Pricing a Put Option Under a Change to the Forward Measure

Summary

The document presents a question about pricing a put option in a discrete-time interest-rate framework. The stated setup expresses the option value as a discounted conditional expectation under the original probability measure, then introduces a density process built from a numeraire-related quantity and the bond price. The author asks why this particular process defines a new measure and what practical benefit the measure change provides.

The displayed transformation rewrites the put price as the bond price multiplied by an expectation of the payoff under the new measure. This illustrates a standard pricing idea: choosing a numeraire can absorb discounting into the probability measure and yield a simpler expectation. The document itself does not include an answer, derive the density normalization, or discuss when the transformed expectation is easier to evaluate. Its value is therefore mainly as a concise statement of the measure-change setup and the conceptual motivation that a full explanation needs to address.

Key ideas

• A put price is first represented as a discounted conditional expectation under the original measure.
• A density process associated with a bond numeraire defines a transformed probability measure.
• Under the transformed measure, the price is expressed as the bond price times an expected payoff.
• The document poses the motivation for this transformation but gives no derivation or answer.

Tags

# Change of measure price put option


# Change of measure price put option












I hope you can help me out. I'm really stuck understanding this. In my lecture notes we calculated the price of a put option (maturity m,with strike price $(1+i)^m$, where i is some interest rate) as follows: $A_t^{(m)}=A_t (Put^{(m)}(I, (1+i)^m))=1/{\phi_t^G} E(\phi_m^G ((1+i)^m-I_m)_{+}|\mathfrak{G}_t)$.

Then they defined a density process $\xi_t=\phi_t^G\cdot P(t,m)$ which is a normalized (P,G)-martingale. All good up to here.

Then they define a new measure $P^*$ with

$\frac{dP^*}{dP}|_{\mathfrak{G}_m}=\xi_m=\phi_m^G$

Here is my first question: Why do they define it in exactly this way?

Then they say $A_t^{(m)}=\frac{P(t,m)}{\xi_t} E(\xi_m\cdot ((1+i)^m-I_m)_+|\mathfrak{G}_t)$, using the change of measure they get $A_t^{(m)}=P(t,m)E^{*}(((1+i)^m-I_m)_+|\mathfrak{G}_t)$

I understand the steps but I don't understand why we are doing the change of measure? Why is it better to have a price in a non-real world measure $P^*$ instead of in $P$?

I would really appreciate any kind of help or hints.

Thank you

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.