depends on parameters describing the variability of the log-exchange rate. The implications of the model for the valuation of American currency options and optimal exercise strategies are examined by applying numerical methods.     

SELECTIVE HEDGING OF BANK ASSETS WITH TREASURY BILL FUTURES CONTRACTS

The recent volatility of interest rates, the associated profit pressures imposed on banks, and the surge in the development of new contracts have stimulated a desire to understand and apply financial futures hedging to banking operations. This paper models interest rate futures contracts in a theory of bank behavior to illustrate the hedging of bank loans as well as government securities. The model predicts the hedge will be greater (1) the greater the expected rise in interest rates and (2) the greater the effect of disintermediation on bank deposits. A simulation of the financial futures trading strategy is reported for banks of various asset sizes using data from the Eleventh Federal Reserve District. Depending on bank risk aversion and interest rate expectations, hedging the bank's total interest rate exposure with T-bill futures reduces the variability of unhedged profits by 80 percent.     

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.     

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.     

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.     

An Equilibrium Model of Catastrophe Insurance Futures and Spreads

This article presents a valuation model of futures contracts and derivatives on such contracts, when the underlying delivery value is an insurance index, which follows a stochastic process containing jumps of random claim sizes at random time points of accident occurrence. Applications are made on insurance futures and spreads, a relatively new class of instruments for risk management launched by the Chicago Board of Trade in 1993, anticipated to start in Europe and perhaps also in other parts of the world in the future. The article treats the problem of pricing catastrophe risk, which is priced in the model and not treated as unsystematic risk. Several closed pricing formulas are derived, both for futures contracts and for futures derivatives, such as caps, call options, and spreads. The framework is that of partial equilibrium theory under uncertainty.     

DURATION,IMMUNIZATION, AND HEDGING WITH INTEREST RATE FUTURES

The recent advent of the interest rate futures markets has greatly enriched the hedging opportunities of market participants faced with undesired interest rate risk. The variety of futures contracts presently spans a number of instruments with different risk, maturity, and coupon characteristics. This paper modifies the concept of duration and extends the duration hedging approach to cases where futures contracts are used as the hedging instrument. The derived hedge ratios take into account differences in coupon, maturity, and risk for three different regimes. Usage of these hedge ratios should lead to more efficient hedging of interest rate risk.     

PRICING CURRENCY OPTIONS UNDER STOCHASTIC INTEREST RATES AND JUMP‐DIFFUSION PROCESSES

We investigate the effects of stochastic interest rates and jumps in the spot exchange rate on the pricing of currency futures, forwards, and futures options. The proposed model extends Bates's model by allowing both the domestic and foreign interest rates to move around randomly, in a generalized Vasicek term‐structure framework. Numerical examples show that the model prices of European currency futures options are similar to those given by Bates's and Black's models in the absence of jumps and when the volatilities of the domestic and foreign interest rates and futures price are negligible. Changes in these volatilities affect the futures options prices. Bates's and Black's models underprice the European currency futures options in both the presence and the absence of jumps. The mispricing increases with the volatilities of interest rates and futures prices. JEL classification: G13     

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.     

The volatility of temperature and pricing of weather derivatives

We propose an Ornstein–Uhlenbeck process with seasonal volatility to model the time dynamics of daily average temperatures. The model is fitted to approximately 45 years of daily observations recorded in Stockholm, one of the European cities for which there is a trade in weather futures and options on the Chicago Mercantile Exchange. Explicit pricing dynamics for futures contracts written on the number of heating/cooling degree-days (so-called HDD/CDD futures) and the cumulative average daily temperature (so-called CAT futures) are calculated, along with a discussion on how to evaluate call and put options with these futures as underlying.     

THE PROPOSED INTRODUCTION OF FUTURES‐STYLE MARGINING IN THE UNITED STATES: AN AUSTRALIAN COMPARISON

We extend the quadratic approximation method to examine American‐style options traded using futures‐style margining and show that an early exercise premium can exist when the cost of carry is negative. Empirical results based on a reduced form of the model using futures‐style call options traded on the Australian All Ordinaries Share Price Index are consistent with previous research: call option early exercise premiums are economically zero. Full option prices are examined by comparing observed futures‐style with theoretical stock‐style values. We find futures‐style values exceed stock‐style values and argue that the increase results from improvements in liquidity. The findings are particularly relevant given the pending decision at the Commodity Futures Trading Commission to introduce a futures‐style system in the United States. JEL classification: G13, C13