Numeraire Changes for Bond Option Valuation

Summary

The document formulates an expectation involving a payoff at a future date, discounted by the savings account, and asks how it can be rewritten under a forward measure associated with a zero-coupon bond maturing on that date. The proposed density factor uses the savings account and bond prices. The question is how this factor defines a valid change of measure when a ratio appearing in the expression simplifies to one at the terminal date, and how that relates to the usual stochastic-exponential form of a density process.

No answer or derivation is included, so the measure change is not justified within the document. In general, a forward measure is defined using a normalized tradable numeraire relative to the money-market account; the density process is a time-indexed ratio, not merely its terminal value. The text provides no interest-rate model, integrability conditions, or worked option valuation, limiting conclusions about validity.

Key ideas

• The valuation question concerns rewriting a discounted bond-option payoff under a forward measure.
• The proposed measure change uses a zero-coupon bond as numeraire relative to the savings account.
• A density process must be understood as a time-indexed ratio, even if a terminal-date ratio simplifies.
• The document poses the issue but does not provide a derivation or establish the conditions for validity.

Tags

# Measure change in a bond option problem


# Measure change in a bond option problem












This is not a homework or assignment exercise.

I'm trying to evaluate $\displaystyle \ \ I := E_\beta \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}}\big]$, where $\beta$ is the savings account, $B$ is the present value (at some time $t$) of the zero coupon bond paying $\$1$ at maturity, $\mathbf{1}$ is the indicator function, $r(s)$ is the short rate, and $f(t,T)$ is the instantaneous forward rate (i.e. $f(t,T,T)$). Expanding on this a bit more:

$$ \displaystyle \ \ \beta(T_0) = e^{\int_0^{T_0} r(s)d(s)}$$

$$ \displaystyle \ \ B(t,T) = e^{-\int_t^T f(t,s)d(s)}$$

The first step is to do a measure change as follows:

$$\displaystyle \ \ I = E_{T_0} \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}} \frac{\beta(T_0)B(0,T_0)}{\beta(0)B(T_0,T_0)}\big]$$

My question is why is this correct? For a density process to be a valid change of measure we need to to be a Doleans-Dade exponential local martingale, but I can't see why this is the case in this example, because:

$$\displaystyle \ \ \frac{\beta(0)B(T_0,T_0)}{\beta(T_0)B(0,T_0)} = 1$$

which is not in the usual form of a Doleans exponential?

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