Problem Set 2

Stochastic Foundations for Finance

Solutions

Instructions: This problem set is due on 9/20 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Scan your work and submit a PDF file.

Problem 1 Suppose that X is a normally distributed random variable with mean \mu=12 and standard deviation \sigma=20.

  1. What is the probability that X \leq 0?
  2. What is the probability that X \leq -4?
  3. What is the probability that X > 8?
  4. What is the probability that 4 < X \leq 10?

Problem 2 Suppose that X is a normally distributed random variable with mean \mu=10 and standard deviation \sigma=20. Compute the 90%, 95%, and 99% confidence interval for X.

Problem 3 Suppose that X=\ln(Y) is a normally distributed random variable with mean \mu=3.9 and standard deviation \sigma=15.

  1. What is the probability that Y \leq 6?
  2. What is the probability that Y > 4?
  3. What is the probability that 3 < Y \leq 12?
  4. What is the probability that Y \leq 0?

Problem 4 Suppose that X is a normally distributed variable with mean \mu=3.70 and standard deviation \sigma=0.80. If Y=e^{X}, what is the probability that Y is greater than 45?

Problem 5 Let Y = e^{\mu + \sigma Z} where \mu = 1, \sigma = 2 and Z \sim \mathcal{N}(0, 1). Compute:

  1. \operatorname{E}(Y)
  2. \operatorname{SD}(Y) = \sqrt{\operatorname{E}(Y^{2}) - \operatorname{E}(Y)^{2}}
  3. \operatorname{E}(Y^{0.3})
  4. \operatorname{E}(Y^{-1})

Problem 6 Consider a stock whose price at time T is given by S_{T} such that, \ln(S_{T}) \sim \mathcal{N}(\ln(S_{0})+(\mu-0.5\sigma^{2})T, \sigma^{2} T). The expected return is 12% per year and the volatility is 35% per year. The current spot price is $100.

  1. Compute the expected price in 2 years from now.
  2. Compute the mean and standard deviation of the log-spot price in 2 years from now.
  3. Compute the probability that the spot price is less than $100 in 2 years from now.
  4. Compute the probability that the spot price is greater than $120 in 2 years from now.

Problem 7 Consider a stock whose price at time T is given by S_{T} such that, \ln(S_{T}) \sim \mathcal{N}(\ln(S_{0})+(\mu-0.5\sigma^{2})T, \sigma^{2} T). The expected return is 18% per year and the volatility is 32% per year. The current spot price is $60.

  1. Compute the expected price 9 months from now.
  2. Compute the mean and standard deviation of the log-spot price 9 months from now.
  3. Compute the 95% confidence interval of \ln(S_{T}) 9-months from now, and report the corresponding values for S_{T}.