Problem Set 4

Topics in Quantitative Finance

Instructions: This problem set is due on 9/26 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 Consider a non-dividend paying stock whose returns follow a diffusion such that \frac{dS}{S} = \mu dt + \sigma dB, where B is a one-dimensional Browninan motion. The risk-free rate is denoted by r. If \Lambda is a stochastic discount factor such that \frac{d\Lambda}{\Lambda} = -r dt - \frac{\lambda}{\sigma} dB, what should be the sign of \left(\frac{d\Lambda}{\Lambda}\right) \left(\frac{dS}{S}\right) if \mu < r?

Problem 2 Suppose that you have two independent Brownian motions B_{1} and B_{2}. How can you build a new Brownian motion B_{3} using B_{1} and B_{2} so that (dB_{2})(dB_{3}) = \rho dt.

Problem 3 Suppose the SDF is given by \frac{d\Lambda}{\Lambda} = -r dt - \lambda_{1} dB_{1} - \lambda_{2} dB_{2} - \ldots - \lambda_{n} dB_{n}, where B_{1}, B_{2}, \ldots, B_{n} are independent Brownian motions. What does it mean if one of the \lambda’s is equal to zero?

Problem 4 Consider two assets whose price processes are given by \frac{d\mathbf{S}}{\mathbf{S}} = \pmb{\mu} dt + \pmb{\sigma} d\mathbf{z}, where d\mathbf{z} 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 5 Consider a stochastic discount factor in continuous time given by \frac{d\Lambda}{\Lambda} = - r_{f} dt - \lambda_{1} dB_{1} - \lambda_{2} dB_{2}, where B_{1} and B_{2} are independent Brownian motions. Suppose that you have two non-dividend paying assets with the following dynamics: \frac{dS_{1}}{S_{1}} = \mu_{1} dt + \sigma_{1} dB_{1}, and \frac{dS_{2}}{S_{2}} = \mu_{2} dt + \sigma_{21} dB_{1} + \sigma_{22} dB_{2}. Suppose that r_{f} = 0.05, \mu_{1} = 0.15, \mu_{2} = 0.20, \sigma_{1} = 0.3, \sigma_{21} = 0.5, and \sigma_{22} = -0.1.

  1. Compute the instantaneous correlation between the returns of each asset.
  2. Determine \lambda_{1} and \lambda_{2}.

Problem 6 In this problem we consider a probability space (\Omega, \mathcal{F}, \operatorname{P}) in which time goes from 0 to T. Consider two non-dividend paying assets S_{1} and S_{2} that follow geometric Brownian motions \begin{aligned} \frac{dS_{1}}{S_{1}} & = \mu_{1} dt + \sigma_{1} dB_{1}, \\ \frac{dS_{2}}{S_{2}} & = \mu_{2} dt + \sigma_{2} dB_{2}, \end{aligned} where B_{1} and B_{2} are two potentially correlated Brownian motions such that dB_{1} dB_{2} = \rho_{1, 2} dt. There is a stochastic discount factor \Lambda so that each process \Lambda S_{i} is a local martingale for i = 1, 2. The SDF is characterized by \frac{d\Lambda}{\Lambda} = -r dt - \lambda dB, where B is a Brownian motion correlated with B_{1} and B_{2} so that dB dB_{i} = \rho_{i} for i = 1, 2, and \lambda is the market price of risk of B. The money-market account is denoted by \beta and grows geometrically at the risk-free rate r so that \frac{d\beta}{\beta} = r dt.

  1. Show that \lambda = \frac{\mu_{i} - r}{\rho_{i} \sigma_{i}} \text{ for } i = 1, 2. Explain why \rho_{i} = 1 for efficient assets.
  2. Let \operatorname{P}^{*} denote the risk-neutral measure defined by \frac{d\operatorname{P}^{*}}{d\operatorname{P}} = \mathcal{E}_{T}, where \mathcal{E} = \Lambda \beta. Compute B_{1}^{*} and B_{2}^{*} so that both terms are Brownian motions under \operatorname{P}^{*}.
  3. Compute the instantaneous correlation between B_{1}^{*} and B_{2}^{*}.
  4. If you were to form a zero-cost portfolio in which you go long S_{1} and you go short S_{2}, what should be the risk-neutral drift of such portfolio?