Problem Sets 9 and 10 Q&A

Here are a couple of Q&A’s that I had late Wednesday with Finance 4335 students about Problem Sets 9 and 10:

Student 1 Problem Set 9 Question:

  • “Regarding Problem Set 9, I would like to know if I can use Goal Seek on Excel to find out the standard deviation on problem 1d. Or if there is another way to find out the standard deviation? Additionally, for problem 2, I was wondering if I got my call and put prices wrong, since they had a great disparity. Do we use BSM here or do I misunderstand the value of T ? Thank you in advance for your help.”
Dr. Garven answer to Student 1:
  • Using Goal Seek (or Solver) is a really good idea for finding the standard deviation in problem 1D. That there is a big disparity between call and put prices in problem 2 is to be expected, since the call option starts out-of-the-money whereas the put option starts in-the-money.  Furthermore, the call is only in-the-money at one terminal node, whereas the put is in-the-money at three of the four terminal nodes.  I recommend that you apply the binomial formula to calculate the call price, and then use the put-call parity equation to calculate the put price.
Student 2 Problem Set 9 and 10 Questions:
“I’m working on problem sets 9 and 10 and ran into a few questions. On problem set 10, for question 3 if the bonds are riskless does this just mean to redo the calculations from the prior questions but just with a sigma of 0%? Also on problem set 9 problem 1 part D I’m a little lost on how to extract the sigma from the d1 equation given all the other variables. Is there a simpler formula to use to get sigma? Also this may relate to the same issue, but on question 2 part C, with the increase in sigma should we redo all the work from the prior questions using the new sigma and assess the change on the prices from there or is there some formula/analysis that is easier to see how to change in sigma directly corresponds to the price of call/put options?”
Dr. Garven answer to Student 2:
  • If bonds are riskless, this implies that the yield-to-maturity on such bonds equals the riskless rate of interest.  If this is not the case, then there would be a riskless arbitrage opportunity; e.g., if a riskless bond has a yield-to-maturity exceeding the riskless rate of interest, then the investor could earn riskless arbitrage profits by borrowing at the riskless rate of interest to purchase the cheap bond.
  • Regarding sigma (i.e., the standard deviation), this can be done by trial and error, either manually or using Solver or Goal Seek in Excel.  Since 1) all parameter values for Scenarios A and D are the same except for option prices and volatility, and 2) Scenario D option prices are lower than Scenario A option prices, this implies that volatility must also be lower under Scenario D compared with Scenario A; note that both call and put option prices are positively related to volatility.  I explain this property of option prices in my video, and it is also explained on page 9 of the Derivatives and Options assigned reading.  Also, the spreadsheet referenced on the last page of the Derivatives Theory, Part 2 lecture note provides a link to a spreadsheet that solves Black-Scholes-Merton option pricing problems like this; you might consider downloading and working with that spreadsheet (the URL for which is http://fin4335.garven.com/fall2021/CRR-vs-BSM.xlsm).

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