# 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

• For the question of "why is it better to have a price in a non-real world measure", can you think of any examples where something may be worth more to one person than to another? If you can, then you can think of it as these two people having different measures. In which case, what is the real world measure? is it the first person's? how about the second? maybe it's some other measure. Maybe everyone has their own measure, depending on their own circumstances... – will Jan 27 at 0:29
• What will said about multiple measures is true. But when evaluating arbitrage operations, one measure is convenient and most frequently used: the risk-neutral measure. – noob2 Jan 29 at 0:47
• Hi Noob2 thanks a lot for your reply. :-) Ah now, I see. Since P* ist the risk neutral measure we have the martingale property which is powerful! That helped, thanks. – Wombat Jan 30 at 6:52