(State price density process). Show that the risk-neutral...

(State price density process). Show that the risk-neutral pricing formula $(5.2 .30)$ may be rewritten as $$ D(t) Z(t) V(t)=\mathbb{E}[D(T) Z(T) V(T) \mid \mathcal{F}(t)] $$ Here $Z(t)$ is the Radon-Nikodým derivative process (5.2.11) when the market price of risk process $\Theta(t)$ is given by (5.2.21) and the conditional expectation on the right-hand side of (5.9.1) is taken under the actual probability measure 252 5 Risk-Neutral Pricing $\mathbb{P}$, not the risk-neutral measure $\tilde{\mathbb{P}}$. In particular, if for some $A \in \mathcal{F}(T)$ a derivative security pays off $\mathbb{I}_A$ (i.e., pays 1 if $A$ occurs and 0 if $A$ does not occur), then the value of this derivative security at time zero is $\mathbb{E}\left[D(T) Z(T) \mathbb{I}_A\right]$. The process $D(t) Z(t)$ appearing in (5.9.1) is called the state price density process.

(State price density process). Show that the risk-neutral pricing formula $(5.2 .30)$ may be rewritten as
$$
D(t) Z(t) V(t)=\mathbb{E}[D(T) Z(T) V(T) \mid \mathcal{F}(t)]
$$

Here $Z(t)$ is the Radon-Nikodým derivative process (5.2.11) when the market price of risk process $\Theta(t)$ is given by (5.2.21) and the conditional expectation on the right-hand side of (5.9.1) is taken under the actual probability measure
252
5 Risk-Neutral Pricing
$\mathbb{P}$, not the risk-neutral measure $\tilde{\mathbb{P}}$. In particular, if for some $A \in \mathcal{F}(T)$ a derivative security pays off $\mathbb{I}_A$ (i.e., pays 1 if $A$ occurs and 0 if $A$ does not occur), then the value of this derivative security at time zero is $\mathbb{E}\left[D(T) Z(T) \mathbb{I}_A\right]$. The process $D(t) Z(t)$ appearing in (5.9.1) is called the state price density process.

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