Problem Set 5

Stochastic Foundations for Finance

Instructions: This problem set is due on 12/10 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 (A Random Walk) Let \{X_{n}\} be a sequence of independent random variables, each taking values 1 or -1 with equal probability 1/2. For n \geq 1, define S_{n} = S_{0} + \sum_{i=1}^{n} X_{i}, and \tau = \min\{n > 0 : S_{n} = A \text{ or } S_{n} = -B\}. In the following, assume that P(\tau = \infty \mid S_{0} = 0) = 0.

  1. Compute P(S_\tau = A \mid S_{0} = 0) by defining a recursion on f(k) = P(S_\tau = A \mid S_{0} = k).
  2. Compute \operatorname{E}(\tau \mid S_{0} = 0) by defining a recursion on g(k) = \operatorname{E}(\tau \mid S_{0} = k).

Problem 2 (Three Martingales) A martingale is a stochastic process \{M_{n}\} that satisfies \operatorname{E}(M_{n} \mid \mathcal{F}_{n - 1}) = M_{n - 1}, where \mathcal{F}_{n-1} represents all information available up to time n-1. In the following, \mathcal{F}_{n} represents the information set generated by the sequence \{X_{0}, X_{1}, \ldots, X_{n}\}.

  1. Let \{X_{n}\} be a sequence of independent random variables such that \operatorname{E}(X_{n}) = 0 for all n \geq 1. Define the partial sum process by S_{0} = 0 and S_{n} = X_{1} + X_{2} + \cdots + X_{n} for n \geq 1. Show that \{S_{n}\} is a martingale.

  2. Suppose now the \{X_{n}\} are independent random variables with \operatorname{E}(X_{n}) = 0 and \operatorname{V}(X_{n}) = \sigma^{2} for all n \geq 1. Define M_{0} = 0 and M_{n} = S_{n}^{2} - n \sigma^{2} for n \geq 1, where S_{n} = X_{1} + X_{2} + \cdots + X_{n} for n \geq 1. Prove that \{M_{n}\} is a martingale.

  3. Finally, consider independent random variables \{X_{n}\} such that X_{n} > 0 and E(X_{n}) = 1 for all n \geq 1. Let M_{0} = 1 and set M_{n} = X_{1} X_{2} \cdots X_{n} for n \geq 1. Show that \{M_{n}\} is a martingale.

Problem 3 (The Probability Generating Function) The probability generating function (PGF) of a discrete random variable X \ge 0 is defined as G_{X}(z) = \operatorname{E}(z^{X}) = \sum_{k=0}^{\infty} z^{k} \, p_{X}(k), where z is a complex number with |z| \leq R, R denotes the radius of convergence, and p_{X}(x) denotes the probability mass function.

  1. A discrete random variable X is said to have the Poisson distribution with parameter \lambda if it has a probability mass function given by p_{X}(k) = \operatorname{P}(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}, and we write X \sim \text{Poisson}(\lambda). Find G_{X}(z).

  2. Compute G'_{X}(1) and G''_{X}(1), and show that \operatorname{E}(X) = \operatorname{V}(X) = \lambda.

If X and Y are two independent non-negative integer-valued random variables, then we have that G_{X + Y}(z) = G_{X} G_{Y}.

  1. Show that if X_{1} \sim \text{Poisson}(\lambda_{1}) and X_{2} \sim \text{Poisson}(\lambda_{2}) are independent, then X_{1} + X_{2} \sim \text{Poisson}(\lambda_{1} + \lambda_{2}).

  2. Show that in general if \{X_{n}\} are independent and X_{i} \sim \text{Poisson}(\lambda_{i}) for i = 1, \ldots, n, then X_{1} + \ldots + X_{n} is also distributed Poisson with parameter \lambda_{1} + \ldots + \lambda_{n}.

Problem 4 (The Characteristic Function) If X is a real-valued random variable with density function f_{X}(x), the characteristic function of X is defined as \phi_{X}(u) = \operatorname{E}(e^{iuX}) = \int_{-\infty}^{\infty} e^{iux} \, f_{X}(x) dx.

  1. If X \sim \mathcal{N}(\mu, \sigma^{2}), find \phi_{X}(u).

If X and Y are two independent real-valued random variables, we also have that \phi_{X + Y}(u) = \phi_{X}(u)\phi_{Y}(u).

  1. Let X \sim \mathcal{N}(\mu_{X}, \sigma_{X}^{2}) and Z \sim \mathcal{N}(\mu_{Z}, \sigma_{Z}^{2}) two independent random variables, and let Y = X + Z.

    1. Show Y \sim \mathcal{N}(\mu_{Y}, \sigma_{Y}^{2}), where \mu_{Y} = \mu_{X} + \mu_{Z} and \sigma_{Y}^{2} = \sigma_{X}^{2} + \sigma_{Z}^{2}.
    2. Compute \operatorname{Cov}(Y, X).
    3. Show that X + Y is also normally distributed.

Problem 5 (Stock Prices are Lognormal) Consider a probability space (\Omega, \mathcal{F}, \operatorname{P}) and a Brownian motion \{B_{t}\} generating a filtration \{\mathcal{F}_{t}\}_{t \ge 0}. Consider a non-dividend paying stock whose price S follows a geometric Browninan motion \frac{\mathop{}\!\mathrm{d}S}{S} = \mu \mathop{}\!\mathrm{d}t + \sigma \mathop{}\!\mathrm{d}B, where \mu and \sigma are constants.

  1. Let X = \ln(S). Compute the process followed by \{X_{t}\} as a function of \mu, \sigma and \{B_{t}\}.
  2. Show that S_{t} = S_{0} \exp\left(\left(\mu - \tfrac{1}{2} \sigma^{2}\right) t + \sigma B_{t}\right).
  3. Deduce that \ln(S_{t}) \sim \mathcal{N}\left(\ln(S_{0}) + \left(\mu - \frac{1}{2} \sigma^{2}\right) t, \sigma^{2} t\right).

Problem 6 (Ito’s Lemma) In the following, assume that r, \mu and \sigma are constants.

  1. Suppose that the stock price follows a geometric Brownian motion (GBM) with drift \mu and instantaneous volatility \sigma, i.e., \mathop{}\!\mathrm{d}S= \mu S \mathop{}\!\mathrm{d}t+ \sigma S \mathop{}\!\mathrm{d}B. Show that Y = 1 / S also follow a GBM and determine the drift and volatility as a function of \mu, and \sigma.
  2. Assume that \mathop{}\!\mathrm{d}S= r S \mathop{}\!\mathrm{d}t+ \sigma S \mathop{}\!\mathrm{d}B. Derive the process followed by the futures price F(T) = S e^{r T} where we interpret T as time-to-maturity.
  3. A process S_{t} is a martingale if \operatorname{E}(S_{T} \mid \mathcal{F}_{t}) = S_{t} for t < T. Show that \mathop{}\!\mathrm{d}S= S \sigma \mathop{}\!\mathrm{d}B is a martingale.

Problem 7 (Multivariate Ito’s Lemma)  

  1. Consider the product H = XY where X and Y are Itô diffusions. Derive the stochastic differential equation for H and verify that \frac{\mathop{}\!\mathrm{d}H}{H} = \frac{\mathop{}\!\mathrm{d}X}{X} + \frac{\mathop{}\!\mathrm{d}Y}{Y} + \left(\frac{\mathop{}\!\mathrm{d}X}{X}\right) \left(\frac{\mathop{}\!\mathrm{d}Y}{Y}\right). Under which conditions \left(\frac{\mathop{}\!\mathrm{d}X}{X}\right) \left(\frac{\mathop{}\!\mathrm{d}Y}{Y}\right) = 0?
  2. Suppose a process \beta satisfies \frac{\mathop{}\!\mathrm{d}\beta}{\beta} = r \mathop{}\!\mathrm{d}t, and a process \Lambda satisfies \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} = -r \mathop{}\!\mathrm{d}t- \lambda \mathop{}\!\mathrm{d}B. Define \xi = \Lambda \beta and show that \xi is a martingale by proving it has zero drift.
  3. Consider a stock price process satisfying \frac{\mathop{}\!\mathrm{d}S}{S} = \mu \mathop{}\!\mathrm{d}t+ \sigma \mathop{}\!\mathrm{d}B, and a deflator process satisfying \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} = -r \mathop{}\!\mathrm{d}t- \lambda \mathop{}\!\mathrm{d}B. Find the value of \lambda such that the deflated stock price \xi^{S} = \Lambda S is a martingale.

Problem 8 (The Stochastic Discount Factor) Consider an economy with two tradable assets. The first is a risk-free money-market account \beta satisfying \frac{\mathop{}\!\mathrm{d}\beta}{\beta} = r \mathop{}\!\mathrm{d}t, where \beta_{0} = 1, and r denotes the risk-free rate. The second is a non-dividend paying stock S following geometric Brownian motion: \frac{\mathop{}\!\mathrm{d}S}{S} = \mu_{S} \mathop{}\!\mathrm{d}t+ \sigma_{S} \mathop{}\!\mathrm{d}B, where B is a standard Brownian motion under the physical measure \mathbb{P}.

Under the no-arbitrage condition, there exists a strictly positive stochastic discount factor (SDF) \Lambda such that any asset price, when discounted by \Lambda, becomes a martingale. The SDF satisfies \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} = - r \mathop{}\!\mathrm{d}t- \lambda \mathop{}\!\mathrm{d}B, where \lambda is the market price of risk. Throughout, we assume standard integrability conditions ensuring that Itô processes with zero drift are martingales with respect to the filtration \{\mathcal{F}_{t}\}.

  1. Let Y to be any traded asset such as the stock or a derivative written on the stock, such that \frac{\mathop{}\!\mathrm{d}Y}{Y} = \mu_{Y} \mathop{}\!\mathrm{d}t+ \sigma_{Y} \mathop{}\!\mathrm{d}B. Find \lambda such that \Lambda Y is a martingale. Explain the intuition of the result.

Consider now a European call C(S,t) with strike K and maturity T.

  1. Compute \mu_{C} and \sigma_{C} in \frac{\mathop{}\!\mathrm{d}C}{C} = \mu_{C} \mathop{}\!\mathrm{d}t+ \sigma_{C} \mathop{}\!\mathrm{d}B.

From question (a), we know that \lambda = \frac{\mu_{S} - r}{\sigma_{S}} = \frac{\mu_{C} - r}{\sigma_{C}}.

  1. Derive the Black–Scholes PDE \frac{1}{2}\sigma_{S}^{2} S^{2} \frac{\partial^{2} C}{\partial S^{2}} + r S \frac{\partial C}{\partial S} + \frac{\partial C}{\partial t} - r C = 0, with terminal condition C(S, T) = \max(S - K, 0).

Problem 9 (The Risk-Neutral Measure) Consider a stochastic discount factor \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} = - r \mathop{}\!\mathrm{d}t- \lambda \mathop{}\!\mathrm{d}B, where \lambda is the market price of risk, and define B^{*} by \mathop{}\!\mathrm{d}B^{*} = \mathop{}\!\mathrm{d}B+ \lambda \mathop{}\!\mathrm{d}t. Since \xi = \Lambda \beta is a strictly positive \mathbb{P}-martingale, Girsanov’s theorem implies that B^{*} is a standard Brownian motion under the risk-neutral measure \mathbb{P}^{*}, where \frac{\mathop{}\!\mathrm{d}\mathbb{P}^{*}}{\mathop{}\!\mathrm{d}\mathbb{P}} = \xi_{T}.

  1. Show that for any traded asset Y, the discounted price process Y / \beta is a martingale under the risk-neutral measure \mathbb{P}^{*}.

  2. Under the risk-neutral measure \mathbb{P}^{*}, derive the stochastic processes governing the stock price S and the call option price C.

  3. Show that a long forward contract with delivery price K and expiring at T satisfies the Black-Scholes differential equation. What is the boundary condition in this case?

  4. Show that a zero-coupon bond with face value 1 and expiring at T satisfies the Black-Scholes differential equation. What is the boundary condition in this case?