Question

The quantity X tfollows an Arithmetic Brownian motion with drift 3 and volatility 2. Suppose X0 = 100. What is the probability that X1 is at least 100? Recall that for an Arithmetic Brownian motion with drift μ and volatility σ, the change in time interval τ is normally distributed with mean μτ and variance σ2τ.'

Answer 1

To find the probability that X1 is at least 100 in an Arithmetic Brownian motion with drift 3 and volatility 2, use the formula X1 - X0 = μτ + σW, where X0 is the initial value, μ is the drift, σ is the volatility, and W is a standard normal random variable. Set X0 = 100, μ = 3, σ = 2 and τ = 1 to calculate the probability.

Step-by-step explanation:

To find the probability that X1 is at least 100, we need to calculate the probability of X1 being greater than or equal to 100.

Since X follows an Arithmetic Brownian motion with drift 3 and volatility 2, we can use the formula for the change in X over a time interval τ: X1 - X0 = μτ + σW, where W is a standard normal random variable.

In this case, X0 = 100, μ = 3, σ = 2, and τ = 1, so the equation becomes X1 - 100 = 3(1) + 2W.

To find the probability that X1 is at least 100, we need to find the probability that X1 - 100 is greater than or equal to 0. This can be written as P(X1 - 100 ≥ 0), which is equivalent to P(3 + 2W ≥ 0). This probability can be found using the standard normal distribution table or a calculator.

Answer 2

The question asks to calculate the probability that a continuous random variable following an Arithmetic Brownian motion with specified drift and volatility is at least 100 at a given time.

Step-by-step explanation:

The question involves calculating the probability that a continuous random variable X, which follows an Arithmetic Brownian motion with a drift of 3 and volatility of 2 and an initial value X0 of 100, will be at least 100 at time X1. To solve this, recall that the increment of an Arithmetic Brownian motion over a time interval τ is normally distributed with a mean of μτ and variance σ2τ. Since we are interested in the value at X1, τ is 1, which gives the increment a mean of 3(1) = 3 and a variance of 22(1) = 4. The outcome sought is the probability P(X ≥ 100) at time 1 given these parameters.