Pricing of foreign exchange options under the MPT stochastic volatility model and the CIR interest rates

We consider an extension of the model proposed by Moretto, Pasquali, and Trivellato [2010. “Derivative Evaluation Using Recombining Trees under Stochastic Volatility.” Advances and Applications in Statistical Sciences 1 (2): 453–480] (referred to as the MPT model) for pricing foreign exchange (FX) options to the case of stochastic domestic and foreign interest rates driven by the Cox, Ingersoll, and Ross dynamics introduced in Cox, Ingersoll, and Ross [1985. “A Theory of Term Structure of Interest Rates.” Econometrica 53(2): 385–408]. The advantage of the MPT model is that it retains some crucial features of Heston's stochastic volatility model but, as demonstrated in Moretto, Pasquali, and Trivellato [2010. “Derivative Evaluation Using Recombining Trees under Stochastic Volatility.” Advances and Applications in Statistical Sciences 1 (2): 453–480], it is better suited for discretization through recombining lattices, and thus it can also be used to value and hedge exotic FX products. In the model examined in this paper, the instantaneous volatility is correlated with the exchange rate dynamics, but the domestic and foreign short-term rates are assumed to be mutually independent and independent of the dynamics of the exchange rate. The main result furnishes a semi-analytical formula for the price of the FX European call option, which hinges on explicit expressions for conditional characteristic functions.     

Skew-Brownian motion and pricing European exchange options

This article derives a closed-form pricing formula for European exchange options under a non-Gaussian framework for the underlying assets, intending to resolve mispricing associated with a geometric Brownian motion. The dynamics of each of the two correlated underlying assets are assumed to be governed by the exponential of a skew-Brownian motion, which is specified as a sum of a standard Brownian motion and an independent reflected Brownian motion. The proposed pricing formula does not incur additional computational costs than the standard Black–Scholes framework, which one can quickly recover as a particular case of the proposed framework. Finally, we present some numerical experiments followed by a valuable discussion on the results.     

Foreign exchange intervention and equilibrium real exchange rates

Monetary authorities intervene in the currency markets in order to pursue a monetary rule and/or to smooth exchange rate volatility caused by speculative attacks. In the present paper we investigate for possible intervention effects on the volatility of nominal exchange rates and the estimated equilibrium behaviour of real exchange rates. The main argument of the paper is that omission of intervention effects – when they are significant – would bias the ability to detect any PPP-based behaviour of the real exchange rates in the long run. Positive evidence for this argument comes from the experience of six Central and Eastern European economies, whose exchange markets are characterised by frequent interventions.     

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.     

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     

关于境内外人民币期权市场与汇价波动性的分析

目前在境内外人民币外汇市场上主要有三种人民币期权产品,即境内美元兑人民币期权（CNY期权）、境外可交割美元兑人民币期权（CNH期权）和境外不可交割美元兑人民币期权（NDO）。文章分析比较了这三个人民币期权市场的发展特点,探讨了境内外人民币汇价波动性的变动趋势和相互联系,旨在为境内人民币期权市场的定价及风险管理提供参考;文章同时分析了当前影响境内人民币期权市场活跃度的制约因素,并就进一步推动市场健康发展提出相关政策建议。     

The pricing of exchange rate risk and stock market segmentation: The Canadian case

Empirical evidence by Eun and Resnick (1988), among others, has demonstrated the significance of exchange rate risk in the international asset allocation and they have noted that the risk is nondiversifiable. Yet, exchange rate risk was found by Jorion (1991) to be a risk factor that is not priced in the U.S. stock market. This study reexamines such counterintuitive results using data from the Toronto Stock Exchange. The evidence here weakly supports the pricing of the exchange rate risk. Further, the sample period in this study coincides with Jorion's to ensure that both studies examine the pricing of the exchange rate risk in the same global economic environment. The significant pricing of exchange rate risk in Canada and the insignificant pricing in the U.S. imply the possibility of market segmentation.     

Sequential American Exchange Property Options

Property development activities often occur in stages, which are appropriately modeled as sequential American exchange property options, where there are interim expenditures required in order to keep the property development options “alive”. Normally American exchange options require a numerical solution, but herein there is a new closed-form approximate solution, which is computationally efficient and accurate. This method combines repeats of Margrabe European exchange and Geske compound option solutions with tight upper boundaries of either American perpetuities or European exchange options with a high volatility. Illustrations are provided of the sensitivity of the real sequential options and optimal timing to changes in several parameters, which provide a framework for property policy (tax, subsidy and regulatory) guidelines and for property development strategy evaluation. There are several plausible applications of these real option models in commercial and residential property development, within commercial property leases, with regard to switching tenants, and agricultural alternatives.     

关于当前人民币有效汇率合理水平的思考

2011年下半年欧债危机恶化以来,在人民币有效汇率大幅升值的同时,欧美日等国从中国的进口增速均出现了低于其整体进口增速的现象,这表明当前中国出口不景气并不完全归因于外需不足,人民币有效汇率大幅升值至历史高位也是出口增速下滑的重要影响因素。鉴于保持出口稳定是“稳增长”的重要内容之一,文章建议应进一步增强汇率弹性,实现人民币有效汇率的基本稳定,充分发挥汇率政策工具在调节宏观经济内外均衡方面的重要作用。     

Technical analysis profitability when exchange rates are pegged: A note

Abstract

This note extends earlier results which concluded that generally technical analysis trading rules were profitable when applied to several US dollar exchange rates. These results were linked to the presence of long swings in the dollar series, and here, it is tested whether they still hold in a different setting, with a quasi-fixed exchange rate system. Applying non-parametric and parametric tests to the main European currencies does not allow to confirm, in this case, the profitability of these rules. These results strengthen the likelihood of the hypothesis of a causal link from the exchange rate DGP to the profitability of technical analysis trading rules, as already highlighted in several articles.     

The analysis of foreign exchange data using waveform dictionaries

This paper uses waveform dictionaries to decompose the signals contained within three foreign exchange rates using tick-by-tick observations obtained world wide. The three exchange rates examined are the Japanese Yen and the German Deutsche Mark against the U.S. dollar and the Deutsche Mark against the Yen. The data were provided by Olsen Associates.A wabeform dictionary is a class of transforms that generalizes both windowed Fourier transforms and wavelets. Each waveform is parameterized by location, frequency, and scale. Such transforms can analyze signals that have highly localized structures in either time or frequency space as well as broad band structures; that is, waveforms can, in principle, detect everything from shocks represented by Dirac Delta functions, to short bursts of energy within a narrow band of frequencies, to the presence of frequencies that occur sporadically, and finally to the presence of frequencies that hold over the entire observed period. Waveform dictionaries are most useful in analyzing data that are not stationary and non-stationarity up to second order is well recognized in the context of foreign exchange rates.     

Foreign exchange risk and the cross-section of stock returns

We examine the relation between the cross-section of US stock returns and foreign exchange rates during the period from 1973 to 2002. We find that stocks most sensitive to foreign exchange risk (in absolute value) have lower returns than others. This implies a non-linear, negative premium for foreign exchange risk. Sensitivity to foreign exchange generates a cross-sectional spread in stock returns unexplained by existing asset-pricing models. Consequently, we form a zero-investment factor related to foreign exchange-sensitivity and show that it can reduce mean pricing errors for exchange-sensitive portfolios. One possible explanation for our findings includes Johnson's [2004. Forecast dispersion and the cross-section of expected returns. Journal of Finance, 59, 1957–1978] option-theoretic model in which expected returns are decreasing in idiosyncratic cashflow volatility.     

On the valuation of an option to exchange one interest rate for another

This paper studies the problem of pricing a European option on the difference of two interest rates, which is analogous to an option to exchange one asset for another, studied in Margrabe (1978). It is first shown that such option may be valued as exchanging two interest rates implied in relevant futures prices through an extended Black (1976) model, and then by a two-factor Heath-Jarrow-Morton (HJM) model, which shows that introduction of imperfect interest rates movements is essential for pricing such options, for which a one-factor model such as Ho and Lee (1986) should not be applied.     

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.     

European monetary integration and persistance of real exchange rates

The aim of this paper is to analyze whether the persistence properties of the European real exchange rates changed when their currencies joined the euro or during the monetary integration process. More specifically, we investigate whether, as a result of the single currency or the previous macroeconomic stability, nominal price rigidities have decreased and the persistence of real exchange has fallen. We test for stationarity against a change in the integration order on different competitiveness measures during the period that runs from the middle of the seventies to nowadays. The results show that the real exchange rates of the European periphery (Spain, Italy, Portugal, Greece and Finland) underwent a change in their order of integration from I(1) to I(0) at some time around the middle of the 1990s. On the other hand, the real exchange rates of the Central European countries, with a greater stability in the 1980s and 1990s, changed their integration order earlier, if at all, mostly during the 1980s. So, the euro seems to have had, on the whole, little influence on the persistence of real exchange rates. Only for a few countries do our findings detect a significant decrease in persistence related with the nominal convergence process.     

Pricing and Hedging Discount Bond Options in the Presence of Model Risk

This paper focuses on pricing and hedging options on a zero-coupon bond in a Heath–Jarrow–Morton (1992) framework when the value and/or functional form of forward interest rates volatility is unknown, but is assumed to lie between two fixed values. Due to the link existing between the drift and the diffusion coefficients of the forward rates in the Heath, Jarrow and Morton framework, this is equivalent to hedging and pricing the option when the underlying interest rate model is unknown. We show that a continuous rangeof option prices consistent with no arbitrage exist. This range is bounded by the smallest upper-hedging strategy and the largest lower-hedging strategy prices, which are characterized as the solutions of two non-linear partial differential equations. We also discuss several pricing and hedging illustrations.     

Pricing and Hedging Discount Bond Options in the Presence of Model Risk

This paper focuses on pricing and hedging options on a zero-couponbond in a Heath—Jarrow—Morton (1992) framework whenthe value and/or functional form of forward interest rates volatilityis unknown, but is assumed to lie between two fixed values.Due to the link existing between the drift and the diffusioncoefficients of the forward rates in the Heath, Jarrow and Mortonframework, this is equivalent to hedging and pricing the optionwhen the underlying interest rate model is unknown. We showthat a continuous range of option prices consistent with noarbitrage exist. This range is bounded by the smallest upper-hedgingstrategy and the largest lower-hedging strategy prices, whichare characterized as the solutions of two non—linear partialdifferential equations. We also discuss several pricing andhedging illustrations.     

An Empirical Analysis of the Pricing of Bank Issued Options versus Options Exchange Options

Since 1998, large investment banks have become active as issuers of options, generally referred to as call warrants or bank‐issued options. This has led to an interesting situation in the Netherlands, where simultaneously call warrants are traded on the stock exchange, and long‐term call options are traded on the options exchange. Both entitle their holders to buy shares of common stock. We start with a direct comparison between call warrants and call options, written on the same stock and with the same exercise price, but where the call option has a longer time to maturity. In 13 out of 16 cases we find that the call warrants are priced higher, which is a clear violation of basic option pricing rules. In the second part of the analysis we use option pricing models to compare the pricing of call warrants and call options. If implied standard deviations from options are used to price the call warrants, we find that the call warrants are strongly overpriced during the first five trading days. The average overpricing is between 25 and 30%. Only a small part of the overpricing can be explained by rational arguments such as transaction costs. We suggest that the overvaluation can be explained by a combination of an active financial marketing by the banks and the framing effect.     

American option pricing under the double Heston model based on asymptotic expansion

This paper focuses on pricing American put options under the double Heston model proposed by Christoffersen et al. By introducing an explicit exercise rule, we obtain the asymptotic expansion of the solution to the partial differential equation for pricing American put options. We calculate American option price by the sum of the European option price and the early exercise premium. The early exercise premium is calculated by the difference between the American and European option prices based on asymptotic expansions. The European option price is obtained by the efficient COS method. Based on the obtained American option price, the double Heston model is calibrated by minimizing the distance between model and market prices, which yields an optimization problem that is solved by a differential evolution algorithm combined with the Matlab function fmincon.m. Numerical results show that the pricing approach is fast and accurate. Empirical results show that the double Heston model has better performance in pricing short-maturity American put options and capturing the volatility term structure of American put options than the Heston model.