On exact pricing of FX options in multivariate time-changed Lévy models

Abstract

In this paper we discuss foreign-exchange option pricing in conditionally Gaussian models, namely in the variance-gamma and in the normal-inverse Gaussian models. It happens that in the both models closed-form pricing is attainable. The used method developes the one of the work by Madan et al. (Eur Finance Rev 2:79-105, 1998) where the price of the European call is primarily derived. The obtained formulas are based on values of the Gauss and the Appell hypergeometric functions.

On risk measuring in the variance-gamma model

“…We introduce analytical formulas for the static risk measures VaR, ES and ERM for such a portfolio. The results of the paper proceed the direction of investigations by Madan, Carr and Chang [27], Ivanov and Ano [21] and Ivanov and Temnov [22] where closed-form expressions in the variance-gamma model were obtained. Let us notice that the mentioned method of the deriving of explicit formulas is applicable also in the normal-inverse Gaussian model; see Ivanov [20] and Ivanov and Temnov [23].…”

“…Namely, it can be supposed that the normal random variables are correlated, but the gamma ones are independent, see Eberlein and Madan [12]. Also, a dependence between normal distributions is assumed in Ivanov and Ano [21]. In comparison, if the gamma random variables are suggested to be dependent, the normal ones are assumed not being correlated; see the papers by Semeraro [37], Luciano and Semeraro [26] and Guillaume [17].…”

“…The most recent statistical applications of the use of the variance-gamma distribution for the modeling of stock returns are made in Daal and Madan [9] and Rathgeber, Stadler and Stöckl [33]. The method of closed-form solutions by Madan, Carr and Chang [27] was further developed in Ivanov and Ano [21] and Ivanov and Temnov [22].…”

A Credit-Risk Valuation under the Variance-Gamma Asset Return

“…The integral (79) is the same as the one in the bottom of p. 207 of Ivanov and Ano (2016), which was partly computed for some values of the parameters therein. Further, we briefly summarize the results of Ivanov and Ano (2016) and calculate (79) for extra ratios between the parameters of the model.…”

“…Further, we briefly summarize the results of Ivanov and Ano (2016) and calculate (79) for extra ratios between the parameters of the model. Case 1. a = b.…”