Why Forward-Curve Simulations Use the Maturity-Matched Forward Measure

Summary

The document raises a question about simulating future interest-rate forward curves under a T-forward measure. It contrasts centering a simulated rate at its expectation under the measure associated with time t with using the forward rate observed at the initial time. It also states a relationship between the expected average rate under a measure associated with the end of the accrual period and the corresponding initial forward rate.

The central issue is the choice of numeraire: the question asks why a bond maturing at t is used instead of one maturing at t plus the rate period. The text supplies notation for zero-coupon bond prices, average rates, and forward rates, but offers no answer, derivation, simulation results, or caveats beyond noting that the notation may be unnecessary. It therefore frames a fixed-income pricing question rather than resolving it.

Key ideas

• The document distinguishes expectations taken under different forward measures when simulating future rates.
• It states that the expected average rate under the measure tied to the accrual end date equals the initial forward rate.
• It asks why the numeraire is a bond maturing at the simulation time rather than at the accrual end date.
• No derivation or answer to the numeraire question is included.

Tags

# Reason for choosing the T-forward measure to calculate expected value of forward curves


# Reason for choosing the T-forward measure to calculate expected value of forward curves












## Setup

I read that when simulating forward curves $(r_t(s_i))_i$ at some future time $t>0$, one is supposed to center them not around $F(0;t,t+s_i)$, but around $$\mathbb E^{\mathbb Q_{t}}[r_t(s_i)],$$ which is not equal to $F(0;t,t+s_i)$.

## Question

Why is that the case? What makes the numeraire $B(\cdot,t)$ so special? Is there simple intuitive reason why we choose $B(\cdot,t)$ over $B(\cdot,t+s_i)$ for example?

### Notation (most probably unnecessary, but I'll still state it just in case I used non-standard notation)

Given some financial market $\mathcal S=(S_0,S_1,\dots,S_n)$ of tradeable assets $S_i$ we define the T-forward measure $\mathbb Q_T$ in such a way that $\frac{S_i(\cdot)}{B(\cdot,T)}$ is a $\mathbb Q_T$-martingale, where $B(t,T)$ is the prize at time $t$ of a T-zero coupon bond.

Now let $r_t(s):=\frac{1}{s}\int_t^{t+s}r(u)du$ be the average interest rate over $[t,t+s]$, then we assume that $\mathbb E^{\mathbb Q_{t+s}}[r_t(s)] = F(0;t,t+s)$, where $F(u;t,t+s) = \frac{r_u(t+s)(t+s)-r_u(t)t}{s}$ is the continuously compounded (from $t$ to $t+s$, as seen from time $u\leq t$) forward rate.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.