Problem Set 3
Stochastic Foundations for Finance
Instructions: This problem set is due on 11/13 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 Find the derivative of f(x) with respect to x.
- f(x) = 10x
- f(x) = 4x-5
- f(x) = 100x - 3x^{2}
- f(x) = -10x + 6x^{2} - \frac{2}{3}x^{3}
- f(x) = 5 x^{\frac{1}{3}}
- f(x) = \frac{1}{x^{\frac{1}{5}}}
- f(x) = e^{-\frac{1}{2} x}
- f(x) = \ln\left(\frac{1}{x}\right)
Problem 2 Compute the following integrals.
- \displaystyle\int_{0}^{1} e^{-0.1 t} \mathop{}\!\mathrm{d}t
- \displaystyle\int_{0}^{1} 3 x^{5} \mathop{}\!\mathrm{d}x
- \displaystyle\int_{0}^{1} \frac{1}{x + 5} \mathop{}\!\mathrm{d}x
Problem 3 Consider an asset that pays a continuous cash flow c e^{g t} \mathop{}\!\mathrm{d}t from time 0 up to time T. The interest rate is r with continuous compounding
- Compute the value of the asset at time 0.
- Compute the value of the asset at time t < T.
- What should be the value of the asset at time T?
Problem 4 Let S be the price of TESLA stock that follows a geometric Brownian motion such that \mathop{}\!\mathrm{d}S = \mu S \mathop{}\!\mathrm{d}t + \sigma S \mathop{}\!\mathrm{d}W. Your sales team would like to launch a new product called TESLA Quadro that tracks the price of TESLA to the power 4. In other words, the value of this instrument is given by Y = S^{4}. What is the process followed by Y?
Problem 5 Suppose that the stock price follows a geometric Brownian motion (GBM) with drift \mu and instantaneous volatility \sigma. Show that Y = S e^{-\mu t} also follows a GBM and determine the drift and volatility as a function of \mu and \sigma.
Problem 6 Suppose that the stock price follows a geometric Brownian motion (GBM) with drift r and instantaneous volatility \sigma, where r is the risk-free rate. Consider the futures price of S at time t and expiring at T, given by F = S e^{r (T -t)}. Show that F has zero drift and hence is a martingale.