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Problem Set 4

Stochastic Foundations for Finance

Instructions: This problem set is due on 11/19 at 11:59 pm Central Time and is an individual assignment. All problems must be handwritten. Write full sentences to answer questions that require an explanation. Scan your work and submit a PDF file.

Problem 1 Let \{B_{t}\} be a Brownian motion, and let s = 0.7 and t = 2.

  1. Compute \operatorname{P}(B_{s} \leq -0.5), \operatorname{P}(B_{t} > 1), and \operatorname{P}(|B_{t}| \leq 1.5).
  2. Compute \operatorname{P}(|B_{t} - B_{s}| > 2).
  3. Compute the correlation between B_{s} and B_{t}.
  4. Find the distribution of B_{s} + B_{t}, and compute \operatorname{P}(B_{s} + B_{t} > 2). (Hint: write B_{s} + B_{t} = 2 B_{s} + (B_{t} - B_{s}).)

Problem 2 The logarithm of the value of a firm’s assets, relative to their current value, follows a Brownian motion with drift X_{t} = \mu t + \sigma B_{t}, with \mu = 0.05 and \sigma = 0.25 per year.

  1. The firm defaults if the value of its assets falls by 30%. Compute the probability that the firm ever defaults.
  2. Compute the probability that the value of the assets falls by 30% before it rises by 50%.

Problem 3 The log price of a stock follows X_{t} = \mu t + \sigma B_{t}. An analyst estimates the volatility from n daily log returns over one year, T = 1, using the realized variance Q_{n} = \sum_{j = 0}^{n - 1} (X_{t_{j + 1}} - X_{t_{j}})^{2}, \qquad t_{j} = j / n.

  1. Assume first that \mu = 0. Show that \operatorname{E}(Q_{n}) = \sigma^{2} and \operatorname{V}(Q_{n}) = 2 \sigma^{4} / n.
  2. With \sigma = 0.20 and n = 252, compute the standard deviation of Q_{n}, and compare it with \sigma^{2}.
  3. Now let \mu \neq 0. Show that the drift adds \mu^{2} / n to \operatorname{E}(Q_{n}), and explain why the realized variance measures the volatility precisely while the drift cannot be estimated from a year of data in the same way.

Problem 4 Let \{B_{t}\} be a Brownian motion and T > 0.

  1. Compute the mean and the variance of I_{T} = \int_{0}^{T} s \mathop{}\!\mathrm{d}B_{s}, and find its distribution.
  2. Compute the mean and the variance of J_{T} = \int_{0}^{T} B_{s} \mathop{}\!\mathrm{d}B_{s} using the properties of the Itô integral, and check your answer using J_{T} = \frac{1}{2} (B_{T}^{2} - T).
  3. Apply Itô’s formula to B_{t}^{3}, and use the result to write \int_{0}^{T} B_{s}^{2} \mathop{}\!\mathrm{d}B_{s} in terms of B_{T} and an ordinary integral of B_{s}.
  4. Use Itô’s formula to show that B_{t}^{3} - 3 t B_{t} has zero drift, and conclude that it is a martingale.

Problem 5 Consider an Ornstein-Uhlenbeck (or mean-reverting) process for an interest rate y such that \mathop{}\!\mathrm{d}y_{t} = \kappa (\theta - y_{t}) \mathop{}\!\mathrm{d}t+ \sigma \mathop{}\!\mathrm{d}B_{t}, where \kappa > 0, \theta, and \sigma > 0 are constants.

  1. Apply Itô’s formula to e^{\kappa t} y_{t} to show that y_{t} = e^{-\kappa t} y_{0} + \theta\left(1 - e^{-\kappa t}\right) + \sigma e^{-\kappa t} \int_{0}^{t} e^{\kappa s} \mathop{}\!\mathrm{d}B_{s}.
  2. Compute \operatorname{E}(y_{t}) and \operatorname{V}(y_{t}), and explain why y_{t} is normally distributed.
  3. Compute the limits of \operatorname{E}(y_{t}) and \operatorname{V}(y_{t}) as t \to \infty, and explain how they show that y_{t} reverts to \theta. Can the interest rate become negative in this model?

Problem 6 Consider two assets whose price processes are given by \frac{\mathop{}\!\mathrm{d}\mathbf{S}}{\mathbf{S}} = \pmb{\mu} \mathop{}\!\mathrm{d}t+ \pmb{\sigma} \mathop{}\!\mathrm{d}\mathbf{B}, where \mathop{}\!\mathrm{d}\mathbf{B} is a vector of three independent Brownian motions B_{1}, B_{2}, and B_{3}, and \pmb{\sigma} = \begin{pmatrix} 0.3 & -0.1 & 0.2 \\ 0.15 & 0.2 & -0.05 \end{pmatrix}.

  1. Compute the instantaneous volatility of each asset and the instantaneous correlation between their returns.
  2. Use the product rule to find the process followed by the ratio Y = S_{1} / S_{2}, the price of asset 1 measured in units of asset 2. (Hint: first find \mathop{}\!\mathrm{d}(1 / S_{2}) with Itô’s formula.)
  3. Find a Brownian motion B_{4} = a_{1} B_{1} + a_{2} B_{2} + a_{3} B_{3} whose increments are independent of the returns of the two assets.

Problem 7 Let \{B_{t}\} be a Brownian motion under \operatorname{P}, let \theta = 0.5, and define the probability measure \operatorname{P}^{*} by \operatorname{P}^{*}(A) = \operatorname{E}(Z_{1} \mathbf{1}_{\{A\}}), where Z_{1} = e^{-\theta B_{1} - \frac{1}{2} \theta^{2}}.

  1. Use Girsanov’s theorem to find the distribution of B_{1} under \operatorname{P}^{*}, and compute \operatorname{P}(B_{1} > 0) and \operatorname{P}^{*}(B_{1} > 0).
  2. Verify your answer for the mean by computing \operatorname{E}^{*}(B_{1}) = \operatorname{E}(Z_{1} B_{1}) directly. (Hint: differentiate the normal moment generating function \operatorname{E}(e^{c B_{1}}) = e^{\frac{1}{2} c^{2}} with respect to c.)
  3. A stock follows \mathop{}\!\mathrm{d}S_{t} / S_{t} = \mu \mathop{}\!\mathrm{d}t+ \sigma \mathop{}\!\mathrm{d}B_{t} with \mu = 12\% and \sigma = 20\%, and the risk-free rate is r = 4\%. Find the value of \theta that makes the discounted stock price a martingale under \operatorname{P}^{*}, and write the process followed by S_{t} under \operatorname{P}^{*}.

Normal Distribution Table

The table below computes \Phi(z) = \operatorname{P}(Z \leq z) if Z \sim \mathcal{N}(0, 1) and z \geq 0. The rows denote the first decimal whereas the columns denote the second decimal. If you need \Phi(-z), you can always use \Phi(-z) = 1 - \Phi(z).

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990