MN30507 Financial Markets — Derivatives Workshop 2
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MN30507
Financial Markets — Derivatives
Workshop 2
1. B(t) is a standard Brownian motion.
a. What is the expectation and variance of the B(t)? What is the expectation and variance of the increment, B(t) - B(s)?
b. Please construct a Brownian motion with drift process and derive the expectation and variance.
2. B(t) is a Brownian motion. What is the expected position at time t = 2? What is the variance of its position at time t = 2?
3. B(t) is a Brownian motion. The position at time t = 2 is 1.
a. What is the expected position at time t = 3?
b. What is the expected position at time t = 5?
4. Suppose a security follows a Brownian motion process B(t),
a. What is the probability of its position at time t = 2 greater than 3, P(B(2)>3)?
b. What is the probability that B(4) is greater than B(2), P(B(4) > B(2))?
5. If a generalised Brownian motion process is dx = μdt + σeVdt , where e ~ N(0,1), then what is the distribution of dx?
6. Assume the price of a stock follows the process of X(t) = μt + σB(t), where B(t) is Brownian motion, drift (μ) is -3, variance (σ2) is 9. Given that the price is 5 at time t = 3, what is the probability that the price is below 1 at time t = 4?
7. Consider a stochastic process dx = μdt + σdz, drift is 2 and variance is 9 in the first 3 years and drift is 3 and variance is 16. What is the probability distribution of the value at the end of year 6?
2023-04-22