Problem Set 5

Topics in Quantitative Finance

Instructions: This problem set is due on 10/13 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 In this problem all parameters are constant, and the time interval is [0, T]. A non-dividend paying stock S follows a GBM \frac{dS}{S} = \mu_{S} dt + \sigma_{S} dB. The stochastic discount factor is such that \frac{d\Lambda}{\Lambda} = -r dt - \lambda dB_{\Lambda}, where dB dB_{\Lambda} = \rho_{S, \Lambda} dt. As always, the money-market account \beta satisfies d\beta = r \beta dt.

  1. Compute dB^{*} = dB - \frac{d\mathcal{E}}{\mathcal{E}} dB, where \mathcal{E} = \Lambda \beta.
  2. Determine the dynamics of S under the risk-neutral measure \operatorname{P}^{*} defined as \frac{d\operatorname{P}^{*}}{d\operatorname{P}} = \mathcal{E}_{T}.
  3. Compute dB^{S} = dB - \frac{d\mathcal{E}^{S}}{\mathcal{E}^{S}} dB, where \mathcal{E}^{S} = \Lambda S.
  4. Determine the dynamics of S under the alternative measure \operatorname{P}^{S} defined as \frac{d\operatorname{P}^{S}}{d\operatorname{P}} = \mathcal{E}_{T}^{S}.
  5. In class we saw that the price of a European call option written on S with strike price K and expiring at T is C = S \operatorname{P}^{S}(S_{T} > K) - K e^{-r T} \operatorname{P}^{*}(S_{T} > K). In the formula, explain why the risk-adjusted probability of the first term is different from the risk-adjusted probability of the second term.

Problem 2 In the model of Schwartz and Smith (2000), the spot price of a commodity is modelled as S = e^{x + y}, where \begin{aligned} dx & = \mu dt + \sigma_{x} dB_{x}, \\ dy & = -\kappa y dt + \sigma_{y} dB_{y}, \end{aligned} and dB_{x} dB_{y} = \rho_{x, y} dt. In class we saw that we can solve for x_{T} and y_{T} as \begin{aligned} x_{T} & = x_{0} + \mu T + \sigma_{x} \int_{0}^{T} dB_{xt}, \\ y_{T} & = y_{0} e^{-\kappa T} + \sigma_{y} e^{-\kappa T} \int_{0}^{T} e^{\kappa t} dB_{yt}, \end{aligned}

  1. Explain why x captures permanent shocks to S.
  2. Explain the mechanism that makes shocks to y to be mean-reverting.
  3. Explain intuitively why x_{T} and y_{T} are jointly normally distributed.

Problem 3 Suppose that a non-dividend paying stock S follows a geometric Brownian motion under the risk-neutral measure such that \frac{dS}{S} = r dt + \sigma dB^{*}, where r and \sigma are constants. Compute the futures price of S expiring at time T as F(T) = \operatorname{E}^{*}(S_{T}).