Stochastic Models and Processes · 19. November 2023

Conditional Expectation in Simple Terms

Before delving into a practical financial scenario, let's quickly revisit what a \( \sigma \)-algebra is. A \( \sigma \)-algebra, denoted as \( \mathcal{F} \) or \( \mathcal{G} \), is a collection of subsets of a given set (typically the set of all possible outcomes, \( \Omega \)) that:

Includes the universal set \( \Omega \).
Is closed under complementation.
Is closed under countable unions.

This structure is essential for assigning probabilities to events.

In quantitative finance, an interesting property arises when two integrable random variables \( X \) and \( Y \) exist, and \( X \) is \( \mathcal{G} \)-measurable. The property is expressed as:

\( \mathbb{E}[XY | \mathcal{G}] = X \cdot \mathbb{E}[Y | \mathcal{G}] \)

This identity simplifies calculations involving conditional expectations and is widely used in risk management.

Application in Hedging:

Consider a hedging scenario in a financial market:

\( \Omega \): All possible market scenarios.
\( \mathcal{F} \): A \( \sigma \)-algebra representing all events in the market.
\( P \): Probability measure on these events.
\( \mathcal{G} \): A sub-\( \sigma \)-algebra of \( \mathcal{F} \), capturing information available up to the last trading day.
\( X \): A \( \mathcal{G} \)-measurable random variable representing a position in a risk-free asset.
\( Y \): A variable representing the market return of a risky asset.

The goal is to calculate \( \mathbb{E}[XY | \mathcal{G}] \), the expected value of your portfolio's return given information available on the last trading day.

Since \( X \) is constant within \( \mathcal{G} \), the property simplifies the expectation to:

\( \mathbb{E}[XY | \mathcal{G}] = X \cdot \mathbb{E}[Y | \mathcal{G}] \)

Here, \( \mathbb{E}[Y | \mathcal{G}] \) represents the expected return of the risky asset, enabling us to focus only on the conditional expectation of \( Y \).

This approach is fundamental in risk management and derivative pricing, as it treats known, fixed assets and uncertain returns efficiently.

Key Takeaway: Treating a \( \mathcal{G} \)-measurable function as constant simplifies integration and expectation calculations, making it easier to model hedging strategies.