The pricing of foreign currency options under jump‐diffusion processes

In this article, the authors derive explicit formulas for European foreign exchange (FX) call and put option values when the exchange rate dynamics are governed by jump‐diffusion processes. The authors use a simple general equilibrium international asset pricing model with continuous trading and frictionless international capital markets. The domestic and foreign price level are introduced as state variables that contain jumps caused by monetary shocks and catastrophic events such as 9/11 or Hurricane Katrina. The domestic and foreign interest rates are stochastic and endogenously determined in the model and are shown to be critically affected by the jump risk of the foreign exchange. The model shows that the behavior of FX options is affected through the impact of state variables and parameters on the nominal interest rates. The model contrasts with those of M. Garman and S. Kohlhagen (1983) and O. Grabbe (1983), whose models have exogenously determined interest rates. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:669–695, 2007     

Currency barrier option pricing with mean reversion

This article develops a barrier option pricing model in which the exchange rate follows a mean‐reverting lognormal process. The corresponding closed‐form solutions for the barrier options with time‐dependent barriers are derived. The numerical results show that barrier option values and the corresponding hedge parameters under the proposed model are different from those based on the Black‐Scholes model. For an up‐and‐out call, the mean‐reverting process keeps the exchange rate in a small range around the mean level. When the mean level is below the barrier but above the strike price, the risk of the call to be knocked out is reduced and its option value is enhanced compared with the value under the Black‐Scholes model. The parameters of the mean‐reverting lognormal process therefore have a material impact on the valuation of currency barrier options and their hedge parameters. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:939–958, 2006     

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     

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     

Spot‐futures spread,time‐varying correlation,and hedging with currency futures

This article investigates the effects of the spot‐futures spread on the return and risk structure in currency markets. With the use of a bivariate dynamic conditional correlation GARCH framework, evidence is found of asymmetric effects of positive and negative spreads on the return and the risk structure of spot and futures markets. The implications of the asymmetric effects on futures hedging are examined, and the performance of hedging strategies generated from a model incorporating asymmetric effects is compared with several alternative models. The in‐sample comparison results indicate that the asymmetric effect model provides the best hedging strategy for all currency markets examined, except for the Canadian dollar. Out‐of‐sample comparisons suggest that the asymmetric effect model provides the best strategy for the Australian dollar, the British pound, the deutsche mark, and the Swiss franc markets, and the symmetric effect model provides a better strategy than the asymmetric effect model in the Canadian dollar and the Japanese yen. The worst performance is given by the naïve hedging strategy for both in‐sample and out‐of‐sample comparisons in all currency markets examined. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:1019–1038, 2006     

Knock‐in American options

A knock‐in American option under a trigger clause is an option contract in which the option holder receives an American option conditional on the underlying stock price breaching a certain trigger level (also called barrier level). We present analytic valuation formulas for knock‐in American options under the Black‐Scholes pricing framework. The price formulas possess different analytic representations, depending on the relation between the trigger stock price level and the critical stock price of the underlying American option. We also performed numerical valuation of several knock‐in American options to illustrate the efficacy of the price formulas. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:179–192, 2004     

The interrelation of price volatility and trading volume of currency options

This article examines the interrelations between future volatility of the U.S. dollar/British pound exchange rate and trading volume of currency options for the British pound. The future volatility of the exchange rate is approximated alternatively by implied volatility and by IGARCH volatility. The results suggest the presence of strong contemporaneous positive feedbacks between the exchange rate volatility and the trading volume of call and put options. Previous option volumes have significant predictive power with respect to the expected future volatility of the dollar/pound exchange rate. Similarly, lagged volatilities jointly have significant predictive power for option volume. Although option volume (volatility) responds somewhat differently to individual volatility (volume) terms under the two volatility measures, the overall volume‐volatility relations are broadly similar between the implied and IGARCH volatilities. The results generally support the hypothesis that the information‐based trading explains more of the trading volume in currency options on the U.S. dollar/British pound exchange rate than hedging. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:681–700, 2003     

Price‐level co‐movements within currency unions: An alternative integration metric

This paper analyses currency union integration by testing whether price levels in member countries possess a common stochastic trend. The trace statistic test for cointegration proposed by (Johansen, 1995) demonstrates the presence of such a trend for most unions. A disaggregated analysis identifies a common stochastic trend for several though fewer than half of country pairs within a union. Some unions such as the Eurozone have small shares of cointegrated country pairs. Yet, the share of cointegrated country pairs is large relative to countries outside currency unions. Comparison to a control group (country pairs where one country belongs to a given union and the other country does not) indicates that the cointegration found within a currency union is a union‐specific trait and not a feature of the individual countries within the union. These results provide an alternative metric to intraunion trade for gauging the extent of currency union integration.     

Regime‐dependent smile‐adjusted delta hedging

We introduce several regime‐dependent smile‐adjusted deltas and compare their efficiency with the smile‐adjusted deltas that are popular with option traders. Using years of daily option prices, out‐of‐sample hedging performance tests for options of all moneyness and maturities and daily, weekly, or fortnightly rebalancing show that even the simplest regime‐dependent smile‐adjustment consistently outperforms implied BSM delta hedging and local volatility and minimum variance smile‐adjustments. Markov‐switching deltas offer the best performance, with delta‐hedging errors often half the size of implied BSM hedging errors. During volatile markets risk reduction from regime‐dependent delta hedging is much greater than during tranquil periods.     

Distributions implied by American currency futures options: A ghost's smile?

A new and easily applicable method for estimating risk‐neutral distributions (RND) implied by American futures options is proposed. It amounts to inverting the Barone‐Adesi and Whaley method (BAW method) to get the BAW implied volatility smile. Extensive empirical tests show that the BAW smile is equivalent to the volatility smile implied by corresponding European options. Therefore, the procedure leads to a legitimate RND estimation method. Further, the investigation of the currency options traded on the Chicago Mercantile Exchange and OTC markets in parallel provides us with insights on the structure and interaction of the two markets. Unequally distributed liquidity in the OTC market seems to lead to price distortions and an ensuing interesting “ghost‐like” shape of the RND density implied by CME options. Finally, using the empirical results, we propose a parsimonious generalization of the existing methods for estimating volatility smiles from OTC options. A single free parameter significantly improves the fit. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:147–178, 2004     

Pricing American options on foreign currency with stochastic volatility,jumps, and stochastic interest rates

By applying the Heath–Jarrow–Morton (HJM) framework, an analytical approximation for pricing American options on foreign currency under stochastic volatility and double jump is derived. This approximation is also applied to other existing models for the purpose of comparison. There is evidence that such types of jumps can have a critical impact on earlyexercise premiums that will be significant for deep out‐of‐the‐money options with short maturities. Moreover, the importance of the term structure of interest rates to early‐exercise premiums is demonstrated as is the sensitivity of these premiums to correlation‐related parameters. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:867–891, 2007     

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     

State dependent pricing, invoicing currency, and exchange rate pass-through

We analyze exchange rate pass-through and volatility of import prices in a dynamic framework where firms are subject to menu costs and decide on price adjustments in response to exchange rate innovations. The exchange rate pass-through and import price volatility then depend on the invoicing currency in combination with functional forms of cost and demand functions. In particular, there is lower pass-through, less frequent price adjustments, and lower price volatility when prices are set in the importer's currency than when prices are set in the exporter's currency.     

Risk–return relationships in foreign‐currency futures following macroeconomic announcements

This study uses the tick data for foreign‐currency futures to examine risk–return relationships on macroeconomic announcements. This study—different from previous studies—examines the risk–return relationship by capturing the announcement effect on returns with announcement surprises and on volatilities with announcement dummies simultaneously in a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. Strong risk–return relationships are detected for the first min after the announcements. Furthermore, the return–risk tradeoff ratios differ across currencies and across macroeconomic indicators. The same information can be more profitable when acted on the more liquid currency futures. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22: 729–764, 2002     

Option prices and risk‐neutral densities for currency cross rates

The theoretical relationship between the risk‐neutral density (RND) of the euro/ pound cross rate and the bivariate RND of the dollar/euro and the dollar/pound rates is derived; the required bivariate RND is defined by the dollar‐rate marginal RNDs and a copula function. The cross‐rate RND can be used by banks, international businesses, and central bankers to assess market expectations, to measure risks, and to value options, without relying on over‐the‐counter markets, which may be either non‐existent or illiquid. Empirical comparisons are made between cross‐rate RNDs estimated from several data sets. Five one‐parameter copula functions are evaluated and it is found that the Gaussian copula is the only one‐parameter copula function that is ranked highly in all of the comparisons we have made. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:324–360, 2010