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Let :

$P$ and $Q$ two equivalent probability measures and $Z$ the Radon Nikodym derivative : $Z = \frac{dQ}{dP}$ .

$Z(t)$ is the expectation of the Radon Nikodym derivative $Z(t) = E^P \left[Z|F(t) \right]$, and $Y$ is an $F(t)-$ measurable process.

Following this definition we know that $E^Q(Y) = E^P(YZ)$. And we can prove that $E^Q(Y) = E^P(YZ_t)$.

My question : In the equality (Equality from Shreve - Stochastic Calculus for Finance II) below :

$$E_Q\left[1_A\frac{1}{Z(s)}E^P\left[ Y Z(t) | F(s) \right]\right] = E_P\left[1_AE^P\left[ Y Z(t) | F(s) \right]\right]$$

We apply $E^Q(Y) = E^P(YZ_s)$. Why choosing $Z_s$ and not $Z_t$ or others ? I mean how do we know the indice of $Z$ when doing a change of measure as we just did? $s$? $t$? ..

here's the attached context of my question :

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darkcpp
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user30614
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1 Answers1

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Make sure to look up the measure theoretic version of the Bayes theorem.

Define $Z = \frac{ d \mathbb{Q} }{ d \mathbb{P}}$, let $H$ be an arbitrary event we are interested in and let us use the notation $\mathbb{E}^{\mathbb{Q}}_{H}[X] = \mathbb{E}^{\mathbb{Q}}[X | H]$.

Bayes's theorem tells us that

$\mathbb{E}^{\mathbb{Q}}_{H}\left[ X\right]= \frac{\mathbb{E}^{\mathbb{P}}_{H}\left[ X Z \right]}{\mathbb{E}^{\mathbb{P}}_{H}\left[Z \right]}$.

Let $X$ be a European pay-off maturing at some time $t$ and let us consider the risk neutral price of this payoff at some time $s \in [0,t]$. Let $H = \mathcal{F}_s = \sigma \left\lbrace X_u : u \leq s \right\rbrace $. Then,

$\mathbb{E}^{\mathbb{Q}}_{\mathcal{F}_s}\left[ X_t \right]= \frac{\mathbb{E}^{\mathbb{P}}_{\mathcal{F}_s}\left[ X_t Z_t \right]}{\mathbb{E}^{\mathbb{P}}_{\mathcal{F}_s}\left[Z_t \right]}$.

We know that $Z$ is a $\mathbb{P}$ martingale, therefore $Z_s = \mathbb{E}^{\mathbb{P}}_{\mathcal{F}_s}\left[Z_t \right]$ and we get that

$\mathbb{E}^{\mathbb{Q}}_{\mathcal{F}_s}\left[ X_t \right]= \frac{\mathbb{E}^{\mathbb{P}}_{\mathcal{F}_s}\left[ X_t Z_t \right]}{Z_s}$.

Often we take $s=0$, i.e., we are interested in the price of the derivative today.

Take a look at how the forward measure is used when calculating European options on zero coupon bonds - the different subscripts will make sense.

AXH
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