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

Abstract

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.

Valuation of European Style Compound Option Written on European Style Currency and Power Options

“…To deal with the situations where the foreign currency fluctuates with high magnitude i.e. volatility several attempts have been made under the assumption that foreign currency follows a stochastic differential driven various version of sophisticated jump-diffusion processes, for instance, see Hung [18,2007], Li, Peng [19,2013], Ming [21,2019] and David Liu [20,20].…”

Using Multivariate Densities to Assign Lattice Probabilities When There Are Jumps

“…The usual suspects are models that include multiple state variables for the underlying spot, stochastic interest rates, convenience yields, and stochastic volatility. Guo and Hung (2007) propose a setup for currencies that includes both an exponential and a normal jump amplitude. In most applications, the jump magnitude is either normal or lognormal.…”

“…However, the double exponential jump model studied by Kou (2004) and Ramezani and Zeng (2007) also has nice theoretical properties and empirical support. Guo and Hung (2007) propose a setup for currencies that includes both an exponential and a normal jump amplitude. Huang and Wu (2004) consider time-changed Lévy processes and find that a high-frequency jump structure outperforms the low-frequency compound Poisson jump specification.…”

The Behavior Comparison between Mean Reversion and Jump Diffusion of CDS Spread

“…To adequately model the extreme return outliers in equity markets, more recent theoretical and empirical research also argues in favor of jumps in the price process, 6 for example, currency options (Xiao et al 2010), double-jump currency option model (Xu et al 2013;Guo and Hung, 2007), commodity prices (Eydel and and Geman, 1998;Schwartz and Smith, 2000;Escribano et al 2002), option pricing (Merton, 1976;Duffie et al 2000), risk 5 The downside of moving to the log-double-exponential distribution is that it uses more parameters than the log-normal distribution. Moreover, many theoretical results are only valid for the log-normal distribution.…”