Category Archives: Assignments

Today’s class problem solutions for parts A-D applying risk neutral valuation

Today, we discussed the risk neutral valuation approach to pricing options, and today’s class problem assignment was to work parts A through D of the  Option Pricing Class Problem, relying solely upon the risk neutral valuation approach.

As I pointed out during today’s class meeting, the replicating portfolio and delta hedging approaches both imply that a risk neutral valuation exists between an option (both the call and put varieties) and its underlying asset.  This is analytically shown in sections 5 and 6 (located on pp. 8-9) of my Binomial Option Pricing Model (single-period) teaching note which was assigned for October 23.  A particularly useful advantage of the risk neutral valuation approach (compared with the replicating portfolio and delta hedging approaches) is that it is computationally simpler and particularly well suited for modeling multi-timestep option pricing problems.

Here are the solutions for parts A through D (click on the image for a full-size PDF version that you can print out):

Today’s class problem and solutions for parts A and B

Today, we worked on (among other things), parts A and B of the Option Pricing Class Problem. Here are the solutions for parts A and B (click on the image for a full-size PDF version that you can print out):

Be sure to bring your class problem with you to class next Tuesday. I will introduce a third method for pricing called the risk-neutral valuation approach which not only simplify the pricing problem for one timestep but also make it easier to calculate option prices for multiple timesteps. Spoiler alert – as we let the number of timesteps become arbitrarily large for a given discrete time interval, the famous Black-Scholes option pricing formula obtains.

Problem Set 6 Hints and Spreadsheet

I just posted a new problem set on the course website; specifically, Problem Set 6, which is due at the beginning of class on Thursday, October 11.

In question 1, part C of Problem Set 6, I ask you to “Find the maximum price which the insurer can charge for the coinsurance contract such that profit can still be earned while at the same time providing the typical Florida homeowner with higher expected utility from insuring and retrofitting. How much profit will the insurer earn on a per policy basis?” Here are some hints which you’ll hopefully find helpful.

For starters, keep in mind that in part A, insurance is compulsory (i.e., required by law), whereas, in parts B and C, insurance is not compulsory. In part B of question 1, you are asked to show that the homeowner retrofits while choosing not to purchase insurance. However, in part C, the homeowner might retrofit and purchase coverage if the coinsurance contract is priced more affordably. Therefore, the insurer’s problem is to figure out how much it must reduce the price of the coinsurance contract so that the typical Florida homeowner will have higher expected utility from insuring and retrofitting compared with the part B’s alternative of retrofitting only. It turns out that this is a fairly easy problem to solve using Solver. If you download the spreadsheet located at, you can find the price at which the consumer would be indifferent between buying coinsurance and retrofitting compared with self-insurance and retrofitting. You can determine this “breakeven” price by having Solver set the D28 target cell (which is expected utility for coinsurance with retrofitting) equal to a value of 703.8730 (which is the expected utility of self-insurance and retrofitting) by changing cell D12, which is the premium charged for the coinsurance. Once you have the “breakeven” price, all you have to do to get the homeowner to buy the coinsurance contract is cut its price slightly below that level so that the alternative of coinsurance plus retrofitting is greater than the expected utility of retrofitting only. The insurer’s profit then is simply the difference between the price that motivates the purchase of insurance less the expected claims cost to the insurer (which is $1,250).

Guidelines for completing parts B and D on Problem Set 5

For parts B and D on Problem Set 5, you may solve these problems via either calculus or a spreadsheet model.

If you decide to implement a spreadsheet model, then you must email your spreadsheet model to “” prior to the start of class on Tuesday. In the problem set that you turn in at the beginning of class on Thursday, please reference your spreadsheet when you explain your answers for parts B and D problem. However, if you rely upon the calculus for maximizing expected utility, then no spreadsheet is necessary, although you might consider validating the result that you obtain via calculus with a spreadsheet model anyway. Or, you could validate your spreadsheet model with the calculus.

In order to solve this problem via spreadsheet, you’ll need to use the so-called Solver Add-in. The instructions for loading the Solver add-in into Excel are provided at the following webpage:

Problem Set #4 – new (replacement) version uploaded as of 6:15 p.m. this evening

Upon closer inspection, I noticed that my first problem on Problem Set 4 was identical to the Risk Aversion Class Problem that we worked in class last Thursday, so as of 6:15 p.m. this evening, I uploaded a new (replacement) version of Problem Set 4, which is due at the beginning of class on Thursday, 9/20. Therefore, if any of you downloaded this problem set  prior to 6:15 p.m. today, be sure to replace it with this new Problem Set 4, which looks like this:

Risk pooling spreadsheet – solutions for yesterday’s class problem

Here’s a screenshot from a spreadsheet that I coded for the risk pooling class problem (linked below) that we discussed during class yesterday. We found in class yesterday (and now find in this spreadsheet today), that if risks are independent and identically distributed, then by increasing the number of policies in the risk pool, the probability that the average loss exceeds $1,500 declines as we add policies. Without risk pooling, the probability of a “large” loss of $1,500 is 30.85%; with 5 policies, it is 13.18%, and with 10 policies it is 5.69%. However, if risks are positively correlated, then both unique and systematic risks influence this calculation. For example, with 10 policies that have .1 correlation, the probability that the average loss exceeds $1,500 is 12.57% (compared with 5.69% when there is zero correlation):

Risk Pooling Spreadsheet.xlsx

Problem Set 1 Hint…

Problem Set 1 is due at the beginning of class tomorrow. Here is a hint for solving the 4th question on this problem set.

The objective is to determine how big a hospital must be so that the cost per patient-day is minimized. We are not interested in minimizing the total cost of operating a hospital; a sure-fire way to minimize total cost would be to not even have a hospital in the first place. Indeed, if you were to differentiate the total cost function given by C = 4,700,000 + 0.00013X2 with respect to X, this is what the math would tell you.

In part “a” of the 4th question, you are asked to “derive” a formula for the relationship between cost per patient-day and the number of patient days; in other words, what you are interested in determining is what is the most cost-efficient way to scale a hospital facility such that the cost per patient-day is minimized. Once you obtain that equation, then you’ll be able to answer the question concerning optimal hospital size.

Lagrangian Multipliers

There is a section in the assigned “Optimization” reading tomorrow on pp. 74-76 entitled “Lagrangian Multipliers” which (as noted in footnote 9) may be skipped without loss of continuity.  The primary purpose of this chapter is to re-acquaint students with basic calculus and how to use the calculus to solve so-called optimization problems.  Since the course only requires solving unconstrained optimization problems, there’s no need for Lagrangian multipliers.

Besides reading the articles entitled “Optimization” and “How long does it take to double (triple/quadruple/n-tuple) your money?” in preparation for tomorrow’s meeting of Finance 4335, make sure that you fill out and email the student information form as a file attachment to prior to the beginning of tomorrow’s class.  As I explained during yesterday’s class meeting, this assignment counts as a problem set, and your grade is 100 if you turn this assignment in on time (i.e., sometime prior to tomorrow’s class meeting) and 0 otherwise.

How to know whether you are on track with Finance 4335 assignments

At any given point in time during the upcoming semester, you can ensure that you are on track with Finance 4335 assignments by monitoring due dates which are published on the course website. See for due dates pertaining to reading assignments, and for due dates pertaining to problem sets. Also keep in mind that short quizzes will be administered in class on each of the dates indicated for required readings. As a case in point, since the required readings entitled “Optimization” and ” How long does it take to double (triple/quadruple/n-tuple) your money?” are listed for Thursday, August 23, this means that a quiz based upon these readings will be given in class on that day.

Important assignments due on the second class meeting of Finance 4335 (scheduled for Thursday, August 23) include: 1) filling out and emailing the student information form as a file attachment to, 2) subscribing to the Wall Street Journal, and 3) subscribing to the course blog. A completed student information form is graded as a problem set and receives 100 points; if you fail to turn in a student information form, then you will receive a 0 for this “problem set”. Furthermore, tasks 2 and 3 listed above count toward your class participation grade in Finance 4335.

Regarding the student information form, I prefer that you complete this form (by either typing or writing) and email it to prior to the beginning of class on Thursday.  However, if you prefer, you may turn in a hard copy instead at the beginning of class on Thursday.

Instructions for subscribing to the Risk Management Course Blog

A course blog has been established for Finance 4335 at the address; it is also linked from the “Course Blog” button located on the course website. I recommend that you follow the risk management course blog regularly via email, RSS, Facebook, and/or Twitter.

The risk management course blog provides me with a convenient means for distributing important announcements to the class. Topics covered on the course blog typically include things like changes in the course schedule, clarifications and hints concerning problem sets, information about upcoming exams, announcements concerning extra credit opportunities, and short blurbs showing how current events relate to many of the topics which we cover in Finance 4335.

If you already are familiar with RSS, this is a great way to subscribe to the options, futures, and other derivatives course blog. By going to the webpage, you can subscribe by using Firefox’s Live Bookmarks feature, Internet Explorer’s RSS feed subscription feature, or an RSS reader. If you are either a Facebook or Twitter user, everything that is posted on the options, futures, and other derivatives course blog is automatically posted to Facebook and “tweeted”, so you can also subscribe by “liking” the Finance 4335 Facebook page or by “following” @fin4335 on Twitter. Finally, you can also subscribe via email. The remainder of this blog entry explains how to subscribe to the risk management course blog via email.

Email Subscription Instructions:

Email Subscription Instructions: If you would like to receive the risk management course blog via email, you can do this by going to and entering your email address in the form provided on the left hand side of that webpage:

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After clicking “Subscribe”, the following information will appear on the screen:Screen Shot 2016-06-16 at 3.45.08 PM
Next, check for an email from “Risk Management Blog <> ”:

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Next, simply click the “Confirm Follow” button. This will cause you to receive the following email:

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From that point forward, whenever I post to the course blog, you will immediately receive a nicely formatted version of the blog posting via email. Also, you can opt to change your delivery preferences, or even cancel your subscription.