Problem 10 - A Course in Financial Calculus

Question

Brownian motion is not going to be adequate as a stock market model. First, it has constant mean, whereas the stock of a company usually grows at some rate, if only due to inflation. Moreover, it may be too 'noisy' (that is the variance of the increments may be bigger than those observed for the stock) or not noisy enough. We can scale to change the 'noisiness' and we can artificially introduce a drift, but this still won't be a good model. Here is one reason why. Suppose that \(\left\\{W_{t}\right\\}_{t \geq 0}\) is standard Brownian motion under \(\mathbb{P} .\) Define a new process \(\left\\{S_{t}\right\\}_{I \geq 0}\) by \(S_{t}=\mu t+\sigma W_{t}\) where \(\sigma>0\) and \(\mu \in \mathbb{R}\) are constants. Show that for all values of \(\sigma>0, \mu \in \mathbb{R}\) and \(T>0\) there is a positive probability that \(S_{T}\) is negative.

Step-by-Step Solution

Answer

There is a positive probability that \(S_T\) is negative because the normal distribution has a nonzero probability density for all real numbers.

Step 1: Understanding the Process

We are given a process \{S_t\}_{t \ge 0} defined as the linear transformation of a standard Brownian motion \{W_{t}\}_{t \geq 0}: S_t = \mu t + \sigma W_t. Here, \(\mu\) is the drift term representing a constant rate of change, and \(\sigma\) is the volatility term scaling the noise.

Step 2: Finding the Distribution of $S_T$

For any fixed time \(T > 0\), the process \(S_T\) can be written as \(S_T = \mu T + \sigma W_T\). Given that \(W_T\) is a normal distribution with mean 0 and variance \(T\), \(\sigma W_T\) has a normal distribution with mean 0 and variance \(\sigma^2 T\).

Step 3: Calculating the Probability of $S_T$ Being Negative

To find the probability that \(S_T < 0\), we note that \(S_T = \mu T + \sigma W_T\). The new random variable \(\sigma W_T\) follows a normal distribution with mean 0 and variance \(\sigma^2 T\), allowing \(S_T\) to be normally distributed with mean \(\mu T\) and variance \(\sigma^2 T\). The probability that a standard normal variable is less than \(-\frac{\mu T}{\sigma \sqrt{T}}\) must be calculated.

Step 4: Finding the Standard Normal Cumulative Distribution Function (CDF) Value

The probability we need is \(\mathbb{P}(S_T < 0) = \mathbb{P}(\mu T + \sigma W_T < 0) = \mathbb{P}(\sigma W_T < -\mu T) = \mathbb{P}(W_T < -\frac{\mu T}{\sigma})\). Simplifying gives \(\mathbb{P}\left(Z < -\frac{\mu \sqrt{T}}{\sigma}\right)\), where \(Z\) is a standard normal variable. Since the standard normal distribution has tails reaching infinity, this probability is positive for any finite negative argument.

Key Concepts

Brownian Motion

Brownian motion is a fundamental concept in stochastic processes, describing the random, continuous movement of particles suspended in a fluid (a liquid or a gas). It is named after Robert Brown, who first observed it in 1827. Formally, a Brownian motion is a continuous-time stochastic process with these key properties:

• Starts at zero: The process begins at zero, meaning at time zero, the position is zero.
• Independent increments: The changes in the process over non-overlapping intervals are independent of each other.
• Normally distributed increments: For any time interval, the change in values follows a normal distribution.
• Continuous paths: The trajectory of the motion is continuous over time.

Mathematically, if we denote Brownian motion by \(W_t\), at any time \(t\), \(W_t\) is normally distributed with mean 0 and variance \(t\). This property of variance growing with time makes Brownian motion an excellent tool to model random movements, such as the chaotic path of a small particle in a fluid. In the context of financial markets, Brownian motion is often adapted to model stock prices or exchange rates due to its inherent randomness and unpredictability.

Stock Market Models

Stock market models aim to represent the behavior of stock prices using mathematical frameworks. An essential model is the geometric Brownian motion (GBM), which is often used due to its simplicity and realistic properties. Unlike standard Brownian motion, which can dip into negative values, the GBM ensures stock prices remain positive.

A key reason for transforming Brownian motion into GBM for stocks is to incorporate the factors such as:

• **Exponential growth**: Stocks are typically expected to grow steadily over time, reflecting economic factors like inflation and business growth.
• **Volatility**: This element captures large swings in stock prices, reflecting market uncertainty.

In a GBM setting, the stock price \(S_t\) at time \(t\) is modeled as being influenced by a deterministic "drift" term (exponential growth) and a stochastic term (random volatility). Despite being underpinned by Brownian motion, the model takes this further to better reflect the real-world attributes and behaviors of stock prices, making it a favorite among financial analysts and traders.

Probability Distributions

Probability distributions describe how the values of a random variable are spread or distributed. In the context of stochastic processes like Brownian motion or stock market models, probability distributions are crucial to understanding how likely it is for variables to take on particular values.

For a standard Brownian motion, the increments are normally distributed, meaning that for any interval of time, the change in the process has a normal distribution with specific mean and variance. This means:

• **Normal Distribution**: Characterized by its bell-shaped curve, where most of the data falls around the mean. It is symmetrical, with 68% of data within one standard deviation up or down from the mean.
• **Variance and Standard Deviation**: These measures indicate how much the values can deviate from the mean. Higher variance means more widespread data; more applicable in more volatile markets.

This understanding plays a crucial role in finance as it aids in predicting potential changes in financial variables such as stock prices, based on past dynamics and inherent randomness.

Drift and Volatility

Drift and volatility are two fundamental aspects that define the path of a stock price in mathematical models.

• **Drift**: This term refers to the expected rate of return or the average change in a stock's price over time. It represents the deterministic part of the stock price model, implying consistent growth or decline. For instance, in the equation \(S_t = \mu t + \sigma W_t\), \(\mu\) is the drift coefficient.
• **Volatility**: Unlike drift, volatility captures the randomness or the fluctuations of stock prices. It represents the uncertainty and risk involved in the stock's movements. In the above equation, \(\sigma\) (sigma) signifies the volatility factor that scales the noise of the Brownian motion.

Together, drift and volatility help in crafting realistic financial models by balancing expected returns with the inherent unpredictability of the market. Understanding and calculating these components enable investors to make informed decisions and model the risks related with their investments.