Category: Helpful Hints

I just posted the solutions for Problem Set 6, which include the most concise and informative explanations I’ve ever written on replicating portfolio, delta hedging, and risk-neutral valuation approaches to options pricing. You can access them via this link.

Here are some tips to effectively prepare for Midterm 2 with the limited time remaining:

1. Rest Well: Ensure you get a good night’s sleep and arrive at Foster 303 on time for the exam tomorrow, starting at 2 p.m.
2. Formula Sheet: Study the formula sheet thoroughly, ensuring you understand everything listed on this two-page document.
3. Problem Set Solutions: Visit the “Problem Set Solutions” page and confirm your understanding of the solutions for Problem Sets 6-9, the Sample Midterm 2 exam, and the Wiener Processes class problems.
4. Lecture Notes: If possible, review the lecture notes on Binomial Trees, Early Exercise of American Call and Put Options on Non-Dividend Paying Stocks, Effects of Dividends on the Pricing of European and American Options, Wiener Processes and Ito’s Lemma, Applying Ito´s Lemma to determine the probability distribution parameters for the continuously compounded rate of return and Actual versus Risk Neutral Probability of a Call Option Expiring in-the-Money.

Good luck, and I will look forward to seeing all of you tomorrow at the exam.

Exam formula sheet:

The formula sheet, also included in the exam booklet, is available at http://fin4366.garven.com/spring2024/formulas_part2.pdf.

Exam Instructions:

I recommend using the “Pricing of American Options on Non-Dividend Paying Stocks spreadsheet I uploaded earlier today to help you solve Problem Set 8.  It would also be worthwhile to spend a few minutes reviewing the Early Exercise of American Call and Put Options on Non-Dividend Paying Stocks lecture note before working on this problem set.

Upon opening the spreadsheet, you will likely see the following security alert due to the spreadsheet’s embedded Visual Basic code. If you trust the source of this file (hopefully you do! :-)), you will enable macros to proceed any further:

Since no dividends are considered in any of these spreadsheets, you’ll obtain the not-surprising result that the value of the American call is the same as that of the European call in all three cases.  However, in-the-money American puts may have higher values than otherwise identical European puts.

Feel free to use this spreadsheet when you are working on problem set 8, but DON’T turn the spreadsheet in as part of your PDF; instead, explain the reasons for the results you obtain in plain English.  You can solve both Problems 1 and 2 using the 2-period put worksheet that is included in this spreadsheet.  The trial-and-error aspect of Problem 2 can be accomplished by using Solver.

Lastly, I would like to note that it is also feasible to solve Problem 2 analytically – no calculus – just basic algebra. Extra credit will be awarded to anyone who successfully does so.

The lecture not upon which yesterday’s lecture, “American Options, Part 2,” is based (Effects of Dividends on the Pricing of European and American Options) , uses some notation that may seem unfamiliar.  Specifically, I am referring to the notation for terminal node call option payoffs that use the following syntax: C(T) = (S(T)-K)⁺. This is an abbreviation for C(T) = Max(0, S(T)-K). Furthermore, this PDF uses discrete rather than continuous compounding.

 

The deadline for Problem Set 6 has been moved from 2 pm today to 5 pm tomorrow.

In problem set 6, problems 1 and 2 feature the same time to expiration, T = 1 year. For problem 1, we define , while for problem 2, .  Broadly, this sets the stage for examining the dynamics between the discrete-time Cox-Ross-Rubinstein (CRR) model and the continuous-time Black-Scholes-Merton (BSM) model, where , and n corresponds to the number of time-steps that occur from the beginning of the binomial tree to the array of terminal nodes that exist at the expiration date T.  As you will see toward the end of today’s lecture, CRR model prices and probabilities converge to  BSM model prices and probabilities as the number of time steps becomes arbitrarily large.

Furthermore, as the number of steps, n, increases, becomes smaller. This leads to a corresponding decrease in the risk-neutral probability q and adjustments to the up (u) and down (d) factors.

Problem Set 2 is available from the course website at http://fin4366.garven.com/spring2024/ps2.pdf; its due date is Tuesday, January 30.

Problem Set 2 consists of two problems. The first problem requires calculating expected value, standard deviation, and correlation, and using this information as inputs into determining expected returns and standard deviations for 2-asset portfolios. The second problem involves using the standard normal probability distribution to calculate the probabilities of earning various levels of return by investing in risky securities and portfolios; see pp. 13-19 of the http://fin4366.garven.com/spring2024/lecture4.pdf lecture note for coverage of that topic.