Pricing a Cross-Currency Swap Coupon Under Different Forward Measures
Summary
The document explains how to value a foreign-currency floating-rate coupon in domestic currency using either the foreign or domestic T-forward measure. Under covered interest rate parity, converting the foreign-measure value at spot is equivalent to applying the FX forward and domestic discount factor. Both routes give the same present value when the payoff is handled consistently across currencies and measures.
The key warning is that expectations of FX and the foreign payoff cannot generally be split into a product: they may be dependent. Even if split, the resulting expectation of the foreign payoff under the domestic measure is not the foreign-measure forward rate used in the original valuation. The document supplies a measure-change identity to show the equivalence of the two correct valuation methods. It assumes a frictionless market with exact covered interest rate parity and does not discuss basis, transaction costs, or other market imperfections.
Key ideas
- A foreign coupon can be valued under its currency's forward measure and converted to domestic currency at spot.
- Covered interest rate parity rewrites that value using the FX forward and domestic discount factor.
- The domestic-measure expectation of the FX-converted payoff matches the foreign-measure valuation after conversion.
- The expectation of an FX rate times a payoff cannot generally be split into separate expectations.
- The payoff expectation must use the measure appropriate to the valuation relationship.
Tags
Full text
# Cross currency Swap Pricing under T- forward measure
# Cross currency Swap Pricing under T- forward measure
Let's assume we are in a frictionless market where the covered interest rate parity exactly holds i.e.
\begin{align} F^X(0,T) &=X_0 \frac{P^f(0,T)}{P^d(0,T)} \end{align}
where $X_0$ is the FX spot, $P^f(0,T)$ and $P^d(0,T)$ the present value of a foreign and a domestic ZCB.
Let's assume we want to price a single coupon libor payment $L_f(S,T)$ at $T$ of the foreign leg of a XCCY swap reported in Domestic currency.
One way to do it is:
\begin{align*} PV_d = X_0 P^f(0,T)E_f^T( L_f(S,T)\mid \mathcal{F}_t) &= X_0 P^f(0,T)F_f(0,S,T) \end{align*}
i.e we do all the calculation under the foreign $T$ - forward measure and at $t=0$ we convert everything to domestic currency using the FX spot $X_0$
This also implies(using the definition of the fx forward):
\begin{align*} PV_d = F^X(0,T)P^d(0,T)F_f(0,S,T) \tag{1} \end{align*}
Does (1) hold?
The other way to compute the PV is basically convert to domestic currency at T and now under domestic T-forward measure we have: \begin{align*} PV_d = P^d(0,T)E_d^T(X_TL_f(S,T)\mid \mathcal{F}_t) \end{align*}
In order to reach (1) can we simply split the product inside the expectation?
i.e $E_d^T(X_T\mid \mathcal{F}_t)E_d^T(L_f(S,T)\mid \mathcal{F}_t)$
## Answer by Kurt G. (score 1)
https://quant.stackexchange.com/a/81018
- Regarding your question if (1) holds: did you not just prove it?
- Now to the question at the bottom of OP: Even if you could split the product inside the expectation $$\tag1 PV_d = P^d(0,T)\,\mathbb E_d^T\big[X_T\,L_f(S,T)\big] $$ you would get \begin{align}\tag2 PV_d &= P^d(0,T)\,\underbrace{\mathbb E_d^T\big[X_T\big]}_{F^X(0,T)}\,\mathbb E_d^T\big[L_f(S,T)\big]\\[2mm] &=X_0P^f(0,T)\,\mathbb E_{\color{red}d}^T\big[L_f(S,T)\big]\tag3 \end{align} which has a wrong currency in that expectation.
- The solution to the riddle is that the relationship $$\tag4 X_t\,P^f(t,T)\,\mathbb E_f^T[H\mid{\cal F}_t]=P^d(t,T)\,\mathbb E_d^T[X_T\,H\mid{\cal F}_t] $$ says nothing else that the PV of a foreign payoff $H$ must be the same regardless if it is calculated in the foreign measure and converted to $d\,,$ or if it is converted at $T$ to $d$ and calculated in the domestic measure.
- From (4) you could again deduce your equation (1).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.