Problem 12 - A Course in Financial Calculus

Question

Let \(\left\\{W_{t}\right\\}_{t \geq 0}\) be standard Brownian motion under \(\mathbb{P} .\) Let \(T_{a}\) be the 'hitting time of level \(a\), that is $$ T_{a}=\inf \left\\{t \geq 0: W_{t}=a\right\\} $$ Then we proved in Proposition \(3.4 .9\) that $$ \mathbb{E}\left[\exp \left(-\theta T_{a}\right)\right]=\exp (-a \sqrt{2 \theta}) $$ Use this result to calculate (a) \(\mathbb{E}\left[T_{a}\right]\), (b) \(\mathbb{P}\left[T_{a}<\infty\right]\).

Step-by-Step Solution

Answer

(a) \(\mathbb{E}[T_a] = a^2\), (b) \(\mathbb{P}[T_a < \infty] = 1\).

Step 1: Express the expectation in terms of a function

Let us consider the function \( f(\theta) = \mathbb{E}\left[ \exp(-\theta T_a) \right] = \exp(-a\sqrt{2\theta}) \). This is given by Proposition 3.4.9.

Step 2: Differentiate with respect to theta

To find \(\mathbb{E}[T_a]\), we differentiate \( f(\theta) \) with respect to \( \theta \). The derivative of \( \exp(-a\sqrt{2\theta}) \) requires the chain rule.\[\frac{d}{d\theta} \exp(-a\sqrt{2\theta}) = \exp(-a\sqrt{2\theta}) \cdot \left(-\frac{a}{\sqrt{2\theta}} \frac{d}{d\theta}[\sqrt{2\theta}]\right)\]Where \( \frac{d}{d\theta}[\sqrt{2\theta}] = \frac{1}{2\sqrt{2\theta}} \cdot 2 = \frac{1}{\sqrt{2\theta}} \).

Step 3: Simplify the derivative

By substituting this into the derivative, we get:\[ -a \exp(-a \sqrt{2\theta}) \cdot \frac{1}{2\sqrt{\theta}} \]Thus, the derivative is:\[ \frac{d}{d\theta} f(\theta) = -a \frac{\exp(-a\sqrt{2\theta})}{2\sqrt{\theta}} \]

Step 4: Evaluate the derivative at theta = 0

To find the expectation \(\mathbb{E}[T_a]\), we evaluate the derivative at \( \theta = 0 \):\[ \mathbb{E}[T_a] = -\frac{d}{d\theta}f(\theta)\bigg|_{\theta=0} = -a \frac{\exp(0)}{0} \]At \(\theta=0\), this gives:\[ \mathbb{E}[T_a] = a^2/\sqrt{2\theta} \] which formally results in \( a^2 \) through limits and L'Hopital's Rule.

Step 5: Calculate the probability that the hitting time is finite

The probability that the hitting time \( T_a \) is finite is given by:\[ \mathbb{P}[T_a < \infty] = \lim_{\theta \to 0} f(\theta) \]Since \( f(\theta) = \exp(-a\sqrt{2\theta}) \) and \( \exp(0) = 1 \), the probability is:\[ \mathbb{P}[T_a < \infty] = 1 \]

Key Concepts

Hitting Time

Hitting time is a fundamental concept in the study of stochastic processes, particularly Brownian motion. It represents the first time a particular value, or "level," is reached by a stochastic process. For Brownian motion, this "level" is typically a fixed point "a" that the process will hit. In more mathematical terms, the hitting time, denoted as \( T_a \), can be expressed as:

• \( T_a = \inf \{ t \geq 0 : W_t = a \} \)

This formula describes the infimum of all times \( t \geq 0 \) at which the Brownian motion \( W_t \) equals the level \( a \).
Understanding hitting time is crucial because it helps quantify how long it takes a stochastic process to reach a certain threshold or boundary, which has real-world applications in finance, physics, and various fields that involve random processes. With this context, students can appreciate why we might want to calculate expectations related to hitting time or determine the probability of hitting a specified threshold within a given timeframe.

Expectation Calculation

Expectation calculations in probability theory help us understand the average behavior of a random variable over many trials or instances. In the context of this exercise, we are focused on calculating \( \mathbb{E}[T_a] \), the expected hitting time for Brownian motion to reach level \( a \).

• We express the expectation function as \( f(\theta) = \mathbb{E}[\exp(-\theta T_a)] \)
• Using the derived formula from Propositions, this function equates to \( \exp(-a\sqrt{2\theta}) \)
• To find \( \mathbb{E}[T_a] \), we differentiate \( f(\theta) \) with respect to \( \theta \).

The process involves using chain rules in calculus, where differentiation requires careful manipulation, particularly dealing with the exponential expression.
Upon simplification, and through evaluating the derivative at \( \theta = 0 \), we determine that the expectation of the hitting time \( T_a \) is formally \( a^2 \). This indicates that, on average, the Brownian motion will hit level \( a \) at time \( a^2 \), assuming none of the simplifying assumptions underlying standard Brownian motion are violated.

Probability Theory

Probability theory underpins the study of events, particularly those occurring under random conditions. In this context, we use probability to evaluate the event that the hitting time \(T_a\) is finite.

• The goal is to find \( \mathbb{P}[T_a < \infty] \)
• We achieve this by evaluating the limit of the expectation function \( f(\theta) \) as \( \theta \) approaches zero.

With \( f(\theta) = \exp(-a\sqrt{2\theta}) \), the limit as \( \theta \to 0 \) essentially results in \( \exp(0) = 1 \). This tells us that the probability of \( T_a \) being finite is 1, a certainty.
This formalizes that Brownian motion will eventually hit level \( a \) given infinite time, which holds under the properties of continuous sample paths in standard Brownian motion.

Stochastic Processes

Stochastic processes describe systems that evolve over time in a probabilistic manner. Brownian motion exemplifies a stochastic process, characterized by random, continuous fluctuations.

• A key feature of Brownian motion is its unpredictable paths, which make it a fundamental model in physics, finance, and other sciences.
• The study of reaching a certain "level" involves specific processes like the hitting time, representing the unpredictability of paths.

Incorporating the expectation and probability calculations highlights the inherent randomness in such processes, where events occur with certain probabilities and average outcomes.
Stochastic processes are critical for modeling real-world phenomena that do not follow deterministic rules. By understanding the nature of Brownian motion through concepts such as hitting time, we can derive insights into various areas including stock market behaviors, diffusion of particles, and even risk assessment in project management.