# Cross Currency Swaptions

### Introduction

A **cross currency Bermudan swaption** is an embedded Bermudan option in a cross currency swap. In more detail, it is a contract in which the holder of a cross currency swap is long or short an option to put the swap at certain cash flow payment dates. For example, suppose a fixed leg cross-currency swap payer is long a Bermudan swaption. Then the payer makes periodic payments based on a fixed rate in a certain currency and receives floating rate payments based on the interest rate of another currency. At certain payment dates the payer can cancel the swap, or equivalently, can enter another swap to pay the floating rate and receive the fixed rate, thereby effectively cancelling the original swap. Some deals charge an exercise fee.

Most cross currency swaptions are fixed-for-floating swaptions. However, fixed-for-fixed and floating-for-floating cross-currency swaptions are also popular. Typical swaptions have fixed notionals for both receive and pay legs, and a single fixed coupon rate for the fixed leg and a single spread for the floating leg. Trades with amortizing or variable notionals, and/or variable fixed leg coupons, and/or variable spreads over floating legs are also common.

### Technical Details

The popular methods for valuing Bermudan swaptions are Rubinstein's binomial tree method and its extended version, the Hull-White's trinomial tree method, or other lattice-based numerical methods. These methods are good for one factor and sometimes two factor models. However, as the number of modeling factors increases, the number of nodes of a tree or grids of a lattice grows exponentially. Such a characteristic greatly limits the power of tree- or lattice-based numerical methods. In the case of a cross-currency swaption, the minimum number of factors is three: one factor for each currency and one for the FX rate. If two or more factor models are to be used for the interest rates, the tree- or lattice-based methods are no longer useful.

The natural way to break this computational barrier is to resort to **Monte Carlo simulation**. Historically, Monte Carlo methods were considered to be too difficult to use in Bermudan or American option valuation. There are now ways to extend the power of Monte Carlo simulation. Among these methods, the least squares Monte Carlo method proposed by Longstaff and Schwartz has received the most attention due to its generality, simplicity and relatively good convergence.

### Analysis Supported

FINCAD cross currency swaption functions can be used for the following:

- Calculate by Monte Carlo simulation fair value and risk statistics of a Bermudan style fix-for-float cross currency swaption. The underlying swap may have a custom structure (variable notional, variable coupon, generated dates, etc.). The Hull-White one factor term structure model is used in each currency.
- Calculate by Monte Carlo simulation fair value and risk statistics of a Bermudan style fix-for-fix cross currency swaption. The underlying swap may have a custom structure (variable notional, variable coupon, generated dates, etc.). The Hull-White one factor term structure model is used in each currency.
- Calculate by Monte Carlo simulation fair value and risk statistics of a Bermudan style float-for-float cross currency swaption. The underlying swap may have a custom structure (variable notional, variable coupon, generated dates, etc.). The Hull and White one factor term structure model is used in each currency.

To evaluate the FINCAD solutions to value various cross currency swaptions, contact a FINCAD Representative.