Problem 16 - A Course in Financial Calculus

Question

Let \(\left\\{\mathcal{F}_{t}\right\\}_{0 \leq t \leq T}\) denote the natural filtration associated to a standard \(\mathbb{P}\)-Brownian motion, \(\left\\{W_{t}\right\\}_{0 \leq t \leq T}\). The result of Lemma 3.4.6.3 can be rewritten as $$ \mathbb{E}\left[\exp \left(\sigma W_{t}-\frac{1}{2} \sigma^{2} t\right) ; A\right]=\exp \left(\sigma W_{s}-\frac{1}{2} \sigma^{2} s\right) \mathbf{1}_{A}, \quad \text { for all } A \in \mathcal{F}_{S} $$ Use differentiation under the integral sign to provide another proof that \(\left\\{W_{t}^{2}-t\right\\}_{t \geq 0}\) is a \(\left(\mathbb{P},\left\\{\mathcal{F}_{t}\right\\}_{t \geq 0}\right)\)-martingale and show that the following are also \(\left(\mathbb{P},\left\\{\mathcal{F}_{t}\right\\}_{t \geq 0}\right)\) martingales: (a) \(W_{t}^{3}-3 t W_{t}\) (b) \(W_{t}^{4}-6 t W_{t}^{2}+3 t^{2}\)

Step-by-Step Solution

Answer

(a) \(W_t^3 - 3tW_t\) and (b) \(W_t^4 - 6tW_t^2 + 3t^2\) are martingales (\(\mathbb{P}, \mathcal{F}_t\)).

Step 1: Understand the Problem

The problem asks us to show that certain processes involving Brownian motion are martingales under a given filtration. Specifically, for parts (a) and (b), we need to show that the processes \(W_t^3 - 3tW_t\) and \(W_t^4 - 6tW_t^2 + 3t^2\) are martingales.

Step 2: Recall Martingale Property

To show that a process \(X_t\) is a martingale with respect to a filtration \(\{\mathcal{F}_t\}\), under a probability measure \(\mathbb{P}\), we need to show that \(\mathbb{E}[X_t | \mathcal{F}_s] = X_s\) for all \(0 \leq s \leq t\), and that \(X_t\) is \(\mathcal{F}_t\)-adapted.

Step 3: Differentiate Process for (a)

For the process \(W_t^3 - 3tW_t\), first find its differential using Itô's Lemma. For \(X_t = W_t^3\), we have \(d(W_t^3) = 3W_t^2 dW_t + 3W_t dt\). Therefore, for \(Y_t = W_t^3 - 3tW_t\), \[d Y_t = (3W_t^2 dW_t + 3W_t dt) - 3(W_t dt + t dW_t) = 3W_t^2 dW_t.\]Since \(dY_t\) has zero drift and is \(\mathcal{F}_t\)-adapted, \(Y_t\) is a martingale.

Step 4: Differentiate Process for (b)

For \(Z_t = W_t^4 - 6tW_t^2 + 3t^2\), apply Itô's Lemma. For \(X_t = W_t^4\), \(d(W_t^4)= 4W_t^3 dW_t + 6W_t dt\). Using this, \[d Z_t = (4W_t^3 dW_t + 6W_t dt) - 6(W_t^2 dt + 2tW_t dW_t + t dt)\].Simplifying gives \[d Z_t = (4W_t^3 dW_t + 6W_t dt) - 6W_t^2 dt - 12tW_t dW_t - 6t dt = 4W_t^3 dW_t - 12t W_t dW_t.\]Hence, the differential has zero drift, showing that \(Z_t\) is a martingale.

Key Concepts

Filtration

In the context of probability theory and stochastic processes, a filtration is essentially a family of sigma-algebras, \(\{\mathcal{F}_t\}_{t \geq 0}\), that represents the information available at each time point. It's like a timeline of information accumulation, where each \(\mathcal{F}_t\) is a set of events whose occurrence or non-occurrence is known by time \t\. Filtrations are crucial in defining adapted processes, which ensures that the process's value at any time depends only on the information up to that time. This property is necessary for processes like martingales, where future values cannot be "predicted" using past information.

Itô's Lemma

Itô's Lemma is a fundamental result in stochastic calculus. It allows us to find the differential of a function that depends on a Brownian motion and potentially time. Itô's Lemma is essentially the stochastic counterpart to the chain rule in classical calculus.

Given a smooth function \(f(t, W_t)\), its differential is characterized by the following rule:

• Determine partial derivatives: \ \frac{\partial f}{\partial t}, \frac{\partial f}{\partial x}, \text{and} \frac{\partial^2 f}{\partial x^2} \.
• Apply Itô's formula: \ df = \left(\frac{\partial f}{\partial t} dt + \frac{1}{2}\frac{\partial^2 f}{\partial x^2} dW^2\right) + \frac{\partial f}{\partial x} dW\.

In practice, this lemma accounts for the fact that Brownian motion has a quadratic variation of \(t\) rather than zero, modifying deterministic calculus rules.

Martingale Property

A stochastic process is a martingale when its conditional expectation, given all past information, is equal to its current value. More formally, a process \(\{X_t\}\) is a martingale with respect to a filtration \ \{\mathcal{F}_t\} \ and a probability measure \ \mathbb{P} \, if for all \(0 \leq s \leq t\), \ \mathbb{E}[X_t | \mathcal{F}_s] = X_s \ and \(X_t\) is \(\mathcal{F}_t\)-adapted.

This property makes martingales incredibly useful in financial mathematics and other fields. Martingales model "fair" games, where there is no advantage to be gained with extra information, and this property is pivotal in risk-neutral valuation and pricing of derivative securities.

Probability Measure

A probability measure is a mathematical function \(\mathbb{P}\) that assigns a probability to each event in a sigma-algebra, \(\mathcal{F}\). It ensures that the probability of the whole sample space is 1, that the probability of any event is between 0 and 1, and that it is countably additive.

Consider the natural filtration associated with a standard Brownian motion under the probability measure \(\mathbb{P}\). It provides the framework to discuss random processes with respect to real-world or "risk-neutral" scenarios. Transforming probability measures, such as using Girsanov's Theorem, is a key tool in mathematical finance, adjusting the original measure to simplify problems or model different conditions.