This article proposes an interest rate model for dual-currency interest rate markets. The model assumes that volatility is a deterministic function of time alone. This volatility structure can reduce the dimension of the required state variables. An important special case is presented, which corresponds essentially to a Vasicek/Hull-White yield curve model in each currency. The model is very useful for pricing cross-currency derivatives.Key Words: interest rate, cross currency, calibration, volatility structure.Introduction:An interest rate reflects the cost of borrowing or the reward of saving. Interest rate curve is the plot of maturities and associated interest rates that illustrates future interest rates in a clear and concise way. Interest rate curve is also called the term structure of interest rates. Interest rate plays critical roles in finance and economy. It affects everyone and every business. One factor model is mathematically tractable, but may be insufficient to capture all dynamics of interest rate curve movements. As such, multi factor models of interest rates arise to have a better explanation of interest rate evolutions. A multi factor model assumes the movement of interest rates is determined by multiple state variables. The most important contracts for calibrating interest rate term structure models are caplets and swaptions. To get a fast and stable model calibration, it is important to have closed from or semi-closed form approximation for their values. There is a rich literature on interest rate modelling. The first analysis is one factor models, such as Vasicek (1977), and Cox, al et. (1985). These models assume that the movement of an interest rate curve is determined by a single state variable. This state variable is usually called the short rate that follows a stochastic diffusion process. Medova al et. (2006) study interest rate data by using a three-factor interest rate curve model and the Kalman filter. The model captures the salient features of the whole term structure in forward simulation. Yu and Ning (2019) propose an interest rate model by means of uncertain differential equations with jumps and derive a closed form price for zero-coupon bond. Verschuren (2019) develops a coherent framework on how to best incorporate negative interest rates in these studies through a single curve stochastic term structure model. Kikuchi (2024) presents a new quadratic Gaussian short rate model with a stochastic lower bound to capture changes in the yield curve including negative interest rates. Akram (2020) presents a long-term interest rate model to reflect the central bank’s actions influence the long-term interest rate primarily through the short-term interest rate. Hansen (2023) presents a term structure model for no-arbitrage bond yields and realized bond market volatility and shows that conditional yield curve covariation is priced in long-term yields. Levrero and Matteo [2019] study the relationship between short- and long-term interest rates and outline an asymmetry in the relationship. Bauer and Hamilton (2019) conclude that conventional tests of whether variables other than the level, slope and curvature can help predict bond returns have significant size distortions. This article presents a new interest rate model for cross-currency fixed income derivatives. These models have the property that all volatility is a deterministic function of time alone. In general, a deterministic volatility structure leads to a model such that if the underlying Brownian motion driving all uncertainty in both economies is of one dimension, then in general three-dimensional state variables are required to completely characterize the yield curve and exchange rate dynamics. The analytic tractability of the constant correlation of the separable deterministic volatility (SDV) model provide closed form formulas for these values as functions of the yield curves and exchange rate volatility (see https://finpricing.com/lib/FxVolIntroduction.html) . By a further judicious choice of volatility structure, one can reduce the dimension of the required state variable. An important special case for applications is presented in which only three state variables are required. This case corresponds essentially to a Vasicek/Hull-White yield curve model in each currency. In this particular case, a general framework for European contingent claim valuation is also worked out. The model concentrates on the evolution of the instantaneous forward rate. We will summarise some standard results and introduce a number of financial variables and concepts such as changes ofnumeraire. The equations describing the dynamics involve a stochastic term which includes a multi-dimensional correlated Brownian motion. When calculating expected values of this Brownian motion it is necessary to specify a probability space and filtration to which the Brownian motion is adapted. The rest of this paper is organized as follows: The model is presented in Section 1; Section 2 studies the relationship with other well-known models. Section 3 elaborates calibration. Numerical results are discussed in Section 4; the conclusions are given in Section 5.
