Pricing a Domestic Payoff Involving a Foreign Forward Rate
Summary
This question considers Monte Carlo valuation of a payoff based on the positive part of the difference between domestic and foreign forward rates. It assumes a domestic zero-coupon bond maturing at the later tenor date is used as numeraire, and describes the forward FX rate and domestic forward rate as martingales under the associated measure. It also assumes the processes are lognormal.
The author questions whether the usual numeraire-based pricing expression applies, since the rate-difference payoff may not itself be a tradable domestic asset and its discounted value may not be a martingale. The document gives no accepted answer or valuation adjustment, so it frames a modeling and measure-consistency issue rather than resolving it. It does not provide simulation results or establish a complete pricing method.
Key ideas
- The proposed payoff depends on the difference between domestic and foreign forward rates.
- The question uses a domestic zero-coupon bond as numeraire and specifies a corresponding pricing measure.
- The author doubts that the payoff can be inserted directly into the standard numeraire pricing expression.
- No solution or Monte Carlo procedure is supplied in the document.
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# 69223
# What adjustments need to be made before a Monte-Carlo simulation can be applied for the exotic option $(L_{\text{domestic}}-L_{\text{foreign}})^{+}$
I just want to reassure myself that I understand why Monte-Carlo is the appropriate tool in computing the fair value prices for different options. Let's say we have a Tenor discretization $T_{0}=0<T_{1}< T_{2}$ and some numeraire $N(T_{2};t)=P(T_{2};t)$, i.e. the domestic zero coupon bond that matures at $T_{2}$.
Say we have a risky domestic forward rate $L_{\text{domestic}}$, foreign forward rate $L_{\text{foreign}} (T_{1},T_{2};t):=L_{\text{foreign}}(t)$ and a forward FX rate $FFX(t)$. $FFX$ and $L_{\text{domestic}}$ are martingales under the equivalent martingale measure associated to $N(T_{2};t):=N(t)$ and that all three of the processes follow a log-normal process under $\mathbb Q^{P(T_{2})}$
What adjustments need to be made before a Monte-Carlo simulation can be applied for the exotic option $(L_{\text{domestic}}(T_{1})-L_{\text{foreign}}(T_{1}))^{+}$ under the measure $\mathbb Q ^{P(T_{2})}$?
My thoughts, $\left((L_{\text{domestic}}(t)-L_{\text{foreign}}(t))^{+}\right)_{0\leq t \leq T_{2}}$ is not even tradeable in the domestic market such that
$$\frac{(L_{\text{domestic}}(t)-L_{\text{foreign}}(t))^{+}}{N(t)}$$ is not a $\mathbb Q^{P(t_{2})}$ martingale. So we cannot use the universal pricing formula:
$V(0) = N(0)\mathbb E ^{\mathbb Q ^{P(T_{2})}}[\frac{(L_{\text{domestic}}(T_{1})-L_{\text{foreign}}(T_{1}))^{+}}{N(T_{1})}]$.
Any ideas on how to progress since I cannot use the above formula?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.