Foreign Exchange Options Research Articles

English

  In 2001, the commission of inquiry into the rapid depreciation of the exchange rate attributed the rapid 72.4% depreciation of the South African Rand from June to December 2001 to the excessive volatility caused by market participants’ expectations. This paper investigates whether the market participants’ expectations implicit in foreign exchange options can provide a signal for currency crises. To achieve this, and to capture the dynamics of nonlinearity of implied volatility in foreign exchange options, the paper uses the Markov regime switching GARCH with time varying probabilities. We find that, using implied volatility in foreign exchange options, the technique manages to identify the crisis dates identified by previous literature and to some extent offers an early signal of the crisis before it occurs.      Key words: Currency crisis, foreign exchange, market participants’ expectations, implied volatility, Markov regime switching, generalized autoregressive conditional heteroscedastic  (GARCH), market microstructure, Black-Scholes model, options, European options, at the money, option maturities.

Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order

If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used to fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner.

FX Volatility Smile Construction

The foreign exchange options market is one of the largest and most liquid OTC derivative markets in the world. Surprisingly, very little is known in the academic literature about the construction of the most important object in this market: The implied volatility smile. The smile construction procedure and the volatility quoting mechanisms are FX specific and differ significantly from other markets. We give a brief overview of these quoting mechanisms and provide a comprehensive introduction to the resulting smile construction problem. Furthermore, we provide a new formula which can be used for an efficient and robust FX smile construction.

Calibration of implied volatility for the exchange rate for the Chinese Yuan from its derivatives

Revised implied volatility curves and surfaces for the Chinese Yuan (CNY) exchange rate are obtained from market quotations for CNY non-deliverable options by solving an inverse problem of foreign exchange option pricing, which is calculated using a regularization approach in an optimal control framework. To take account of the market expectation for the CNY exchange rate, a stochastic adjusted factor is applied that follows a Vasicek model with parameters fitted from market quotations for CNY non-deliverable forwards. A well-posed numerical scheme is implemented.

Pricing Kernels for Barrier Options

This paper proposes a pricing methodology for barrier options using integral kernels, which is completely different from the standard approach that postulates a model.Our methodology is more attractive when the model of the standard approach can not be calibrated to the market price of a no-touch option, the most liquid exotic option in the the foreign exchange options market.Prices by our methodology are surely in the pricing bounds which are model-independently derived from super-/sub-replication using plain-vanilla options and no-touch options, while the mis-calibrated model may produce prices out of the bounds. This is demonstrated by the numerical examples.

Understanding Foreign Exchange Option Returns: The Information Content of Volatility

According to general asset pricing theory, options should reward their holders for the systematic risk they are bearing. In this paper, we study the returns of foreign exchange options. We find that, by sorting options according to the distance of their implied volatility from the historical volatility, we obtain portfolios with positive average monthly returns. These returns are not explained by standard aggregate risk factors, which suggest either that additional risk factors should be accounted for, or that investors behavior differs from the traditional paradigm of rational agents.

An Fx options model that incorporates 25-delta strangles and 25-delta risk reversals

The paper suggests a new class of models (Q-Phi) to capture the information that the foreign exchange options market provides through the 25-delta strangles and 25-delta risk reversals. The model is able to capture the stochastic movements of a full strike structure of implied volatilities. We argue that extracting information through this model and pricing path-dependent and non-benchmark strike options is a better methodology than using a constant implied volatility. The model can be used to price exotic options and hedge them robustly with benchmark European options. It is easy to calibrate and the model parameters lend themselves to intuitive interpretation by a trader managing an Fx options book.

How important is the term structure in implied volatility surface modeling? Evidence from foreign exchange options

We claim that previously proposed parametric specifications that linearly approximate the term structure of the implied volatility surface (IVS) in option prices fail to capture important information regarding the expectations of market participants. This paper proposes a parametric specification for describing the IVS that allows flexible modeling of the term structure through a Nelson and Siegel (1987) factorization, recently proposed by Diebold and Li (2006) in the context of yield curve modeling. The specification is tested on implied volatilities from the over-the-counter foreign exchange options market, where contracts with long expiries are actively traded and thus the term structure dimension of the surface should be very important. We first show that the proposed volatility specification can consistently and remarkably improve our ability to describe the surface on any given day. We then establish the economic relevance of the incremental information captured by our proposed specification by showing that it can produce more accurate forecasts of implied volatility that can support long-term profitable trading strategies in the absence of transaction costs.

A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options

Vanilla (standard European) options are actively traded on many underlying asset classes, such as equities, commodities and foreign exchange (FX). The market quotes for these options are typically used by exotic options traders to calibrate the parameters of the (risk-neutral) stochastic process for the underlying asset. Barrier options, of many different types, are also widely traded in all these markets but one important feature of the FX options markets is that barrier options, especially double-no-touch (DNT) options, are now so actively traded that they are no longer considered, in any way, exotic options. Instead, traders would, in principle, like to use them as instruments to which they can calibrate their model. The desirability of doing this has been highlighted by talks at practitioner conferences but, to our best knowledge (at least within the realm of the published literature), there have been no models which are specifically designed to cater for this. In this paper, we introduce such a model. It allows for calibration in a two-stage process. The first stage fits to DNT options (or other types of double barrier options). The second stage fits to vanilla options. The key to this is to assume that the dynamics of the spot FX rate are of one type before the first exit time from a ‘corridor’ region but are allowed to be of a different type after the first exit time. The model allows for jumps (either finite activity or infinite activity) and also for stochastic volatility. Hence, not only can it give a good fit to the market prices of options, it can also allow for realistic dynamics of the underlying FX rate and realistic future volatility smiles and skews. En route, we significantly extend existing results in the literature by providing closed-form (up to Laplace inversion) expressions for the prices of several types of barrier options as well as results related to the distribution of first passage times and of the ‘overshoot’.

Distribution and Statistical Behavior of Implied Volatilities

In this paper we look at the distribution and statistical behavior of what is known as implied volatilities in the foreign exchange (FX) market. We describe the statistical properties of over-the-counter foreign exchange (OTC FX) options implied at-the-money volatilities. The data set is unique because it presents continuous daily observations of implied volatility. Trying to do the same type of analysis for almost any other option market is very difficult because the data will be “unclean” for this purpose. This is explained in more detail. We find the distribution of implied volatilities has a high peak and fat tails relative to the Gaussian distribution. We also find that the short-term implied volatilities are more volatile than long-term implied volatilities. The data show that the volatility of implied volatility is unstable over time and the correlation between the implied volatilities varies considerably over time. The paper discusses the various implications of these issues. An important question one needs to ask is: What are implied volatilities measuring? The answer to this question is not quite clear in the literature and will affect interpretation of the data.

FX risk-neutral valuation relationships for the SU jump-diffusion family

This paper derives preference-free pricing formulae for foreign exchange options, which are consistent with a general equilibrium representative agent economy. These risk-neutral valuation relationships (RNVR's) are obtained for the SU jump-diffusion family. Call and put options are particular cases of our general model. These option pricing formulae nest Merton's (1976) jump-diffusion equations. Our option valuation formulae are able to generate symmetric and asymmetric volatility smiles and skews with similar shapes to those observed in the foreign exchange options market, and they solve several pricing biases of Black (1976) and Garman and Kohlhagen (1983) models. Copyright © 2010 John Wiley & Sons, Ltd.

FX Risk Neutral Valuation Relationships for the Su Jump-Diffusion Family

This paper derives preference-free pricing formulae for foreign exchange options which are consistent with a general equilibrium representative agent economy. These risk neutral valuation relationships (RNVR's) are obtained for the Su jump-diffusion family. Call and put options are particular cases of our general model. These option pricing formulae nest Merton's (1976) jump-diffusion equations. Our option valuation formulae are able to generate symmetric and asymmetric volatility smiles and skews with similar shapes to those observed in the foreign exchange options market, and they solve several pricing biases of Black (1976) and Garman and Kohlhagen (1983) models.

Influence of Transaction Costs on Foreign Exchange Option Contracts: Intra-Daily Tests

This paper tests the impact of transaction cost specication on deviations from lower boundary and put-call parity properties. Using PHLX traded foreign exchange options, prices for puts and calls are matched to the nearest five minutes. The results indicate how boundaries on the arbitrage profit function determined by alternative measures of transactions costs can impact the interpretation of deviations from distribution free properties of options such as put-call parity.

Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order

If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner.

Valuation of time charter contracts for ships

This article describes a valuation method for time charter contracts for ships, i.e. leasing contracts for ships with embedded Bermudan options for buying the ship and extending the contract. As there often are embedded foreign exchange options in the buy options on the ship, a two factor stochastic model is developed and it is shown how the price can be determined applying techniques from contingent claim analysis such as dynamic programming.

Duru-Kleinert Asymptotic Expansions for Long-Term Foreign Exchange and Swaptions Implied Volatility Smile

In this paper, we develop asymptotic formulas for long-dated Foreign Exchange (FX) and swaptions implied volatilities. We extend the method exposed in Decamps and De Schepper (2009b) to a generic model with time-dependent parameters. Imposing a condition on the skew, we derive averaging formulas for the parameters. The method is applied to the pricing of FX options when the domestic and foreign interest rate curves are driven by Gaussian short-term rate models and to the pricing of swaptions in the Libor market model.

Computational aspects of pricing foreign exchange options with stochastic volatility and stochastic interest rates

This paper presents an analytic result for the price of a European call option on a foreign exchange currency rate. Market volatility is assumed correlated with the exchange rate and interest rates, domestic and foreign, are assumed to be stochastic. Integrals involving interest rates are derived, characteristic functions are produced, and, with evaluation, the nature of the integrals involved in Fourier inversion is examined. By comparison with FX market data, some of the effects of the nature of stochastic interest rates upon option prices are examined.

Comparison of Black–Scholes Formula with Fractional Black–Scholes Formula in the Foreign Exchange Option Market with Changing Volatility

In this paper, the authors discuss the fractional option pricing with Black–Scholes formula, deduce the Fractional Black–Scholes formula, show the empirical results by using China merchants bank foreign exchange call option price, and find when the volatility is smaller, the asymptotic mean squared error of Fractional Black–Scholes is bigger than the Traditional Black–Scholes’, while the volatility is bigger—the market mechanism has a full play, the result is reverse. Namely when the market mechanism is given a full scope, the estimating effect of Fractional Black–Scholes is better than Traditional Black–Scholes’.

Foreign exchange, fractional cointegration and the implied–realized volatility relation

Almost all relevant literature has characterized implied volatility as a biased predictor of realized volatility. In this paper we provide new time series techniques to investigate the validity of this finding in several foreign exchange options markets, including the Euro market. First, we develop a new fractional cointegration test that is shown to be robust to both stationary and non-stationary regions. Second, we employ both intra-day and daily data to measure realized volatility in order to assess the relevance of data frequency in resolving the bias. Third, we use data on implied volatility traded on the market. In contrast to previous studies, we show that the frequency of data used for measuring realized volatility within a fractionally cointegrating framework is important for the results of unbiasedness tests. Significantly, for many popular exchange rates, the use of intra-day rather than daily data affects the emergence of a different bias, as the possibility of a fractionally integrated risk premium admits itself!

Cross-dynamics of volatility term structures implied by foreign exchange options

This paper examines the cross-dynamics of volatility term structure slopes implied by foreign exchange options. The empirical findings demonstrate that a few principal components can explain a vast proportion of the variation in volatility term structure slopes across the major exchange rates. Furthermore, the results indicate that the euro is the dominant currency, as the implied volatility term structure of the euro is found to affect all the other volatility term structures, while the term structure of the euro appears to be virtually unaffected by the other currencies. Finally, our results provide evidence of significant nonlinearities in the relationship between the euro and the Swiss franc.