Options for Collateral Options

Alexandre Antonov and Vladimir V Piterbarg

Credit support annexes specify rules for posting collateral. If multiple currencies are allowed, then the party posting collateral has, now and at each future point in time, a choice of which currency to post. This choice leads to optionality that needs to be accounted for when valuing even the most basic of derivatives, such as forwards or swaps (see Piterbarg 2012).

In this chapter, we consider the important case of two currencies, under the assumption of full substitution rights (see Piterbarg 2013). In this case, the adjustment to the discount factor applied to a cashflow paid at time T reduces to calculating an expression of the form

$$D\left(T\right)\stackrel{\Delta }{=}\text{E}\left(\exp \left(-{\displaystyle {\int }_0^{T}q{\left(s\right)}^{+}}\text{d}s\right)\right),T>0$$

(12.1)

where the stochastic process q(·) represents the so-called collateral basis, that is, the difference between foreign exchange-adjusted collateral rates in the two currencies. Here and throughout we use the notation x⁺ $\stackrel{\Delta }{=}$ max(x, 0). The collateral basis is typically modelled as a Gaussian process (see Piterbarg 2012; McCloud 2013a), a choice we adopt here, too.

Even with the Gaussian assumption in place, the exact calculation of the expected value in Equation 12.1 in