Deriving the T Forward Measure from the Bond Numeraire
Summary
The note explains how the T forward measure arises when a zero coupon bond maturing at T replaces the risk free account as numeraire. It corrects a suspected expression in the question and identifies the density that changes probability measures: the bond price divided by the risk free asset, normalized by its value at the pricing time. Since the discounted bond price is a martingale under the risk neutral measure, this normalized quantity has expectation one and can serve as a Radon Nikodym derivative.
Under the resulting measure, a payoff at T is priced by multiplying its expected value by the bond price at the current time. This reframes risk neutral valuation in terms of the selected bond numeraire. The explanation relies on the martingale property and a payoff at the bond's maturity; it gives no numerical example or detailed discussion of technical conditions. It also notes that practitioners often specify asset dynamics directly under the forward measure.
Key ideas
- The T forward measure uses a bond maturing at T as its numeraire.
- The discounted bond price is a martingale under the risk neutral measure.
- Normalizing the bond numeraire by its current value gives the measure change density.
- A payoff at maturity can be priced as the bond price times its expectation under the forward measure.
- Asset dynamics are often modeled directly under the forward measure in practice.
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Full text
# T-Forward measure
# T-Forward measure
Ref: https://en.wikipedia.org/wiki/Forward_measure
I am trying to understand how to move from risk neutral measure $Q$ to T-Forward measure $Q_T$.
It appears we can move from one measure to another using the "Radon-Nikodym derivative $\frac{dQ_T}{dQ}$, i.e
$$P(t,T) = E_Q [ \frac{B(t)}{B(T)} ] = E_{Q_T} [\frac{B(t)}{B(T)} \frac{dQ_T}{dQ} ] $$
What I dont understand is how you deduce what $\frac{dQ_T}{dQ}$ is. Wikipedia states it is the following, but im not sure how?
$$ \frac{dQ_T}{dQ} = \frac{B(t)P(T,T)}{B(T)P(t,T)} = 1$$
In this example P(t,T) is the price of a zero coupon bond at time t for maturity T
## Answer by user9403 (score 4, accepted)
https://quant.stackexchange.com/a/33418
I think your statement has a typo. I can't find the statement you made in the article you cite.
The forward measure is the measure induced by using a bond as the numeraire instead of the risk free asset. Letting $H(X_T)$ be the payoff function for an asset $X_t$,
$$ \tilde{\mathbb{E}}\left[\frac{B(t)H(X_T)}{B(T)}\right]=P(t, T)\tilde{\mathbb{E}}\left[\frac{B(t)}{B(T)P(t, T)} H(X_T) \right] $$ $$=P(t, T)\tilde{\mathbb{E}}\left[\frac{B(t)P(T, T)}{B(T)P(t, T)} H(X_T) \right]$$ $\frac{P(s, T)}{B(s)}$ is a martingale under the risk neutral measure,and so the following holds:
$$\tilde{\mathbb{E}}\left[\frac{P(T, T)}{B(T)}\right]=\frac{P(t, T)}{B(t)}$$
Rearranging, it becomes clear that $\frac{B(t)P(T, T)}{B(T)P(t, T)} $ is a martingale with expectation one and is thus mathematically able to be a Radon-Nikodym derivative. Hence the pricing formula can be written as follows: $$g(X_t, t)=P(t, T)\hat{\mathbb{E}}\left[H(X_T) \right]$$
In practice, the dynamics of $X_T$ are often postulated under the T-Forward measure without the intermediary risk-neutral step.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.