Distribution of a conditional expectation

Question

I am reading a book on financial mathematics and a Theorem gives a price formula for a Call Option:

$$ \begin{aligned} \pi_{\text {call }}(t) &=P(t, S) q(t, S, \mathcal{I})-K P(t, T) q(t, T, \mathcal{I}) \\ \end{aligned} $$ where $\mathcal{I}=(A(S-T)+\log K, \infty)$, and $q(t, S, d y)$ and $q(t, T, d y)$ denote the $\mathcal{F}_{t}$ conditional distributions of the real-valued random variable $Y=-B(S-T)^{\top} X(T)$ under the $S$ - and $T$-forward measure, respectively.

In the last step of the proof, he comes up with the formula

$$ \pi(t)=P(t, S) \mathbb{Q}^{S}\left[E \mid \mathcal{F}_{t}\right]-K P(t, T) \mathbb{Q}^{T}\left[E \mid \mathcal{F}_{t}\right] $$ for the exercise event $E=\left\{-B(S-T)^{\top} X(T)>A(S-T)+\log K\right\}$.

I am confused, because the first formula looks like a real value (the measure $q(t,T,\cdot)$ evaluated on the set $\mathcal{I}$) and the second formula looks like a random variable to me ( $\mathbb{Q}^{S}\left[E \mid \mathcal{F}_{t}\right]$ is a random variable).

How do $q(t, T, \mathcal{I})$ and $\mathbb{Q}^{T}\left[E \mid \mathcal{F}_{t}\right]$ coincide? What exactly is a $\mathcal{F}_{t}$ conditional distribution of the real-valued random variable $Y=-B(S-T)^{\top} X(T)$ under $T$-forward measure?

We can rewrite $E$

$$E=\{Y> A(S-T)+\log K\}=\{Y\in (A(S-T)+\log K,\infty)\}=\{Y\in\mathcal{I}\}$$

and the conditional distribution

$$\mathbb{Q}^{T}\left[E \mid \mathcal{F}_{t}\right]=\mathbb{Q}^{T}\left[Y\in\mathcal{I} \mid \mathcal{F}_{t}\right]=\mathbb{Q}^{T}\left[\cdot \mid \mathcal{F}_{t}\right](Y^{-1}(\mathcal{I}))$$

I guess $\mathbb{Q}^{T}\left[\cdot \mid \mathcal{F}_{t}\right]$ can be a probability measure, let's say $Q_1$, in some circumstances? And then we would have the distribution of $Y$ under the measure $Q_1$ ?

Answer 1

I found a footnote on this question. One distinguishes between conditional probability and conditional distribution.

Recall that for every $\mathbb{R}^{n}$-valued random variable $Z$ and sub- $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$, there exists a regular conditional distribution $\mu(\omega, d z)$ of $Z$ given $\mathcal{G}$. That is, $\mu(\omega, \cdot)$ is a probability measure on $\mathbb{R}^{n}$ for every $\omega \in \Omega, \omega \mapsto \mu(\omega, E)$ is $\mathcal{G}$-measurable for every $E \in \mathcal{B}\left(\mathbb{R}^{n}\right)$, and $\mathbb{E}[f(Z) \mid \mathcal{G}](\omega)=$ $\int_{\mathbb{R}^{n}} f(z) \mu(\omega, d z)$ for all bounded measurable functions $f$, for a.e. $\omega$. See e.g. [Sect. 44, Bauer,H.:Wahrscheinlichkeitstheorie, 5thedn. deGruyterLehrbuch, ISBN 3-11-017236-4]

Especially in the author's notation, we find

$$ \begin{align} \mathbb{Q}^S[E|\mathcal{F}_t](\cdot)&=\mathbb{Q}^S[Y^{-1}(\mathcal{I})|\mathcal{F}_t](\cdot)\\ &=E_{\mathbb{Q}^S}[1_\mathcal{I}(Y)|\mathcal{F}_t](\cdot)\\ &=\int_\mathbb{R} 1_\mathcal{I}(y)\mu(\cdot,dy)\\ &=\int_\mathbb{R} 1_\mathcal{I}(y)\mathbb{Q}^S[Y^{-1}(dy)|\mathcal{F}_t](\cdot)\\ &=\int_\mathcal{I} \mathbb{Q}^S[Y^{-1}(dy)|\mathcal{F}_t](\cdot)\\ \end{align}$$

where

$$q(t,S,dy)=\mathbb{Q}^S[Y^{-1}(dy)|\mathcal{F}_t]$$

and

$$q(t,S,dy)(\omega)=\mathbb{Q}^S[Y^{-1}(dy)|\mathcal{F}_t](\omega)$$