Multi-Currency Markets

Tutorial Contents

1. Fixed Income Notation
2. Fixed Income Probability Measures
3. Multi-Currency Markets
4. The HJM Framework
5. Basic Fixed Income Instruments
6. Caps, Floors and Swaptions
7. CMS Swaps, Caps and Floors
8. Bermudan Swaptions
9. Structured Notes and Exotic Swaps
10. Callable Libor Exotics and TARNs
11. Volatility Derivatives
12. Yield Curves and Curve Construction
13. Managing Yield Curve Risk
14. Vanilla Models With Local Volatility
15. Vanilla Models With Stochastic Volatility
16. Introduction to Term Structure Models
17. One-Factor Short Rate Models I
18. One-Factor Short Rate Models II
19. Multi-Factor Short Rate Models
20. The Libor Market Model I
21. The Libor Market Model II
22. Risk Management and Sensitivity Computations
23. P&L Attribution
24. Payoff Smoothing
25. Vegas in the Libor Market Model

Some derivatives depend on multiple currencies. If this is the case, the challenge of modeling such derivatives becomes more difficult. In this section, we introduce a framework for modeling such multi-currency derivatives.

Notation

Consider a setup where we have two economies, a domestic economy and a foreign economy, each with their own currency. In the following, we will denote the domestic currency by £ (GBP) and the foreign currency by $ (USD). Note that these choices of currencies are arbitrary and are just used for concreteness.

Let $P_d(t,T)$ and $P_f(t,T)$ denote the time $t$ prices of $T$-bonds in the domestic and foreign economies, respectively. Translating the value of the foreign $T$-bond to the domestic currency can be achieved at time $t$ through the spot FX rate $X(t)$ measured in units of domestic currency per unit of foreign currency. In other words, the value $\hat{P}_d(t,T)$ to a domestic investor of one foreign $T$-bond is

FX Forwards

For no initial cash outlay, it is possible to generate a guaranteed cashflow at time $T$ of one unit of foreign c currency and $-X_T(t)$ units of domestic currency, where $X_T(t)$ represents the time $t$ value of the forward FX rate with maturity $T$. The strategy is as follows:

1. Buy one foreign $T$-bond for $\hat{P}_d(t,T)$ in domestic currency.
2. Finance this purchase by short-selling $\tfrac{\hat{P}_d(t,T)}{P_d(t,T)}$ units of the domestic $T$-bond. Clearly, this strategy is free to enter, and will generate a time $T$ cashflow of $\$1 - £\tfrac{\hat{P}_d(t,T)}{P_d(t,T)}$. To preclude arbitrage, we find that this expression must be zero at time $t$. Hence, we have locked in an exchange rate of
   $X_T(t) := \frac{\hat{P}_d(t,T)}{P_d(t,T)} = X(t)\frac{P_f(t,T)}{P_d(t,T)}$
   at time $t$ for an FX trade at maturity $T$.

Risk Neutral Measures

Let $\beta_d(t)$ and $\beta_f(t)$ denote the continuously compounded money markets in the domestic and foreign economies, respectively. $\beta_d(t)$ and $\beta_f(t)$ induce two separate risk-neutral measures, which we denote by $\mathbb{Q}^d$ and $\mathbb{Q}^f$, respectively. Clearly, these two measures are related. If $g(T)$ is a random payoff at time $T$ made in the foreign currency, in a complete market this payoff is worth

to the foreign investor, where $\mathbb{E}_t^f[\cdot]$ represents the conditional expectation operator under $\mathbb{Q}^f$. From the point of view of a domestic investor, the payoff $g(T)$ must translate to a domestic currency at a rate of $X(T)$ at time $T$, making the effective domestic payoff $g(T)X(T)$. Hence,

where $\mathbb{E}_t^d[\cdot]$ represents the same operator under $\mathbb{Q}^d$. Importantly, to preclude arbitrage, we find that $V_f(t)$ and $V_d(t)$ must also be linked by the spot rate. Specifically, we have

or

If we now consider the arbitrary payoff $Y(T) := \tfrac{g(T)X(T)}{\beta_d(T)}$, the above shows that

By our earlier theory, this shows that the density process linking $\mathbb{Q}^d$ and $\mathbb{Q}^f$ is given by

Because we know that this density process is a martingale under $\mathbb{Q}^d$, if $X(t)$ is to be an Ito process, it must take the form

under $\mathbb{Q}^d$, i.e. its drift is completely determined by the spread in the short-rates between the two economies.

Other Measures

Heuristically, by using a relation analogous to the chain rule applied to Radon-Nikodym derivatives, relations between other domestic and foreign probabilty measures are easy to deduce. Suppose our desired numeraire is $N$, with values $N_d$ and $N_f$ in the domestic and foreign economies respectively. Denote the measures under these numeraires by $\mathbb{Q}^{N, d}$ and $\mathbb{Q}^{N, f}$ under the domestic and foreign currencies respectively. Then, we find that

In particular, if we set $N$ to be the $T$-bond in each currency, by noting the formula for the FX forward rate we see that

This also demonstrates the fact that $X_T(t)$ is a $\mathbb{Q}^{T,d}$-martingale satisfying $X_T(T) = X(T)$.