Problem 5 - A Course in Financial Calculus

Question

Suppose that \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma^{2}\). Calculate $$ \mathbb{E}\left[e^{\theta X}\right] $$ and hence evaluate \(\mathbb{E}\left[X^{4}\right]\).

Step-by-Step Solution

Answer

\(\mathbb{E}[e^{\theta X}] = e^{\mu \theta + \frac{1}{2} \sigma^2 \theta^2}\) and \(\mathbb{E}[X^4] = 3\sigma^4 + 6\sigma^2\mu^2 + \mu^4\).

Step 1: Define the Moment Generating Function (MGF)

The moment generating function (MGF) of a random variable provides a way to find moments. For a normal random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the MGF is \(M_X(t) = \mathbb{E}\left[e^{tX}\right] = e^{\mu t + \frac{1}{2} \sigma^2 t^2}\). This is the starting point to calculate \(\mathbb{E}\left[e^{\theta X}\right]\).

Step 2: Substitute \(\theta\) into the MGF

Substitute \(t = \theta\) into the MGF expression to get \(M_X(\theta) = \mathbb{E}\left[e^{\theta X}\right] = e^{\mu \theta + \frac{1}{2} \sigma^2 \theta^2}\). This gives us the desired expected value of the exponential function.

Step 3: Differentiate the MGF to Find Moments

Finding the moments involves differentiating the MGF. The \(n\)-th moment is obtained by differentiating the MGF \(n\) times with respect to \(t\) and evaluating at \(t = 0\). To find \(\mathbb{E}[X^4]\), we need the fourth derivative of the MGF.

Step 4: Compute the First Four Derivatives

1. First derivative: \( \frac{d}{dt} M_X(t) = (\mu + \sigma^2 t) e^{\mu t + \frac{1}{2} \sigma^2 t^2} \) 2. Second derivative: \( \frac{d^2}{dt^2} M_X(t) = (\sigma^2 + (\mu + \sigma^2 t)^2) e^{\mu t + \frac{1}{2} \sigma^2 t^2} \) 3. Third derivative: \( \frac{d^3}{dt^3} M_X(t) = (3\sigma^2 (\mu + \sigma^2 t) + (\mu + \sigma^2 t)^3) e^{\mu t + \frac{1}{2} \sigma^2 t^2} \) 4. Fourth derivative: \( \frac{d^4}{dt^4} M_X(t) = (3\sigma^4 + 6 \sigma^2 (\mu + \sigma^2 t)^2 + (\mu + \sigma^2 t)^4) e^{\mu t + \frac{1}{2} \sigma^2 t^2} \).

Step 5: Evaluate the Fourth Derivative at 0

Evaluate the fourth derivative at \(t = 0\): \( \frac{d^4}{dt^4} M_X(t) \bigg|_{t=0} = (3\sigma^4 + 6 \sigma^2 \mu^2 + \mu^4) e^{0} = 3\sigma^4 + 6\sigma^2\mu^2 + \mu^4 \). This is \(\mathbb{E}[X^4]\).

Key Concepts

Normal Distribution

Understanding normal distribution is key to comprehending the behavior of many natural phenomena. Also known as the Gaussian distribution, it is a continuous probability distribution that is symmetric about the mean. A normal distribution is characterized by two parameters:

• Mean (\(\mu\)): This is the peak center of the distribution curve and determines its position on the horizontal axis.
• Variance (\(\sigma^2\)): This measures the spread or width of the distribution curve. A larger variance implies a wider curve.

The classic bell-shaped curve of a normal distribution shows that most of the data points cluster around the mean. This distribution is crucial because it describes many real-world variables, such as heights, test scores, and measurement errors.
Statistical methods assume normality because of its convenient mathematical properties, making normal distribution foundational in probability and statistics.

Expectation

The expectation of a random variable, often referred to as the expected value or mean, is a forecast of the variable's probable average outcome. It is denoted by \(\mathbb{E}[X]\), representing the average value you would anticipate \(X\) obtaining per trial if the trials could be repeated infinitely. For a normally distributed random variable, the expectation or mean is \(\mu\).

• This mean serves as the central tendency of the distribution.
• It's a crucial indicator for understanding the distribution's location on the horizontal axis.

In the context of moment generating functions (MGF), the expectation helps identify the mean and simplifies complex probability calculations.
By understanding expectation, one can predict outcomes and make informed decisions based on probabilistic data.

Variance

Variance is a measure of the spread or dispersion within a set of data. It quantifies the degree of variation of data points from the mean. The formula for variance is:\[\sigma^2 = \mathrm{Var}(X) = \mathbb{E}[(X - \mu)^2]\]where:

• \(\sigma^2\) is the variance.
• \(X\) is the random variable.
• \(\mu\) is the mean of the distribution.

For a normal distribution, the variance is provided as a parameter. Variance is essential because it indicates how much the values of a random variable differ from the mean. A higher variance means more spread out data.
Understanding variance is integral for risk assessment and statistical inference, helping to predict and understand variability in datasets.

Higher Moments

Higher moments of a distribution provide deeper insights into its shape and characteristics beyond the basic mean and variance. They capture more details about the distribution's behavior.1. **Third Moment (Skewness):** Skewness measures asymmetry of the probability distribution of a real-valued random variable about its mean. Skewness can be:

• Zero (mean = median = mode) indicating a perfectly symmetrical distribution.
• Positive (right skewed) indicating longer tails to the right.
• Negative (left skewed) indicating longer tails to the left.

2. **Fourth Moment (Kurtosis):** Kurtosis assesses the tailedness or the sharpness of the distribution's peak.

• A higher kurtosis means more of the variance is due to infrequent extreme deviations.
• Normal distribution has a kurtosis of 3 (mesokurtic).

Calculating higher moments often involves using the moment generating function, which simplifies the process by differentiating and evaluating at zero, as in the exercise where \(\mathbb{E}[X^4]\) was derived. Understanding higher moments helps in comprehensively analyzing statistical data.