Problem 22 - A Course in Financial Calculus

Question

Suppose that \(\left\\{S_{t}\right\\}_{t \geq 0}\) is a geometric Brownian motion under \(\mathbb{P}\). Let \(0 \leq t_{1} \leq t_{2} \leq\) \(\cdots \leq t_{n}\) be fixed times and define $$ G_{n}=\left(\prod_{i=1}^{n} S_{t_{i}}\right)^{1 / n} $$ Show that \(G_{n}\) has a lognormal distribution under \(\mathbb{P}\).

Step-by-Step Solution

Answer

\(G_n\) is lognormally distributed because \( \log(G_n) \) is normally distributed.

Step 1: Understanding Geometric Brownian Motion

A geometric Brownian motion (GBM) is characterized by the stochastic differential equation \( dS_t = S_t(\mu \, dt + \sigma \, dW_t) \), where \( \mu \) is the drift, \( \sigma \) is the volatility, and \( W_t \) is a standard Brownian motion under the probability measure \( \mathbb{P} \). This means that \( S_t \) follows a lognormal distribution, and \( \log(S_t) \) is normally distributed with mean \( \log(S_0) + (\mu - \sigma^2/2)t \) and variance \( \sigma^2 t \).

Step 2: Express the Product of GBMs

The given expression for \( G_n \) involves the product \( \prod_{i=1}^{n} S_{t_{i}} \). Each \( S_{t_i} = S_0 \exp \left((\mu - \sigma^2/2)t_i + \sigma W_{t_i} \right) \) due to the solution of GBM. Substitute this expression into the product to get \( \prod_{i=1}^{n} S_{t_i} = S_0^n \exp \left( \sum_{i=1}^{n} (\mu - \sigma^2/2)t_i + \sigma W_{t_i} \right) \).

Step 3: Simplifying the Geometric Mean Expression

Now, simplify the expression for \( G_n \): \[ G_n = \left( S_0^n \exp \left( \sum_{i=1}^{n} (\mu - \sigma^2/2)t_i + \sigma W_{t_i} \right) \right)^{1/n} \].This simplifies to:\[ G_n = S_0 \exp \left( \frac{1}{n} \sum_{i=1}^{n} (\mu - \sigma^2/2)t_i + \sigma W_{t_i} \right). \]

Step 4: Determine the Distribution of the Exponent

The term \( \frac{1}{n} \sum_{i=1}^{n} ((\mu - \sigma^2/2)t_i + \sigma W_{t_i}) \) is a sum of independent normally distributed random variables, which is itself normally distributed by the properties of normal distribution. If each \( W_{t_i} \) is an independent increment, the variance term of the sum will be \( \sigma^2 \sum_{i=1}^{n} t_i / n^2 \).

Step 5: Conclude with the Lognormal Distribution

Since \( G_n \) is expressed as the exponential of a normally distributed random variable, \( G_n \) itself has a lognormal distribution. Lognormal distributions arise as exponentiated versions of normally distributed variables, fitting the characteristics of \( G_n \). Therefore, under \( \mathbb{P} \), \( G_n \) indeed follows a lognormal distribution.

Key Concepts

Lognormal Distribution

A lognormal distribution is a probability distribution whose logarithm is normally distributed. This means that if you take the natural logarithm of a variable following a lognormal distribution, you will obtain a normal distribution.
The lognormal distribution is characterized particularly by its asymmetry; it is positively skewed. In finance and other fields, quantities that cannot take a negative value, such as stock prices or the aforementioned geometric Brownian motion, often align with this model.

• The product of positive-valued variables, as in the case of the geometric mean of prices, leads naturally to this distribution.
• Lognormal distributions are used because they ensure that the resulting random variable remains positive, which makes them particularly suitable for modeling stock prices frequently assumed to follow geometric Brownian motion.

Understanding the properties of lognormal distributions helps in predicting and describing phenomena across various fields effectively.

Stochastic Differential Equation

Stochastic differential equations (SDEs) are used to model the dynamics of systems that are subject to random influences, making them essential in understanding processes like geometric Brownian motion. In the context of the exercise, the SDE governing the dynamical system is given by\[ dS_t = S_t (\mu \, dt + \sigma \, dW_t) \]where:

• \( S_t \) is the stock price at time \( t \)
• \( \mu \) represents the drift term or expected rate of return
• \( \sigma \) is the volatility representing the variation or "noisiness" in the process
• \( dW_t \) is the increment of a standard Brownian motion, which introduces the randomness into the equation

The stochastic differential equation thus incorporates both deterministic aspects (through the drift term) and stochastic aspects (through the volatility term).
This equation describes the random nature of the process and predicts probabilities of future prices, allowing us to derive significant properties like the lognormal distribution viewed here.

Normal Distribution

The normal distribution, often referred to as the bell curve due to its shape, forms the basis for many statistical models.
When variables or processes are said to be normally distributed, it implies that outcomes tend to occur around the mean (the peak of the bell curve), with decreasing frequency as you move away from it.

• This distribution is symmetric around its mean, which allows many natural phenomena and random processes to be modeled using it, especially due to the Central Limit Theorem.
• With a standard deviation that measures the spread or "width" of the curve, the normal distribution is a cornerstone in probability theory.

In the case of geometric Brownian motion, the logarithm of the stock price, \( \log(S_t) \), follows a normal distribution.
This is pivotal because it informs the prediction and description of stock price behaviors over time, providing a framework to understand movements and fluctuations within financial markets.

Probability Measure

Probability measure is a fundamental concept that assigns a probability to each event in a given space. In mathematical finance, the choice of probability measure is crucial as it affects the modeling and outcomes of stochastic processes.
Under the standard probability measure \(\mathbb{P}\), events are assigned probabilities based on "real-world" likelihoods.

• This measure ensures the entire probability of all possible outcomes sums to 1, making it a complete space for probabilistic analysis.
• In the context of stock prices and financial models, choosing the correct probability measure allows for more accurate predictions and potentially realistic simulations of asset behavior.

Understanding probability measures helps forecast and quantify risks and returns, guiding decision-making in complex stochastic environments like the one modeled here with geometric Brownian motion.