Section 24.1

Interest Rates and Currencies

Arbitrage from mismatched rates or compounding, triangular currency cycles, forward prices.

Toolkit

No-arbitrage pricing identifies all trading strategies with zero net initial cost and no liabilities later; if such a strategy has positive probability of profit and no probability of loss, it is an arbitrage, and absence of arbitrage forces consistency relations among rates, prices and exchange rates.

Key facts:

Compounding: money grows by factor $e^{rT}$ (continuous) or $(1+r)^T$ (discrete); two riskless rates for the same horizon must coincide, else borrow at the low rate and invest at the high rate for a sure profit of the growth-factor gap per unit borrowed.
Forward price: to have zero initial value, $F_0 = S_0 e^{rT}$ (no dividends); more generally the time- $t$ value of a forward struck at $K$ is $S_t - K e^{-r(T-t)}$ , replicated by holding one unit of the asset and shorting $K$ zero-coupon bonds maturing at $T$ .
Triangular FX arbitrage: with cross rates $q_{ij}$ (units of currency $j$ per unit of currency $i$ ), consistency requires
$q_{12}\,q_{23}\,q_{31}=1$
for every cycle; any deviation gives a riskless profit by trading around the cycle in the profitable direction.

Problems