Mock Midterm 2
Questions
Problem 1 (3 pts) A non-dividend paying stock price is currently $120. It is known that at the end of 8 months it will be either $150 or $90. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of an 8-month European call option with a strike price of $125? Use the replicating portfolio argument and indicate the number of shares required to hedge the position. Use a risk-free bond with face value $100 in your computations.
Problem 2 (3 pts) A stock price is currently $50. It is known that at the end of 4 months it will be either $60 or $40. Analysts at a prestigious bank have determined that there is a 70% probability that the stock will be $60 in four months. The risk-free interest rate is 8% per annum with continuous compounding. The stock pays a dividend yield of 4% per year.
- What is the value of a 4-month European put option with a strike price of $52? Use the risk-neutral probability method to compute the value of the option.
- Using put-call parity, determine the price of a European call with the same characteristics.
Problem 3 (2 pts) The stock of Boulder Corp. trades at $130 per share and pays a dividend yield of 10% per year with continuous compounding. The stock could trade at $160 or $100 in six months, depending on how the economy evolves. The risk-free rate is currently 5% per year. A banker just sold to a client an instrument that pays in six months $5,000 if the stock price goes up and $1,000 otherwise. How many shares of the stock (per instrument sold) does she need to buy or sell to hedge her exposure?
Problem 4 (2 pts) A stock that pays a dividend yield of 4% per year costs $200. Financial analysts at all major investment banks agree there is a 60% probability that the stock will trade for $250 next year if the company succeeds in developing a novel AI algorithm. Otherwise, the stock price could fall to $100. The risk-free rate is 4% per year and is expected to remain constant. What should be the price of a security that pays $0 next year if the stock goes up and $1,000 otherwise?
Problem 5 (4 pts) The current price of a non-dividend-paying stock is $250. Every three months, it is expected to increase or decrease by 20%. The risk-free rate is 10% per year. Compute the price of an American put option with a strike price of $255 and a maturity of six months written on the stock. In each node, explain why you would exercise the put or not.
Problem 6 (2 pts) A friend of yours, who is attending a prestigious university, says: “I learned in class that you should never exercise early an American call option written on a non-dividend paying asset. On the other hand, an American put option with similar characteristics should be exercised early if the stock price is low enough” Determine whether the statement is true or false.
Problem 7 (4 pts) Let S be the price of GOOG stock that follows a geometric Brownian motion such that dS = \mu S dt + \sigma S dW. Your sales team would like to launch a new product called PWR that tracks the price of GOOG to the power 1.5. Traders at your bank are excited but would like to know the dynamics of Y = S^{1.5}, since they would also like to write options on PWR.
- Write down the dynamics of Y, and determine its drift and volatility.
- If GOOG stock is currently 160, compute the probability that PWR is higher than 2,220 in six months. In your computations, use \mu = 0.15 and \sigma = 0.50.
Formula Sheet
In the following, S denotes the stock or spot price of an asset, r is the continuously-compounded risk-free rate expressed per year, \delta denotes the dividend yield, T denotes the time-to-maturity of a forward, futures or an option, and K denotes the strike price of an option or the delivery price of a forward contract.
Binomial Pricing
At any node of a binomial tree in which the stock price can move up to S_{u} = u \times S or down to S_{d} = d \times S, the risk-neutral probability of an up-move is given by q = \frac{S e^{(r - q) \Delta t} - S_{d}}{S_{u} - S_{d}} = \frac{e^{(r - q) \Delta t} - d}{u - d}, where \Delta t denotes the length of each period. To make the tree consistent with the observed volatility of stock returns, we typically choose u = e^{\sigma \sqrt{\Delta t}} and d = 1 / u.
Impact of Dividends Dividends
For European call and put options with strike price K and time-to-expiration T written on a non-dividend paying asset, we have that C - P = S e^{-\delta T} - K e^{-r T}, where C and P denote the call and put prices. Put-call parity implies the following bounds for European call and put options: \begin{aligned} \max(S e^{-\delta T} - K e^{-r T}, 0) & \leq C \leq S, \\ \max(K e^{-r T} - S e^{-\delta T}, 0) & \leq P \leq K e^{-r T}. \end{aligned}
Modeling Stock Prices
Stock prices follow a geometric Brownian motion such that \frac{dS}{S} = \mu dt + \sigma dW, \tag{1} where \mu denotes the drift, \sigma the instantaneous volatility of stock returns and W is a standard Brownian motion. If we take Y = f(S) where f is a twice-differentiable function, then Ito’s lemma implies dY = Y_{S} dS + \frac{1}{2} Y_{SS} (dS)^{2}, where Y_{S} = f'(S) and Y_{SS} = f''(S). We showed in class that (dS)^{2} = \sigma^{2} S^{2} dt.
Equation (1) implies that \ln(S_{T}) is normally distributed with mean \ln(S_{0}) + \left(\mu - \frac{1}{2} \sigma^{2} \right)T and standard deviation \sigma \sqrt{T}.
The Standard Normal Distribution
The following table reports values for \phi(z) = P(Z ≤ z), where Z \sim \mathcal{N}(0, 1).
| z | P(Z ≤ z) | z | P(Z ≤ z) | z | P(Z ≤ z) | z | P(Z ≤ z) |
|---|---|---|---|---|---|---|---|
| -2.37 | 0.0089 | -1.17 | 0.1210 | 0.03 | 0.5120 | 1.23 | 0.8907 |
| -2.32 | 0.0102 | -1.12 | 0.1314 | 0.08 | 0.5319 | 1.28 | 0.8997 |
| -2.27 | 0.0116 | -1.07 | 0.1423 | 0.13 | 0.5517 | 1.33 | 0.9082 |
| -2.22 | 0.0132 | -1.02 | 0.1539 | 0.18 | 0.5714 | 1.38 | 0.9162 |
| -2.17 | 0.0150 | -0.97 | 0.1660 | 0.23 | 0.5910 | 1.43 | 0.9236 |
| -2.12 | 0.0170 | -0.92 | 0.1788 | 0.28 | 0.6103 | 1.48 | 0.9306 |
| -2.07 | 0.0192 | -0.87 | 0.1922 | 0.33 | 0.6293 | 1.53 | 0.9370 |
| -2.02 | 0.0217 | -0.82 | 0.2061 | 0.38 | 0.6480 | 1.58 | 0.9429 |
| -1.97 | 0.0244 | -0.77 | 0.2206 | 0.43 | 0.6664 | 1.63 | 0.9484 |
| -1.92 | 0.0274 | -0.72 | 0.2358 | 0.48 | 0.6844 | 1.68 | 0.9535 |
| -1.87 | 0.0307 | -0.67 | 0.2514 | 0.53 | 0.7019 | 1.73 | 0.9582 |
| -1.82 | 0.0344 | -0.62 | 0.2676 | 0.58 | 0.7190 | 1.78 | 0.9625 |
| -1.77 | 0.0384 | -0.57 | 0.2843 | 0.63 | 0.7357 | 1.83 | 0.9664 |
| -1.72 | 0.0427 | -0.52 | 0.3015 | 0.68 | 0.7517 | 1.88 | 0.9699 |
| -1.67 | 0.0475 | -0.47 | 0.3192 | 0.73 | 0.7673 | 1.93 | 0.9732 |
| -1.62 | 0.0526 | -0.42 | 0.3372 | 0.78 | 0.7823 | 1.98 | 0.9761 |
| -1.57 | 0.0582 | -0.37 | 0.3557 | 0.83 | 0.7967 | 2.03 | 0.9788 |
| -1.52 | 0.0643 | -0.32 | 0.3745 | 0.88 | 0.8106 | 2.08 | 0.9812 |
| -1.47 | 0.0708 | -0.27 | 0.3936 | 0.93 | 0.8238 | 2.13 | 0.9834 |
| -1.42 | 0.0778 | -0.22 | 0.4129 | 0.98 | 0.8365 | 2.18 | 0.9854 |
| -1.37 | 0.0853 | -0.17 | 0.4325 | 1.03 | 0.8485 | 2.23 | 0.9871 |
| -1.32 | 0.0934 | -0.12 | 0.4522 | 1.08 | 0.8599 | 2.28 | 0.9887 |
| -1.27 | 0.1020 | -0.07 | 0.4721 | 1.13 | 0.8708 | 2.33 | 0.9901 |
| -1.22 | 0.1112 | -0.02 | 0.4920 | 1.18 | 0.8810 | 2.37 | 0.9911 |