Midterm Exam 2 information

Midterm 2 will be given during class on Thursday, April 4. This test consists of 4 problems. You are only required to complete 3 problems. At your option, you may complete all 4 problems, in which case I will throw out the problem on which you receive the lowest score.

The questions involve topics which we have covered since the first midterm exam. I expect my Finance 4366 students to demonstrate mastery concerning how arbitrage-free prices for European and American call and put options obtain within the binomial framework. Irrespective of the framework you apply (e.g., delta hedging, replicating portfolio, or risk neutral valuation), prices obtained via these methods are arbitrage-free in the sense that if the market price is not equal to the arbitrage-free price, then you can earn riskless trading profits without having to commit any of your own capital. Other than the binomial model, there’s also a question pertaining to the “Wiener Processes and Ito’s Lemma” readings and related class discussions and problems.

By the way, I have posted the formula sheet that I plan to use on the exam at the following location: http://fin4366.garven.com/spring2019/formulas_part2.pdf.

This coming Tuesday’s will be devoted to a review session for midterm exam 2. If you haven’t already done so, I highly recommend that you review Problem Sets 6-8 and also try working the Sample Midterm 2 Exam prior to coming to class on Tuesday.

Some Problem Set 8 Hints

Here are some “helpful hints” pertaining to problem set 8, which is due at the beginning of class on Tuesday, April 2.

1. Problem 1 asks for probability distributions. Given that the company’s cash position follows a generalized Wiener process with a drift rate of .2 per month and a variance rate of .5 per month (with an initial cash position of $3 million), then the probability distributions for 1 month, 6 months, and 1 year are all normal, and it’s your job to determine what the parameter values are for the mean and variance of the company’s cash position.  Note that the generalized Wiener process implies that the expected value of the cash position will equal the initial value of the cash position ($3 million) plus \mu multiplied by the number of months (1, 6, and 12), and the variance of the cash position will be the monthly variance rate multiplied by the number of months.   The probability of a negative cash position in x months time is calculated by finding the z statistic, i.e., z(x) = (0 – (3 + .2(x))/(\sqrt {{\sigma ^2}x}), and then plugging the z statistic into Excel or relying upon the Standard Normal Distribution Function (“z”) Table.

Perhaps the most challenging aspect of problem 1 is part 3: “At what time is the probability of a negative cash position the greatest?” The trade-off here is that as time passes, the expected value of the cash position increases linearly, whereas the volatility increases in a non-linear fashion. To solve this problem, note that the probability of the cash position being negative is maximized when (3 + .2(x))/(\sqrt {{\sigma ^2}x})) is minimized (note that this will yield the most negative possible value for z(x)).

2. The key to solving problem 2 is to recognize that since the stock price is lognormally distributed, the natural logarithm of the stock price is normally distributed. Equation 8 from my “Applying Ito’s Lemma to determine the parameters of the probability distribution for the continuously compounded rate of return” reading shows the distribution in this case:

From this expression, it follows that the probability distribution for log returns is \ln ({S_T}/S) \sim N\left( {(\mu - .5{\sigma ^2})T,{\sigma ^2}T} \right). Thus, the “true” (i.e., actual as opposed to risk neutral) probability that the option will expire in the money is equal to 1-N(z), where

z = \left( {\ln (K/S) - (\mu - .5{\sigma ^2})T} \right)/\sigma \sqrt T.

Also, as we have previously mentioned in class, the risk neutral probability that the option will expire in the money is given by the N({d_2}) term which appears in the Black-Scholes-Merton call option pricing formula (see equation (30) of Teaching the Economics and Convergence of the Binomial and Black-Scholes Option Pricing Formulas, for a description of this pricing formula).

Upcoming extra credit opportunities in Finance 4366

I have decided to offer the following extra credit opportunities for Finance 4366. You can earn extra credit by attending and reporting on either or both of these talks. I will use the grade(s) you earn on your report(s) to replace your lowest quiz grade(s) in Finance 4366. Each report should be in the form of a 1-2 page executive summary in which you summarize and critique each lecture. In order to receive credit, reports must be submitted via email to options@garven.com in either Word or PDF format by no later than Monday, April 1 at 5 p.m.

  1. John F. Baugh Center for Entrepreneurship & Free Enterprise: American Enterprise Institute President Arthur Brooks will present “Love Your Enemies”. Wednesday, March 27th, 5-6:30 pm in Foster 240.
  2. Roy B. Albaugh Lecture: Ian Hutchinson, Ph.D., professor of nuclear science and engineering at MIT and author of several publications on religion and science, will present “Can a Scientist Believe in Miracles?” Thursday, March 28th, beginning at 4 p.m. in Bennett Auditorium (in Draper Academic Building, 1420 S. Seventh St.).

Finance 4366