Problem 2 - A Course in Financial Calculus

Question

Suppose that \(\left\\{W_{t}^{1}\right\\}_{t \geq 0}\) and \(\left\\{W_{t}^{2}\right\\}_{t \geq 0}\) are independent Brownian motions under \(\mathbb{P}\) and let \(\rho\) be a constant with \(0<\rho<1\). Find coefficients \(\left\\{\alpha_{i j}\right\\}_{i, j=1,2}\) such that $$ \tilde{W}_{t}^{1}=\alpha_{11} W_{t}^{1}+\alpha_{12} W_{t}^{2} $$ and $$ \tilde{W}_{t}^{2}=\alpha_{21} W_{t}^{1}+\alpha_{22} W_{t}^{2} $$ define two standard Brownian motions under \(\mathbb{P}\) with \(\mathbb{E}\left[\tilde{W}_{t}^{1} \tilde{W}_{t}^{2}\right]=\rho t .\) Is your solution unique?

Step-by-Step Solution

Answer

The coefficients \(\alpha_{ij}\) that satisfy the conditions are not unique due to the flexibility in choosing \(\kappa\).

Step 1: Understand the Goal

We need to express two new processes \(\tilde{W}_{t}^{1}\) and \(\tilde{W}_{t}^{2}\) in terms of \(W_{t}^{1}\) and \(W_{t}^{2}\) such that they are standard Brownian motions, and their covariance is \(\rho t\). We are looking for coefficients \(\alpha_{ij}\) that satisfy these conditions.

Step 2: Define the Standard Brownian Motion Conditions

For a process to be a standard Brownian motion, it must have a variance of \(t\). Therefore, \(\text{Var}(\tilde{W}_{t}^{1}) = t\) and \(\text{Var}(\tilde{W}_{t}^{2}) = t\). Also, since \(W_{t}^{1}\) and \(W_{t}^{2}\) are independent, their covariance is zero.

Step 3: Set Up the Variance Equations

Using the expressions for \(\tilde{W}_{t}^{1}\) and \(\tilde{W}_{t}^{2}\), write the variance conditions: 1. \(\text{Var}(\tilde{W}_{t}^{1}) = \alpha_{11}^2 t + \alpha_{12}^2 t = t\).2. \(\text{Var}(\tilde{W}_{t}^{2}) = \alpha_{21}^2 t + \alpha_{22}^2 t = t\).

Step 4: Set Up the Covariance Equation

Write the covariance condition between \(\tilde{W}_{t}^{1}\) and \(\tilde{W}_{t}^{2}\): \(\text{Cov}(\tilde{W}_{t}^{1}, \tilde{W}_{t}^{2}) = (\alpha_{11} \alpha_{21} + \alpha_{12} \alpha_{22}) t = \rho t\).

Step 5: Simplify and Solve Equations

From the variance conditions:1. \(\alpha_{11}^2 + \alpha_{12}^2 = 1\).2. \(\alpha_{21}^2 + \alpha_{22}^2 = 1\).From the covariance condition:3. \(\alpha_{11} \alpha_{21} + \alpha_{12} \alpha_{22} = \rho\).

Step 6: Solve for Coefficients

To solve these, assign \(\alpha_{11} = \sqrt{1 - \kappa^2}\) and \(\alpha_{12} = \kappa\) where \(\kappa\) is a parameter satisfying \(0\leq\kappa\leq1\). From \(\alpha_{11} \alpha_{21} + \alpha_{12} \alpha_{22} = \rho\), find \(\alpha_{21}\) and \(\alpha_{22}\) such that this is satisfied, keeping in mind the third equation.

Step 7: Check Uniqueness

The parameters \(\alpha_{ij}\) are not unique because they depend on the choice of \(\kappa\), which can vary while still satisfying the conditions. Different values of \(\kappa\) yield different valid solutions.

Key Concepts

Standard Brownian Motion

Standard Brownian motion is a fundamental concept in probability theory and statistics. It is a continuous stochastic process that models random motion and is denoted as \(W_t\). For a process to qualify as a standard Brownian motion, it needs to have certain properties:

• It starts at zero: \(W_0 = 0\).
• It has independent increments: The motion between any two time intervals is independent.
• It has normally distributed increments: \(W_t - W_s\) (for \(t>s\)) is normally distributed with mean 0 and variance \(t-s\).
• The paths are continuous: The temporal trajectory of \(W_t\) is a continuous function over time.

These characteristics allow modeling a wide array of phenomena in fields such as finance, physics, and engineering. Understanding these properties is crucial for unraveling the complexities of Brownian motion in practical applications.

Covariance

Covariance is a measure of how two random variables change together. It tells us whether there is a positive or negative relationship between the variables. In mathematical terms, if you have two variables \(X\) and \(Y\), their covariance is given by: \[\text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])] \]Covariance can take any value:

• Positive covariance indicates that as one variable increases, the other tends to increase.
• Negative covariance suggests that as one variable increases, the other tends to decrease.
• A covariance of zero implies no linear relationship between the variables.

In the context of Brownian motions \(\tilde{W}_t^1\) and \(\tilde{W}_t^2\), their covariance is set as \(\rho t\) to capture a pivotal relationship, ensuring that even when they are influenced by the same factors, they vary together in a predictable manner.

Variance

Variance is a statistical measure that represents the degree of spread in a set of values. For a random variable \(X\), the variance provides an expectation of the squared deviation from its mean and is calculated as:\[\text{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2]\]It quantifies the degree to which \(X\) varies, with higher variance indicating that data points are more spread out from the mean.

• A variance close to zero implies data points are very close to the mean.
• A larger variance indicates data points are dispersed over a wider range of values.

For standard Brownian motion \(\tilde{W}_t^1\) and \(\tilde{W}_t^2\), the variance is particularly crucial, typically setting it to \(t\) for these processes. This uniform variance across the timeline ensures the characterization of their movements remains consistent and predictable.

Independence

Independence is a critical concept in probability, indicating that the occurrence of one event does not affect the likelihood of another. For random variables \(X\) and \(Y\), they are independent if:\[\text{P}(X \cap Y) = \text{P}(X)\text{P}(Y)\]For Brownian motions \(W_t^1\) and \(W_t^2\) mentioned in the exercise, independence is vital:

• Their respective paths don't influence each other.
• Their joint behavior does not do anything to alter the expected product of their individual probabilities.

Understanding independence helps streamline calculations and simplify models. In this context, when forming new processes \(\tilde{W}_t^1\) and \(\tilde{W}_t^2\), we want to maintain their independence initially, only correlating them to the degree defined by \(\rho\). This yields the intended mathematical and practical outcomes in various applications.