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.     

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.     

Coupling smiles

The present paper addresses the problem of computing implied volatilities of options written on a domestic asset based on implied volatilities of options on the same asset expressed in a foreign currency and the exchange rate. It proposes an original method together with explicit formulae to compute the at-the-money implied volatility, the smile's skew, convexity, and term structure for short maturities. The method is completely free of any model specification or Markov assumption; it only assumes that jumps are not present. We also investigate how the method performs on the particular example of the currency triplet dollar, euro, yen. We find a very satisfactory agreement between our formulae and the market at one week and one month maturities.     

Equilibrium Valuation of Foreign Exchange Claims

This article studies the equilibrium valuation of foreign exchange contingent claims. Within a continuous-time Lucas (1982) two-country model, exchange rates, interest rates and, in particular, factor risk prices are all endogenously and jointly determined. This guarantees the internal consistency of these price processes with a general equilibrium. In the same model, closed-form valuation formulas are presented for currency options and currency futures options. Common to these formulas is that stochastic volatility and stochastic interest rates are admitted. Hedge ratios and other comparative statics are also provided analytically. It is shown that most existing currency option models are included as special cases.     

Forecasting Emerging Market Exchange Rates from Foreign Equity Options

The use of derivatives to infer future exchange rates has long been a subject of interest in the international finance literature. With the recent currency crises in Mexico, Southeast Asia, and Brazil, work on exchange rate expectations in emerging markets is of particular interest. For some emerging markets, foreign equity options are the only liquid exchange‐traded derivatives with currency information embedded in their prices. Given that emerging markets sometimes undergo currency realignment with discrete jumps in their exchange rate, estimation of risk‐neutral probability density functions from foreign equity option data provides valuable evidence concerning market expectations. To illustrate the use of foreign equity options in estimating market beliefs, we consider Telmex options around the 1994 peso devaluation and find evidence that markets anticipated the change in the Mexican government's foreign exchange policy.     

FOREIGN-CURRENCY OPTIONS, EX ANTE EXCHANGE-RATE VOLATILITY, AND MARKET EFFICIENCY: AN EMPIRICAL TEST

“The currency option's attraction results from the volatility of foreign-exchange markets, where rates can move as much as 3 percent in a day” (M. Sesit, Wall Street Journal, April 20, 1984, p. 25). This study compares the foreign-exchange rate implicit volatility of call options and put options that are written on a foreign currency. These implicit volatilities should be equal, and equal to the volatility of the proportional change in the exchange rate, given that option prices are efficient and that the foreign-currency option pricing model described by Biger and Hull holds. The foreign-currency options that are examined in this study are the British pound, Canadian dollar, Japanese yen, Swiss franc, and West German mark. The results of this study support the notions of market efficiency and put-call parity.     

THE VALUATION OF CURRENCY OPTIONS FOR ALTERNATE STOCHASTIC PROCESSES

This paper compares the ability of four valuation models — the Pure Diffusion model of Black-Scholes-Merton, the Absolute Diffusion and Pure Jump models of Cox-Ross, and the mixed Jump-Diffusion model of Merton — to explain the observed behavior of market prices of foreign currency options. The empirical tests are based on a comparison of the pattern of implied volatilities obtained from option market prices and the Black-Scholes-Merton model with those expected theoretically if exchange rates follow the four stochastic processes specified above. The results of the comparison show that the pattern of implied volatilities is most consistent with the mixed Jump-Diffusion model.     

An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston [(1993) The Review of Financial studies, 6, 327–343], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.     

Implied volatilities,stochastic interest rates,and currency futures options valuation: an empirical investigation

Different models of pricing currency call and put options on futures are empirically tested. Option prices are determined using different models and compared to actual market prices. Option prices are determined using historical as well as implied volatility. The different models tested include both constant and stochastic interest rate models. To determine if the model prices are different from the market prices, regression analysis and paired t-tests are performed. To see which model misprices the least, root mean square errors are determined. It is found that better results are obtained when implied volatility is used. Stochastic interest rate models perform better than constant interest rate models.     

Dynamics of foreign exchange implied volatility and implied correlation surfaces

The prices of currency options expressed in terms of their implied volatilities and the implied correlations between foreign exchange rates at a given point in time depend on option delta and time to maturity. Implied volatilities and implied correlations likewise may thus be represented as a surface. It is well known that these surfaces exhibit both skew/smile features and term structure effects and their shapes fluctuate substantially over time. Using implied volatilities on three currency pairs as well as historical implied correlation values between them, we study the nature of these fluctuations by applying a Karhunen-Loève decomposition that is a generalization of a principal component analysis. We demonstrate that the largest share in the dynamics of these surfaces' fluctuations may be explained by exactly the same three factors, providing evidence of strong interdependences between implied correlation and implied volatility of global currency pairs.     

Valuation of quanto options in a Markovian regime-switching market: A Markov-modulated Gaussian HJM model

We consider the valuation of European quanto call options in an incomplete market where the domestic and foreign forward interest rates are allowed to exhibit regime shifts under the Heath–Jarrow–Morton (HJM) framework, and the foreign price dynamics is exogenously driven by a regime switching jump-diffusion model with Markov-modulated Poisson processes. We derive closed-form solutions for four different types of quanto call options, which include: options struck in a foreign currency, a foreign equity call struck in domestic currency, a foreign equity call option with a guaranteed exchange rate, and an equity-linked foreign exchange-rate call.     

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.     

Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model

The profound financial crisis generated by the collapse of Lehman Brothers and the European sovereign debt crisis in 2011 have caused negative values of government bond yields both in the USA and in the EURO area. This paper investigates whether the use of models which allow for negative interest rates can improve option pricing and implied volatility forecasting. This is done with special attention to foreign exchange and index options. To this end, we carried out an empirical analysis on the prices of call and put options on the US S&P 500 index and Eurodollar futures using a generalization of the Heston model in the stochastic interest rate framework. Specifically, the dynamics of the option’s underlying asset is described by two factors: a stochastic variance and a stochastic interest rate. The volatility is not allowed to be negative, but the interest rate is. Explicit formulas for the transition probability density function and moments are derived. These formulas are used to estimate the model parameters efficiently. Three empirical analyses are illustrated. The first two show that the use of models which allow for negative interest rates can efficiently reproduce implied volatility and forecast option prices (i.e. S&P index and foreign exchange options). The last studies how the US three-month government bond yield affects the US S&P 500 index.     

Biases in option prices : Evidence from the foreign currency option market

This paper examines a European call model of option pricing over a data set which does not suffer from the early exercise problems that have plagued earlier studies of call options on common stocks. We specifically examine a data set of American call prices on spot foreign exchange for which it is plausible to apply an adjusted version of the Garman-Kohlhagen (1983) and Grabbe (1983) European call option model. We make adjustments for interest rate risk and find that the model is nearly unbiased in the valuation of foreign currency options. We conclude that the Geske-Roll (1984) conjecture about dividend uncertainty creating biases in stock option prices holds analogously in the foreign currency option market. Interest rate differential risk (analogous to risky dividends) thus appears to be an important element in the valuation of foreign currency options.     

Keep on smiling? The pricing of Quanto options when all covariances are stochastic

The paper introduces a model for the joint dynamics of asset prices which can capture both a stochastic correlation between stock returns as well as between stock returns and volatilities (stochastic leverage). By relying on two factors for stochastic volatility, the model allows for stochastic leverage and is thus able to explain time-varying slopes of the smiles. The use of Wishart processes for the covariance matrix of returns enables the model to also capture stochastic correlations between the assets. Our model offers an integrated pricing approach for both Quanto and plain-vanilla options on the stock as well as the foreign exchange rate. We derive semi-closed form solutions for option prices and analyze the impact of state variables. Quanto options offer a significant exposure to the stochastic covariance between stock prices and exchange rates. In contrast to standard models, the smile of stock options, the smile of currency options, and the price differences between Quanto options and plain-vanilla options can change independently of each other.     

Currency risk hedging: Futures vs. forward

The objective of this paper is to address the issue of choosing between currency forward and currency futures contracts when hedging against currency risk within a stochastic interest rates environment. We compare between the hedging effectiveness of the two derivative assets both within a narrow sense (i.e., volatility minimization) and within a wide sense (i.e., risk-return trade-off). When judging hedging effectiveness in the narrow sense, forward and futures contracts give identical results even if they do not have identical prices. When judging hedging effectiveness in the wide sense, the choice between the two contracts is determined by the correlation between the domestic and the foreign term structures dynamics.     

International Yield Curves and Currency Puzzles

The currency depreciation rate is often computed as the ratio of foreign to domestic pricing kernels. Using bond prices alone to estimate these kernels leads to currency puzzles: the inability of models to match violations of uncovered interest parity and the volatility of exchange rates. This happens because of the FX bond disconnect, the inability of bonds to span exchange rates. Incorporating innovations to the pricing kernel that affect exchange rates but not bonds helps resolve the puzzles. This approach also allows one to relate news about cross-country differences between international yields to news about currency risk premiums.     

Arbitrage-free market models for option prices: the multi-strike case

This paper studies modeling and existence issues for market models of option prices in a continuous-time framework with one stock, one bond and a family of European call options for one fixed maturity and all strikes. After arguing that (classical) implied volatilities are ill-suited for constructing such models, we introduce the new concepts of local implied volatilities and price level. We show that these new quantities provide a natural and simple parametrization of all option price models satisfying the natural static arbitrage bounds across strikes. We next characterize absence of dynamic arbitrage for such models in terms of drift restrictions on the model coefficients. For the resulting infinite system of SDEs for the price level and all local implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We give explicit examples of volatility coefficients satisfying the required assumptions, and hence of arbitrage-free multi-strike market models of option prices.      

EXACT FORMULAS FOR PRICING BONDS AND OPTIONS WHEN INTEREST RATE DIFFUSIONS CONTAIN JUMPS

I develop Heath‐Jarrow‐Morton extensions of the Vasicek and Jamshidian pure‐diffusion models, extend these models to incorporate Poisson‐Gaussian interest rate jumps, and obtain closed‐form models for valuing default‐free, zero‐coupon bonds and European call and put options on default‐free, zero‐coupon bonds in a market where interest rates can experience discontinuous information shocks. The jump‐diffusion pricing models value the instrument as the probability‐weighted average of the pure‐diffusion model prices, each conditional on a specific number of jumps occurring during the life of the instrument. I extend the models to coupon‐bearing instruments by applying Jamshidian's serial‐decomposition technique.