Problem 3 - A Course in Financial Calculus

Question

Suppose that the price of a certain asset has the lognormal distribution. That is \(\log \left(S_{T} / S_{0}\right)\) is normally distributed with mean \(v\) and variance \(\sigma^{2} .\) Calculate \(\mathbb{E}\left[S_{T}\right] .\)

Step-by-Step Solution

Answer

The expected value \( \mathbb{E}[S_T] = S_0 \cdot e^{v + \sigma^2/2} \).

Step 1: Understand the Problem

We need to calculate the expected value of the final asset price, denoted as \( \mathbb{E}[S_T] \), given that the logarithm of the price ratio \( \log(S_T / S_0) \) follows a normal distribution with mean \( v \) and variance \( \sigma^2 \).

Step 2: Express the Logarithm

Since \( \log(S_T / S_0) \sim \mathcal{N}(v, \sigma^2) \), we can express this as \( \log(S_T) - \log(S_0) \sim \mathcal{N}(v, \sigma^2) \). Therefore, \( \log(S_T) \sim \mathcal{N}(\log(S_0) + v, \sigma^2) \).

Step 3: Recall the Property of Exponential

Since \( \log(S_T) \sim \mathcal{N}(\log(S_0) + v, \sigma^2) \), the distribution of \( S_T \) is lognormal. For a lognormal variable \( X = e^Y \), where \( Y \sim \mathcal{N}(\mu, \sigma^2) \), the expected value is given by \( \mathbb{E}[X] = e^{\mu + \sigma^2/2} \).

Step 4: Apply Lognormal Expectation Formula

In this scenario, \( \mu = \log(S_0) + v \). Thus, \( \mathbb{E}[S_T] = e^{(\log(S_0) + v) + \sigma^2/2} \). This simplifies to \( \mathbb{E}[S_T] = S_0 \cdot e^{v + \sigma^2/2} \).

Key Concepts

Normal Distribution

The normal distribution is fundamental to understanding many statistical and financial models, including the concept of a lognormal distribution, which is crucial in financial mathematics. At its core, the normal distribution is represented as the bell-shaped curve seen in many natural and social phenomena. A normal distribution is characterized by its mean (average) and variance (spread). The mean indicates the center of the distribution, and the variance indicates how much the values spread out from the mean. Key features include:

• Symmetry about the mean: The distribution is perfectly symmetrical, which means half the values lie to the left of the mean, and half lie to the right.
• The empirical rule: About 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
• All values along the X-axis have a probability greater than zero.

Understanding normal distribution is essential because many variables in nature, psychology, and economics tend to cluster around a central mean with predictable variability.

Expected Value Calculation

Expected value is a key concept in probability and statistics, often used to predict the outcome of probabilistic events. It essentially provides an average outcome that you would expect from an experiment or real-world event, considering all possible outcomes and their probabilities.In mathematical terms, the expected value of a random variable is calculated by weighing each possible outcome by its probability of occurrence, then summing all those values.For example, if you were to roll a six-sided die, the possible outcomes are 1, 2, 3, 4, 5, and 6. Assuming a fair die, each outcome has a probability of 1/6. To find the expected value, you would calculate:\[ \text{Expected Value} = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6) = 3.5 \]In financial mathematics, expected value calculations help in making informed decisions, such as assessing the average payoff of an investment, considering the probabilities of differing returns.

Financial Mathematics

Financial mathematics uses advanced mathematical techniques to solve complex problems related to finance. One noteworthy application is the modeling of asset prices, crucial for risk management and financial engineering.A common model used in financial mathematics is the lognormal model for asset prices. This is grounded in the concept that prices cannot be negative and they tend to grow multiplicatively. The lognormal distribution is employed because it allows for such behaviors, under the assumption that the logarithm of asset prices follows a normal distribution.This means:

• If an asset's current price is denoted by \( S_0 \), its future price \( S_T \) can be modeled such that \( \log(S_T/S_0) \) follows a normal distribution.
• This is expressed in terms of a drift \( v \) (deterministic trend) and volatility \( \sigma^2 \) (random fluctuation).
• Calculating the expected value \( \mathbb{E}[S_T] \) involves utilizing the lognormal property: \( \mathbb{E}[S_T] = S_0 \cdot e^{v+\sigma^2/2} \).

These calculations are instrumental in fields like option pricing, stock market analysis, and in constructing financial products.