A Cross-Currency Forward-Measure Expectation with Lognormal Inputs

Summary

The document poses a valuation question for a cash flow paid at a future date and resetting earlier. The expression combines a floating rate and a discount bond price in one currency with a discount bond price denominated in another, under a forward measure associated with the payment currency. It asks how to calculate the expectation when the modeled quantities are assumed lognormal and the denominator belongs to a different currency.

The text supplies definitions for the bond prices, floating rate, and measure, and states that two of the processes are martingales in that measure. It does not provide an answer, derivation, covariance treatment, or numerical example. Consequently, it is best read as a setup for studying cross-currency valuation: a complete solution would need to clarify how the foreign-currency quantity is represented under the chosen measure and how dependence among the stochastic terms affects the expectation. The question alone does not establish that the lognormal assumption is sufficient.

Key ideas

• The payoff combines a floating rate and bond-price ratio across two currencies.
• The expectation is taken under a forward measure associated with the payment currency.
• The question assumes lognormal processes and identifies two as martingales in that measure.
• No valuation formula or treatment of cross-currency dependence is provided.

Tags

# Expectation of expression with two currencies under forward measure


# Expectation of expression with two currencies under forward measure












I'm trying to calculate the expected value, at time $0$, of a cashflow paid at time $T$, resetting at time $t$. The coupon is of the form:

$V_0=\mathbb{E}^{T_2}\left[\frac{A_t^y(T_1,T_2)}{B_t^x(T_1,T_2)}C^y_t(T_1,T_2)\right]$

in which:

$A^y_t(T_1,T_2)$ is the price of a discount bond from $T_1$ paying at $T_2$, observed at $t$, in currency $y$,

$B^x_t(T_1,T_2)$ is the price of a discount bond from $T_1$ paying at $T_2$, observed at $t$, in currency $x$,

$C^y_t(T_1,T_2)$ is a floating rate for the period $T_1$ to $T_2$, observed at $t$ and paying at $T_2$, in currency $y$,

$\mathbb{E}^{T_2}$ represents the expectation taken in the $T_2$ forward measure in currency $y$.

We have that the processes $A$ and $C$ are martingale in the chosen measure. Assuming that the prcesses $A$, $B$ and $C$ are log-normal, how can we calculate the expectation taking account of the fact that $B$ is denominated in the currency $x$?

Any pointers or suggestions would be very much appreciated!

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