Question

Let \(S(t)\) denote the price of a security at time \(t .\) A popular model for the process \(\\{S(t), t \geqslant 0\\}\) supposes that the price remains unchanged until a "shock" occurs, at which time the price is multiplied by a random factor. If we let \(N(t)\) denote the number of shocks by time \(t\), and let \(X_{i}\) denote the \(i\) th multiplicative factor, then this model supposes that $$ S(t)=S(0) \prod_{i=1}^{N(t)} X_{i} $$ where \(\prod_{i=1}^{N(t)} X_{i}\) is equal to 1 when \(N(t)=0 .\) Suppose that the \(X_{i}\) are independent exponential random variables with rate \(\mu ;\) that \(\\{N(t), t \geqslant 0\\}\) is a Poisson process with rate \(\lambda ;\) that \(\\{N(t), t \geqslant 0\\}\) is independent of the \(X_{i} ;\) and that \(S(0)=s\). (a) Find \(E[S(t)]\). (b) Find \(E\left[S^{2}(t)\right]\)

Short answer

In short, the expected values for this model are: (a) \(E[S(t)] = S(0) e^{-\lambda t(\mu - 1)}\) (b) \(E\left[S^{2}(t)\right] = S^2(0) e^{-\lambda t(\frac{2}{\mu^2} - 1)}\)

Step-by-step solution

Define the random variables

First we need to define the random variables that we are going to work with: - \(S(t)\): The price of the security at time t - \(N(t)\): The number of shocks by time t - \(X_i\): The i-th multiplicative factor - \(S(0)\): The initial price of the security, \(s\) The price of the security \(S(t)\) at time t is given by the equation: $$ S(t)=S(0) \prod_{i=1}^{N(t)} X_{i} $$

Expressing \(E[S(t)]\) in terms of \(X_i\) and \(N(t)\)

We need to find the expected value of \(S(t)\). To do this, we can first express \(E[S(t)]\) in terms of the random variables \(X_i\) and \(N(t)\): $$ E[S(t)] = E\left[S(0) \prod_{i=1}^{N(t)} X_{i}\right] $$ Next, we use the condition that \(\\{N(t), t \geqslant 0\\}\) is independent of the \(X_i\)'s. $$ E[S(t)] = E\left[E\left[S(0) \prod_{i=1}^{N(t)} X_{i} \mid N(t)\right]\right] $$ Note that the conditional expectation is taken with respect to the random variable \(N(t)\).

Computing the conditional expectation

Now, we need to compute the conditional expectation \(E\left[S(0) \prod_{i=1}^{N(t)} X_{i} \mid N(t)\right]\): $$ E\left[S(0) \prod_{i=1}^{N(t)} X_{i} \mid N(t)\right] = S(0) E\left[\prod_{i=1}^{N(t)} X_{i} \mid N(t)\right] $$ Since the \(X_i\)'s are independent and identically distributed exponential random variables with rate \(\mu\), their expected value is \(E[X_i] = \frac{1}{\mu}\). Therefore, $$ E\left[S(0) \prod_{i=1}^{N(t)} X_{i} \mid N(t)\right] = S(0) \left(\frac{1}{\mu}\right)^{N(t)} $$

Computing the expectation of \(S(t)\)

We can now compute the expectation of \(S(t)\) using the results from the previous steps: $$ E[S(t)] = E\left[E\left[S(0) \left(\frac{1}{\mu}\right)^{N(t)} \mid N(t)\right]\right] = E\left[S(0) \left(\frac{1}{\mu}\right)^{N(t)}\right] $$ The random variable \(N(t)\) follows a Poisson distribution with rate \(\lambda\). Thus, $$ E[S(t)] = S(0) \sum_{n=0}^{\infty} \left(\frac{1}{\mu}\right)^n P(N(t)=n) = S(0) \sum_{n=0}^{\infty} \left(\frac{1}{\mu}\right)^n \frac{e^{-\lambda t} (\lambda t)^n}{n!} $$ By recognizing this as the Taylor expansion of the exponential function, we obtain: $$ E[S(t)] = S(0) e^{-\lambda t(\mu - 1)} $$

Finding \(E[S^2(t)]\)

For finding \(E[S^2(t)]\), we can follow similar steps by first expressing it in terms of \(X_i\) and \(N(t)\), and using the same properties applied in previous steps: $$ E[S^2(t)] = E\left[S^2(0) \left(\prod_{i=1}^{N(t)} X_{i}\right)^2\right] = E\left[E\left[S^2(0) \left(\prod_{i=1}^{N(t)} X_{i}\right)^2 \mid N(t)\right]\right] $$ Now, compute the conditional expectation: $$ E\left[S^2(0) \left(\prod_{i=1}^{N(t)} X_{i}\right)^2 \mid N(t)\right] = S^2(0) E\left[\left(\prod_{i=1}^{N(t)} X_{i}\right)^2 \mid N(t)\right] $$ Since the \(X_i\)'s are iid exponential random variables with rate \(\mu\), the expected value of \(X_i^2\) is \(E[X_i^2] = \frac{2}{\mu^2}\). Therefore, $$ E\left[S^2(0) \left(\prod_{i=1}^{N(t)} X_{i}\right)^2 \mid N(t)\right] = S^2(0) \left(\frac{2}{\mu^2}\right)^{N(t)} $$ Compute the expectation of \(S^2(t)\): $$ E[S^2(t)] = E\left[E\left[S^2(0) \left(\frac{2}{\mu^2}\right)^{N(t)} \mid N(t)\right]\right] = E\left[S^2(0) \left(\frac{2}{\mu^2}\right)^{N(t)}\right] $$ Using the fact that \(N(t)\) follows a Poisson distribution with rate \(\lambda\), $$ E[S^2(t)] = S^2(0) \sum_{n=0}^{\infty} \left(\frac{2}{\mu^2}\right)^n P(N(t)=n) = S^2(0) \sum_{n=0}^{\infty} \left(\frac{2}{\mu^2}\right)^n \frac{e^{-\lambda t} (\lambda t)^n}{n!} $$ By recognizing this as the Taylor expansion of the exponential function, we obtain: $$ E[S^2(t)] = S^2(0) e^{-\lambda t(\frac{2}{\mu^2} - 1)} $$ So we have found the expected value of \(S(t)\) and \(S^2(t)\): (a) \(E[S(t)] = S(0) e^{-\lambda t(\mu - 1)}\) (b) \(E\left[S^{2}(t)\right] = S^2(0) e^{-\lambda t(\frac{2}{\mu^2} - 1)}\)

Key concepts

Poisson processes

A Poisson process is a stochastic process that models a sequence of events occurring randomly over time. It is characterized by the fact that events occur independently of each other, and the number of events in any given time interval follows a Poisson distribution.

Understanding the Poisson process is crucial for tackling problems involving random occurrences over time, such as the number of shocks affecting a security's price in a financial model. A key parameter of a Poisson process is the rate \(\lambda\), which represents the average number of events occurring per time unit.

For example, in the context of the financial model discussed in the exercise, \(N(t)\) denotes the number of shocks by a certain time \(t\), and as it is a Poisson process, the probability of having \(n\) shocks in time \(t\) is given by the formula:\[ P(N(t)=n) = \frac{e^{-\lambda t} (\lambda t)^n}{n!} \]
These probabilities play a significant role in assessing the expected price of the security over time, considering the random shocks. They describe how frequently the price-affecting events occur and allow for the calculation of expected values for \(S(t)\) over any time period.

Exponential random variables

Exponential random variables are greatly relevant in modeling the time between independent, statistically identical events in a Poisson process. These variables are characterized by a memoryless property, meaning the probability of an event occurring in the future is independent of how much time has already elapsed.

In the exercise, the multiplicative factors \(X_i\) represent the size of the shocks and are described as independent exponential random variables with a rate \(\mu\). The rate \(\mu\) dictates how large an impact each shock has on the security price. Mathematically, the expected value of an exponential random variable is given by \(E[X_i] = \frac{1}{\mu}\), which was utilized to derive the expected value of the security's price \(E[S(t)]\).

Moreover, for the purpose of finding \(E[S^2(t)]\), the squared expectation \(E[X_i^2] = \frac{2}{\mu^2}\) must be used, indicating the variance of the exponential random variables. This measures the extent to which the size of shocks varies and ultimately impacts the variance of the security's price.

Expected value

The expected value, or mathematical expectation, of a random variable is a fundamental concept in probability and statistics. It provides the long-run average value of the variable if the random process could be repeated infinitely many times.

In the financial model provided, finding the expected value of the security's price \(E[S(t)]\) and its square \(E[S^2(t)]\) over time is essential for understanding how the security is expected to perform in the face of random shocks. The expected value incorporates all possible outcomes weighted by their probabilities and is calculated using the probability distributions of the underlying random variables.
To compute \(E[S(t)]\), one applies the law of the total expectation, which involves conditioning on the Poisson-distributed number of shocks, \(N(t)\), then taking the expectation. This technique simplifies a complex problem into more manageable parts and highlights the utility of expected values in quantifying average behaviors in stochastic settings.

Expected values also assist investors in making informed decisions by quantifying the central tendency of the security's future price. This statistical measure communicates critical information that can influence financial strategies in environments filled with uncertainty and random variations.