Chapter 6 Questions for Chapter 6

These questions are just for practice - enjoy!

Warm-up

Exercise 6.1 Find an expression for \(\Var(S_t)\) where \(S_t\), \(t \geq 0\) is a geometric Brownian motion with drift parameter \(\mu\), volatility \(\sigma\) and initial value \(S_0\).

Exercise 6.2 Suppose that \(S_t\), \(t \geq 0\) is a geometric Brownian motion with drift parameter \(\mu = 0.2\) and that \(S_0 = 100\). Using volatility parameter \(\sigma = 0.2\) calculate:

  1. \(\E\bigl( S_{10} \bigr)\),
  2. \(\P\bigl( S_{10} > 100 \bigr)\),
  3. \(\P\bigl( S_{10} < 110 \bigr)\).

Repeat the calculations for volatility parameter values \(\sigma = 0.4\) and \(\sigma = 0.6\).

Exercise 6.3 Find the price of a European put option with strike price 100, expiry date of \(1/2\) (i.e. 6 months) when \(S_0 = 105\), \(r = 0.1\) and \(\sigma = 0.3\).

Main problems

Exercise 6.4 Show, by doing the relevant integral, that if \(X \sim N(m, v^2)\) then \(\E(\e^X) = \e^{m + v^2/2}\). Deduce the formula for \(\E(S_t)\) when \(S_t\), \(t \geq 0\) is a geometric Brownian motion with drift parameter \(\mu\), volatility \(\sigma\) and initial value \(S_0\) so, in particular, \(\log S_t/S_0 \sim N\bigl( (\mu - \sigma^2/2)t, \sigma^2 t \bigr)\).

Exercise 6.5 Suppose the volatility of a stock is \(\sigma = 0.33\) (per year). Let \(S_d(n)\) and \(S_m(n)\) denote the stock prices at the ends of day \(n\) and month \(n\) respectively. What are the standard deviations of

  1. \(\log (S_d(n)/S_d(n-1))\), and
  2. \(\log (S_m(n)/S_m(n-1))\).

Exercise 6.6 The price of a security follows a geometric BM with drift \(\mu = 0.15\) and volatility \(\sigma = 0.24\). With \(S_0 = 40\), what is the chance that a call option with strike price \(K = 42\) and an expiry date of four months, so \(T = 1/3\), will finish in the money i.e. \(S_T > 42\)?

Exercise 6.7 Is it sufficient to know the parameters \(K\), \(T\), \(\sigma\), \(S_0\) and \(r\) to calculate \(\P\bigl( S_T > K \bigr)\)? If not, what else is needed?

Exercise 6.8 What is the cost of a call option with strike price 0? Supply an intuitive argument and check what price the Black-Scholes formula suggests. What happens to the cost of a call option as the expiry date increases to \(\infty\)?

Exercise 6.9 A security’s price \(S_t\) follows a geometric BM with \(\mu = 0.6\) and \(\sigma = 0.3\). The risk free rate of interest is \(r = 0.04\). Find \(\P(S_{1/2} < 0.9S_0)\). Consider a new type of investment that returns 100 at time \(1/2\) if \(S_{1/2} < 0.9S_0\) but 0 otherwise. What is the no-arbitrage price of this investment?