On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates

Abstract

We construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations.We first deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of HullWhite [Hull and White, 1990]. We then extend the framework by modeling the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [Andersen and Andreasen, 2002], which can model an interest rate smile. We provide semi-closed form approximations which lead to efficient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity-FX-interest rate hybrid payoffs.

A Novel Monte Carlo Approach to Hybrid Local Volatility Models

“…Further, the local volatility component reads Table 6. Hull-White model parameters as in Grzelak and Oosterlee (2011) (Set A) and Piterbarg (2006), Grzelak and Oosterlee (2012) (Set B). Table 7.…”

Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate

“…Majority of the researchers focus on either inducing correlation between the stock and interest rate, or between the stock and the volatility. Grzelak and Oosterlee [56] overcame the limitations in [55] regarding interest rate smiles by modeling multi-currency models with smiles for FX rate, domestic and foreign fixed income market, respectively. This was achievable through the Heston-Libor hybrid model which involved the freezing of Libor rates technique due to the non-affine property.…”

Pricing inflation products with stochastic volatility and stochastic interest rates

“…Nowadays, the quadratic Gaussian and Libor Market Models (among others) are becoming increasingly important to model interest rates (see for example Bloch andAssefa (2009), Bloch (2009), Andersen and Andreasen (2002) and Grzelak and Oosterlee (2010)), because they can model an interest rate smile. However, the application of these models is left for our future work.…”

“…The CPI then denotes the exchange rate. See for example Grzelak and Oosterlee (2010) which employs a very similar model as our inflation model to model the exchange rate.…”

“…Such a SDE system consists of economic state variables like stock prices, inflation, nominal and real interest rates and volatility. By imposing a correlation structure (between the Brownian motions) on this system of SDEs one can define so-called hybrid models, and use them for pricing exotic derivatives, see, for example, Grzelak and Oosterlee (2009), Grzelak and Oosterlee (2010) or van Haastrecht and Pelsser (2009)…”