Problem 2 in Problem Set 11 is a rescaled version of the class problem discussed last Thursday (see pages 1719 of the lecture note at http://fin4335.garven.com/fall2023/risk_costly.pdf). In both the lecture note and Problem Set 11, reinvestment has a positive net present value. There’s no underinvestment problem with zero or moderate debt, as in such cases, shareholders capture the full benefit (NPV) from reinvesting. However, excessive debt, leading to default in the loss state, creates the perverse incentive for shareholders to “underinvest” because while they bear the reinvestment cost, the firm’s creditors capture the reinvestment benefit. Coordinating finance and risk management decisions ensures shareholders don’t default on the firm’s promised debt payment while also ensuring that they, and not the creditors, capture the full benefit of the decision to reinvest. In other words, coordinating finance and risk management decisions solves this moral hazard by mitigating the incentive conflict created by limited liability.
Problem Set 9 helpful hints – part 2 of 2
Here are some helpful hints for Problem Set 9, problem 2:
 Scenario A requires solving for call and put option prices using the BlackScholesMerton option pricing formulas. See the Part 2 option pricing lecture note, page 21, for a numerical illustration of how to do this.
 Scenario B requires finding the current price of the underlying asset, where the call, put, and exercise prices are all given. Solve the putcall parity equation () for S.
 Scenario C requires finding the exercise price, where the call, put, and underlying asset prices are all given. Solve the putcall parity equation for K.
 Scenario D requires finding for a call option worth $2.38 and a put option worth $3.60. Feel free to use the BlackScholes spreadsheet from the course website, or better yet, create your own Excel spreadsheet in which you solve for the call and/or the put by varying (this can be accomplished either via trial and error or better yet, by using either Solver or Goal Seek). An important lesson you’ll learn from this part of problem 2 is that call and put option prices are positively related to .
Problem Set 9 helpful hints – part 1 of 2
During last Thursday’s class meeting of Finance 4335, we completed our coverage of the CoxRossRubinstein (CRR) model (as outlined on pages 911 of the Part 2 option pricing lecture note and pages 1113 of the Teaching the Economics and Convergence of the Binomial and BlackScholes Option Pricing Formulas assigned reading). This coming Tuesday’s class meeting will be devoted to 1) showing how CRR model probabilities and prices converge to BSM (BlackScholesMerton) model probabilities and prices as the number of timesteps becomes arbitrarily large, and 2) showing how the BSM model can be applied to the pricing and management of credit risk.
In the meantime, it’s not too early to begin working on Problem Set 9, Problem 1. Here are some helpful hints to consider:
 Since the call option described in Problem 1, Part A is initially outofthemoney (i.e., since S = $18 and K = $20), there will be terminal nodes at which the call option expires inthemoney and others at which it expires outofthemoney. By solving the equation and rounding to the nearest integer greater than b (referred to in the abovereferenced sources as the parameter “a“), this indicates the minimum number of up moves required such that this call option expires in–the–money. Once you have this information, you can consider only those terminal nodes at which the call option expires inthemoney (which are nodes a through n) and calculate the call option price by applying the CRR call option pricing equation:
 Part B: Apply the putcall parity equation () to solve for the put option price.
Summary of the Oct. 31 “Teaching the Economics and Convergence of the Binomial and BlackScholes Option Pricing Formulas” assigned reading
Here is a summary of the key points from the Oct. 31 Teaching the Economics and Convergence of the Binomial and BlackScholes Option Pricing Formulas assigned reading, along with Study Questions and Answers:
Derivatives and OptionsTo download this PDF document, click here.
Summary of the Oct. 31 “Derivatives and Options” assigned reading
Here is a summary of the key points from the Oct. 31 Derivatives and Options assigned reading, along with Study Questions and Answers:
Derivatives and OptionsTo download this PDF document, click here.
Outlines and Study questions
I have prepared a handful of outlines and study questions related to some of the Finance 4335, Part 2 reading assignments:
Midterm 2 Exam Helpful Hints
Students will likely find the study guide helpful in preparing for the second midterm exam in Finance 4335 (scheduled for Thursday, October 26, in class). The exam covers:

 Topics 810 as listed on the course lecture notes page (insurance economics, asymmetric information, portfolio theory, and capital market theory),
 Readings from October 3 – 17 as listed on the course readings page, and
 Problem sets 57 as listed on the course problem sets page.
During the past month, we have also worked on class problems related to the abovementioned topics. The class problems and their solutions, along with solutions for problem sets 57, are available on the Problem Set Solutions page.
A formula sheet will be included on the last page of the exam booklet; this same formula sheet can be downloaded from http://fin4335.garven.com/fall2023/formulas_part2.pdf. On this exam, you must only complete three of four problems. If you complete all four problems on the exam, only the three highestscoring problems will count toward your Midterm Exam 2 grade. Each problem will be worth 32 points, and you will receive 4 points for including your name on the exam booklet. Thus, the maximum number of points possible on Midterm Exam 1 will be 100.
Whenever you take an exam in Finance 4335, it is important to not only show your work but also provide complete answers for each question; i.e., besides producing appropriate numerical results, also clearly explain your results using plain English.
Important Notice: Redownload Problem Set 7 from the Finance 4335 course website!
Earlier this afternoon, I found a typo in Problem 2 of Problem Set 7. I fixed that typo and also lightly edited Problem 2 for greater clarity. I didn’t make any changes to Problem 1.
If you downloaded Problem Set 7 before 4:30 p.m. today, please replace it with this updated version. If you’ve already worked on Problem Set 7, don’t worry about Problem 1, but you might want to check your answers for Problem 2 in light of the edits I made.
An important clarification of the logical principles behind the stochastic dominance model
On problem set 4, part D, most of you had no apparent difficulty in correctly establishing that the sum of the differences between the cumulative distribution (CDF) for risk 2 and the CDF for risk 1 is positive. However, many of you drew the wrong conclusion, claiming that since the sum of differences between and came out to a positive number, it followed that risk 2 second order stochastically dominates risk 1. Actually, this result implies the opposite; i.e., that risk 1 second order stochastically dominates 2. The purpose of this blog posting is to clarify everyone’s understanding of the logic behind the stochastic dominance model.
The onepage exam formula sheet includes section 4 which explains that risk i dominates risk j, in both the first and second cases, when 1) the cumulative distribution function (CDF) of the i^{th} risk is either less than or equal to the CDF of the j^{th} risk for all states (first order dominance), or 2) the sum of the differences between the j^{th} risk CDF and the i^{th} risk CDF for all states is positive (second order dominance):
While the math behind first and second order stochastic dominance is summarized in my optional reading entitled “Technical Note on Stochastic Dominance and Expected Utility”, the intuition for first and second order stochastic dominance can be seen in the figures featured on pages 9 and 12 of my DecisionMaking under Risk and Uncertainty, part 4 lecture note.
In the above figure from page 9 of my DecisionMaking under Risk and Uncertainty, part 4 lecture note, the G risk has 50% of a $0 payoff, and 25% each of a $10 payoff and a $100 payoff. The F risk involves removing 25 percentage points off the $0 payoff and adding 25 percentage points extra to the $100 payoff, and both F and G have a 25% probability of $10 payoffs. Graphically, this ensures that F first order stochastically dominates G; i.e., G(W_{s}) is greater than or equal to F(W_{s}) for all s, which also implies that E_{F}[U(W)] > E_{G}[U(W)]. Intuitively, the picture which gets rendered by this analysis shows that most of the probability mass of the stochastically dominant risk (in this case, F) lies below the probability mass of the stochastically dominated risk (in this case, G). Furthermore, since risk F first order stochastically dominates risk G, risk F also second order stochastically dominates risk G because G(W_{s}) – F(W_{s}) > 0 for $0 and $10 payoffs, and G(W_{s}) – F(W_{s}) = 0 for the $100 payoff.
Next consider the figure from page 12 of my DecisionMaking under Risk and Uncertainty, part 4 lecture note:
Here, G(W_{s}) – F(W_{s}) > 0 for payoffs ranging from 15, G(W_{s}) – F(W_{s}) < 0 for payoffs ranging from 58, and G(W_{s}) – F(W_{s}) = 0 payoffs ranging from 812. Thus, there is no first order dominance. However, since the positive difference between G(W_{s}) – F(W_{s}) for payoffs ranging from 15 exceeds the negative difference between G(W_{s}) – F(W_{s}) for payoffs ranging from 58, the sum of G(W_{s}) – F(W_{s}) over the entire range of payoffs comes out positive. Thus, risk F second order stochastically dominates risk G, which also implies that E_{F}[U(W)] > E_{G}[U(W)].
Midterm Exam 1 Formula Sheet and Helpful Hints
The formula sheet for Midterm Exam 1 is now available for downloading at http://fin4335.garven.com/fall2023/formulas_part1.pdf.
Midterm Exam 1 comprises four problems, with a requirement to solve three out of the four. If all four problems are attempted, only the three highestscoring problems will contribute to your Midterm Exam 1 grade. Each problem is valued at 32 points, and you will receive an additional 4 points for including your name on the exam booklet. Therefore, the maximum attainable score for Midterm Exam 1 is 100 points.
On the exam, it is important not only to neatly and concisely present your analysis, but also to provide thorough explanations that showcase your understanding of the concepts being examined. In other words, be sure to articulate your findings using clear and concise language.
In my estimation, the information I have shared in this blog post, coupled with the guidance provided in the study guide titled “Finance 4335 Midterm 1 Synopsis,” should prove very helpful as you prepare for the upcoming midterm exam.
Good luck!