Problem Set 4

Stochastic Foundations for Finance

Instructions: This problem set is due on 11/24 at 11:59 pm CST 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 Consider an arithmetic Brownian motion process for \{x\} such that \mathop{}\!\mathrm{d}x = \mu_{x} \mathop{}\!\mathrm{d}t + \sigma_{x} \mathop{}\!\mathrm{d}B_{x}, where \mu_{x} and \sigma_{x} are constants.

  1. Show that x_{t} = x_{0} + \mu_{x} t + \sigma_{x} B_{x}(t).
  2. Compute \operatorname{E}(x_{t}) and \operatorname{V}(x_{t}).
  3. Explain why x_{t} \sim \mathcal{N}(\operatorname{E}(x_{t}), \operatorname{V}(x_{t})).

Consider an Ornstein–Uhlenbeck (or mean-reverting) process for \{y\} such that \mathop{}\!\mathrm{d}y = \kappa (\theta - y) \mathop{}\!\mathrm{d}t + \sigma_{y} \mathop{}\!\mathrm{d}B_{y}.

  1. Show that y_{t} = e^{-\kappa t} y_{0} + \theta\left(1 - e^{-\kappa t}\right) + \sigma_{y} e^{-\kappa t} \int_{0}^{t} e^{\kappa s} dB_{y}(s).
  2. Compute \operatorname{E}(y_{t}) and \operatorname{V}(y_{t}).
  3. Explain why y_{t} \sim \mathcal{N}(\operatorname{E}(y_{t}), \operatorname{V}(y_{t})).
  4. Compute \operatorname{Cov}(x_{t}, y_{t}).
  5. Compute \lim_{t \to \infty} \operatorname{E}(y_{t}) and explain how this limit shows that the process \{y\} mean-reverts to \theta.

Problem 2 Consider a stock price S that follows a geometric Brownian motion with stochastic volatility. The stock price evolves according to \mathop{}\!\mathrm{d}S = \mu S \mathop{}\!\mathrm{d}t + \sigma S \mathop{}\!\mathrm{d}B_{1}, where the volatility \sigma follows a mean-reverting process given by \mathop{}\!\mathrm{d}\sigma = -\beta \sigma \mathop{}\!\mathrm{d}t + \delta \mathop{}\!\mathrm{d}B_{2}. The two Brownian motions B_{1} and B_{2} are correlated with (\mathop{}\!\mathrm{d}B_{1})(\mathop{}\!\mathrm{d}B_{2}) = \rho \mathop{}\!\mathrm{d}t.

  1. Explain why the model allows \sigma to be negative even though it represents a volatility process.
  2. Use Itô’s formula to derive the stochastic differential equation for the variance v = \sigma^{2}.
  3. Show that the variance follows a process of the form \mathop{}\!\mathrm{d}v = \kappa (\theta - v) \mathop{}\!\mathrm{d}t + \xi \sqrt{v} \mathop{}\!\mathrm{d}B_{2}, and identify the parameters \kappa, \theta, and \xi in terms of \beta and \delta.

Problem 3 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}. You know that \pmb{\sigma} = \begin{pmatrix} 0.3 & -0.1 & 0.2 \\ 0.15 & 0.2 & -0.05 \end{pmatrix}.

  1. Compute the instantaneous correlation between the returns of each asset.
  2. Find a Brownian motion B_{4} = a_{1} B_{1} + a_{2} B_{2} + a_{3} B_{3} whose increments are independent from the instantaneous returns of the two assets.

Problem 4 The total instantaneous return of an asset consists of two components: \frac{\mathop{}\!\mathrm{d}S + D \mathop{}\!\mathrm{d}t}{S} = \frac{\mathop{}\!\mathrm{d}S}{S} + \frac{D}{S} \mathop{}\!\mathrm{d}t, where \mathop{}\!\mathrm{d}S / S represents the capital gains and D / S is the dividend yield, which can be interpreted as the rate at which new shares are created through dividend reinvestment. Let X(t) represent the total number of shares owned at time t, which grows according to: \frac{\mathop{}\!\mathrm{d}X}{X} = \frac{D}{S} \mathop{}\!\mathrm{d}t.

  1. Show that X_{t} = X_{0} \exp\left( \int_{0}^{t} \frac{D_{u}}{S_{u}} \mathop{}\!\mathrm{d}u \right).

Now define the dividend-reinvested asset price as: V = X S, where V represents the total value of the investment which is equal to the number of shares X multiplied by the current price per share S.

  1. Show that \frac{\mathop{}\!\mathrm{d}V}{V} = \frac{\mathop{}\!\mathrm{d}S}{S} + \frac{D}{S} \mathop{}\!\mathrm{d}t.