Problem Set 1

Stochastic Foundations for Finance

Instructions: This problem set is due on 11/25 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 GoingUp Corp. has attracted significant media attention due to its strong upside potential. Financial analysts agree that the stock price follows a geometric Brownian motion with distribution \ln(S_{t}) \sim \mathcal{N}\left(\ln(S_{0}) + \left(\mu - \frac{\sigma^{2}}{2}\right) t, \sigma^{2} t\right). The stock has a drift of \mu = 25\% per year and a volatility of \sigma = 60\% per year. The current stock price is S_{0} = \$220. Calculate the probability that the stock price exceeds $240 ten months from now.

Problem 2 Consider a stock with price S_{t} at time t that follows a geometric Brownian motion (GBM) such that \ln(S_{t}) \sim \mathcal{N}\left(\ln(S_{0}) + \left(\mu - \frac{\sigma^{2}}{2}\right) t, \sigma^{2} t\right). The stock has an expected return of 20% per year and a volatility of 45% per year. The current stock price is $220.

  1. Calculate the expected stock price 15 months from now.
  2. Calculate the mean and standard deviation of the log-stock price 15 months from now.
  3. Calculate the 95% confidence interval for \ln(S_{T}) 15 months from now, and report the corresponding values for S_{T}.

Problem 3 Suppose an asset with price S_{t} at time t follows a geometric Brownian motion (GBM) with distribution \ln(S_{t}) \sim \mathcal{N}\left(\ln(S_{0}) + \left(\mu - \frac{\sigma^{2}}{2}\right) t, \sigma^{2} t\right). The asset has an expected return of \mu = 10\% per year and a volatility of \sigma = 8\% per year. The current asset price is S_{0} = \$1.4.

  1. Calculate the expected value of S_{t} one year from now.
  2. Calculate the expected value of 1 / S_{t} one year from now.

Problem 4 Suppose the stock price follows a geometric Brownian motion (GBM) characterized by drift \mu and instantaneous volatility \sigma, represented as: \frac{\mathop{}\!\mathrm{d}S}{S} = \mu \mathop{}\!\mathrm{d}t + \sigma \mathop{}\!\mathrm{d}B. If a client wishes to understand the volatility exposure from purchasing the square root of the stock price, what would your response be? Explain your reasoning in full sentences.

Problem 5 Suppose a European put option is written on a non-dividend-paying stock. The current stock price is $100, the strike price is $90, the risk-free interest rate is 5% per year with continuous compounding, the volatility is 35% per year, and the option expires in eight months. Using the Black-Scholes model, calculate the price of this put option.

Problem 6 A non-dividend-paying stock currently trades for $32. The current risk-free rate is 5% per year with continuous compounding. Using the Black-Scholes model, you compute the following quantities for a European call option written on the stock with exercise price $30 and expiring in three months.

\sigma d_{1} d_{2} Call Price
5% 3.094 3.069 $2.37
10% 1.566 1.516 $2.41
15% 1.065 0.990 $2.56
20% 0.820 0.720 $2.76
25% 0.679 0.554 $3.00
30% 0.589 0.439 $3.27
35% 0.528 0.353 $3.54
40% 0.485 0.285 $3.82
45% 0.455 0.230 $4.11
50% 0.433 0.183 $4.40

The current market price of the call option is $3.54.

  1. Determine the implied volatility of the stock.
  2. Calculate the delta of the call option.
  3. Calculate the price of a European put option with the same characteristics as the call.
  4. Would the price of the call be zero if the volatility suddenly drops to zero? Explain.

Problem 7 Consider a European put option expiring in nine months and strike price $100 written on a non-dividend paying stock. The risk-free rate is 5% per year with continuous compounding and the stock price is $98.

  1. What is the minimum price for the put that would allow you to compute its implied volatility?
  2. What would be the value of the put if the volatility is extremely large? Explain your reasoning clearly.

Problem 8 Suppose that the sales team of a trading desk just sold a European call option contract (100 European call options) to an important client. The contract is written on a non-dividend-paying stock that trades for $112, expires in 18 months, and has a strike price of $110. The risk-free rate is 5% per year with continuous compounding. A trader on the desk estimates that the volatility of the stock returns is 40% and expects it to remain constant for the life of the contract.

  1. How many shares of the stock does the trader need to buy or sell initially to hedge the exposure created by the sale of the contract?
  2. How many zero-coupon risk-free bonds with face value $110 and expiring in 18 months does the trader need to buy or sell to ensure that the strategy is self-financing?

Problem 9 The S&P 100 index currently stands at 2,392 and has a volatility of 50% per annum. The risk-free rate of interest is 4% per annum and the index provides a dividend yield of 1% per annum. Calculate the value of a three-month European put with strike price 2,300.

Problem 10 Suppose that the spot price of the Canadian dollar (CAD) is USD 0.72 and that the CAD/USD exchange rate has a volatility of 5% per annum. The risk-free interest rates in Canada and the United States are 2.75% and 4.50% per annum, respectively. Calculate the value of a European option to buy CAD 10,000,000 for USD 7,500,000 in six months. Use put-call parity to calculate the price of a European option to sell CAD 10,000,000 for USD 7,500,000 in six months.

Normal Distribution Table

The table below computes \Phi(z) = \operatorname{P}(Z \leq z) if Z \sim \mathcal{N}(0, 1) and z \geq 0. The rows denote the first decimal whereas the columns denote the second decimal. If you need \Phi(-z), you can always use \Phi(-z) = 1 - \Phi(z).

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990