Category Archives: Assignments

Time to complete teacher evaluations for Finance 4335 (along with other courses in which you are enrolled)!

As I am sure you are well aware, today and tomorrow are “Study Days” which should help you prepare for your Fall 2019 final exams. If you haven’t already done so, now would be a good time to complete teacher evaluations for Finance 4335 (along with your other courses), since the window for completing teacher evaluations officially closes as of Thursday, 12/12 at 10 am.

At Baylor, all of your professors undergo annual evaluations concerning quality of teaching, research, and service.  By completing teacher evaluations, you contribute importantly by providing the University with information that may affect not only faculty compensation and promotion/tenure decisions but also provide faculty with useful information concerning ways to improve teaching. Therefore, I encourage you to complete your teacher evaluation for Finance 4335 and all of your other courses!

Tomorrow’s review session and various other study hints for the final exam in Finance 4335

I plan to devote tomorrow’s Finance 4335 class meeting to a review session for the final exam.  Make sure you read the Finance 4335 (Fall 2019) Course Overview prior to coming to class.  This document summarizes virtually all the topics we covered this semester in Finance 4335.  I designed this document to serve as a study guide to help you prepare for the final exam.

Here are some other helpful study hints to consider:

In closing, keep in mind that if you are enrolled in Finance 4335, Section 1 (11-12:15 TR in Foster 203), then your final exam is scheduled for 9-11 a.m. on Friday, December 13.  However, if you are enrolled in Finance 4335, Section 2 (2-3:15 TR in Foster 314), then you are to take the final exam on Monday, December 16, from 4:30-6:30 p.m.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Problem Set 10, due Thursday, 12/5, is now available

Problem Set 10, due Thursday, 12/5, is now available at http://fin4335.garven.com/fall2019/ps10.pdf.  The first problem in this problem set is based on the tax asymmetry concept discussed in class last Tuesday.  The second problem relates to the so-called “underinvestment” problem, which we will cover during tomorrow’s meeting of Finance 4335.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Option Pricing Class Problem (part 3)

The complete solutions for the Option Pricing Class Problem are available in the Option Pricing Class Problem Solutions document.  Parts A and B feature solutions for put and call prices using the replicating portfolio, delta hedging, and risk neutral valuation methods.  Parts C and D feature solutions for longer dated (2 timesteps instead of just 1) puts and calls.  There, I rely upon the risk neutral valuation approach along with put-call parity.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Option Pricing Class Problem (part 2)

This is the 2nd blog posting about Option Pricing Class Problem that I passed out in class on Thursday, November 7. Here’s the (replicating portfolio) solution to part A, which requires the calculation of the price of a one-year European put option on Ripple, Inc. stock, where = $56, = $60, r = 4%, \delta t = 1, u = 1.3, and d = .9:

According to the Replicating Portfolio Approach for pricing this put option, \Delta = \displaystyle\frac{{{P_u} - {P_d}}}{{uS - dS}} = \displaystyle\frac{{0-9.60}}{{72.80 - 50.40}} = -.4286\; and B = \displaystyle\frac{{u{P_d} - d{P_u}}}{{{e^{r\delta t}}(u - d)}} = \frac{{1.3(9.6) - .9(0)}}{{1.0408(.4)}} = 29.98. Then {V_{RP}} = C = \Delta S + B = -.4286(56) + 29.98 = \$5.98.

As we discussed during yesterday’s Finance 4335 class meeting, if you already know the arbitrage-free price of a call option, then the arbitrage-free price of an otherwise identical (European, same underlying (non-dividend paying) asset, same exercise price, and same time to expiration) put option can also be determined by applying the put-call parity equation:

C + K{e^{ - r\delta t}} = P + S \Rightarrow P = C + K{e^{ - r\delta t}} - S = \$4.33 + \$ 60{e^{ - .04}}  - \$ 56 = \$5.98.

As shown above, the put-call parity equation implies that one can create a “synthetic” put option by purchasing an otherwise identical call worth $C and bond worth K{e^{ - r\delta t}}, while also shorting one unit of the underlying asset (which generates proceeds worth $S). Now suppose that P \ne \$ 5.98; specifically, suppose that actual put is worth more (less) than the synthetic put.  If this were to happen, then one could earn riskless arbitrage profit by selling the actual (synthetic) put and buying the synthetic put (actual put).  Thus, we have reconfirmed that $5.98 is indeed the arbitrage-free price for the put. 

Similarly, “synthetic” versions of the call, the bond, and the underlying asset can be created using the following combinations of the other instruments:

  1. Synthetic Call: C = P + S - K{e^{ - r\delta t}} = \$ 5.98 + \$ 56 - \$ 60{e^{ - .04}} = \$ 4.33;
  2. Synthetic Bond: K{e^{ - r\delta t}} = P + S - C = \$ 5.98 + \$ 56 - 4.33 = \$ 60{e^{ - .04}} = \$ 57.65; and
  3. Synthetic Underlying Asset: S = C + K{e^{ - r\delta t}} - P = \$ 4.33 + \$ 57.65 - \$ 5.98 = \$ 56.

During tomorrow’s Finance 4335 class meeting, we will 1) discuss the Delta Hedging and Risk Neutral Valuation approaches to pricing calls and put (see pp. 25-34 of the Derivatives Theory, part 1 lecture note),  2) move on to a discussion of multi-period option pricing formulas (which are presented in the Derivatives Theory, part 2 lecture note), and 3) complete Parts C and D of the Option Pricing Class Problem.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Option Pricing Class Problem

Here’s the Option Pricing Class Problem that I passed out in class today.  We will be working more on this class problem during our next class meeting when we examine 1) the replicating portfolio approach to pricing a put option, 2) the “delta hedging” and “risk neutral valuation” approaches to pricing calls and puts, and 3) extending these models from a single period to multiple periods.

Today, we focused our attention on replicating portfolio approaches to pricing forward contracts and (single time-step) European call options.  In the class problem, the latter concept appears in part B.  Here’s the solution to part B:

According to the Replicating Portfolio Approach, \Delta = \displaystyle\frac{{{C_u} - {C_d}}}{{uS - dS}} = \displaystyle\frac{{12.80 - 0}}{{72.80 - 50.40}} = .5714\; and B = \displaystyle\frac{{u{C_d} - d{C_u}}}{{{e^{r\delta t}}(u - d)}} = \frac{{1.3(0) - .9(12.80)}}{{1.0408(.4)}} = - 27.67. Then {V_{RP}} = C = \Delta S + B = .5714(56) - 27.67 = \$ 4.33.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Next Tuesday’s meeting of Finance 4335 (review session for Midterm 2)

Next Tuesday’s  Finance 4335 class meeting will be devoted to a review session for Midterm 2, which is scheduled for the following Tuesday, November 5. I highly recommend reviewing problem sets 5-7 and the various class problems (insurance economics, moral hazard, adverse selection, portfolio theory, capital market theory). Solutions for all of these are available at http://risk.garven.com/category/problem-set-solutions. Also, try working the Spring 2019 Midterm 2 before tomorrow’s class if time permits. If you prepare this way, it will be a particularly productive class meeting indeed!

Rothschild-Stiglitz model and related numerical example from last Thursday’s class meeting

This past Thursday, we discussed the logic behind the Rothschild-Stiglitz “separating equilibrium” model (see pp. 22-24 of the Moral Hazard and Adverse Selection lecture note) and provided a numerical illustration of its inner workings .

From our initial study of adverse selection in insurance markets (see pp. 17-21 of the Moral Hazard and Adverse Selection lecture note), we find that low-risk insureds cross-subsidize high-risk insureds when pooled premiums (based upon the average of the expected costs for both risk-types) are charged. In the “dynamic” version where there are many different risk-types (see Adverse Selection Dynamics Class Problem), this results in the so-called “insurance death spiral”. The death spiral begins with the exit of the lowest risk members of the pool, because pooling makes insurance too expensive for them and they are better off self-insuring. Their exit causes premiums for remaining pool members to increase, which motivates even more lower risk members to also exit, further shrinking the risk pool and making it even more expensive. Unchecked, this dynamic ultimately results in the failure of the insurance market.

The purpose of the static (2 risk-types) Rothschild-Stiglitz model is to show how insurance contract design can mitigate the adverse selection problem described in the previous paragraph. By offering full coverage contracts (based on the high-risk loss probability) and partial coverage (based on the low-risk probability), the high-risk and low-risk types credibly confirm whether they are high-risk or low-risk by their contract choices.  Since the full coverage contract provides high-risk types with greater expected utility than the partial coverage contract, and the partial coverage contract provides low-risk types with greater expected utility than the full coverage contract,  voilà – the adverse selection problem goes away because the insurer now knows who’s who!

The problem that we worked on toward the end of Thursday’s class meeting provides a  numerical illustration of the Rothschild-Stiglitz model. Here is the problem description (from pg. 24 of the Moral Hazard and Adverse Selection lecture note):

Note that Policy A represents actuarially fair full coverage based on the high-risk probability, whereas Policy C represents actuarially fair partial coverage based on the high-risk probability.  Without any further calculation, the Bernoulli principle implies that high-risk types will prefer Policy A over Policy C, and that Policy A and Policy C are preferred to self-insurance.  Furthermore, Policy B will never be offered, since high-risk types prefer Policy B over A and the insurer would lose $19.50 ($65-$45.50) per high-risk type if it offered Policy B.

Since we are interested in determining the policy pair which maximizes (expected value of) profit, it all boils down to whether  the insurer offers Policy C or D.  We already know that the high-risk types prefer A over C.  We need to determine whether the low-risk types prefer C or D, and whether there’s any possibility that high-risk types might defect from A to D if D were offered (note that the choice of D over A by high-risk types loses money for the insurer, since the expected cost of 30% coverage of high-risk types costs $19.50, and policy D’s premium is only $13.65).  Furthermore, while we know that high-risk types prefer A to C, we don’t yet know  the preference ordering by low-risk types of self-insurance, Policy C, and Policy D.  Under Policy C, the expected profit per low-risk type is $39 – .6(35) = $18, but it is only $13.65-$10.50 = $3.15 under Policy D.

The following spreadsheet provides with the answers that we need (clicking on the picture below brings up the spreadsheet from which this picture is obtained; see the worksheet labeled as “RS (Class Problem)”):

The various calculations in this worksheet confirm our intuition – the profit maximizing pair is A and C.  If A and C is offered, then the insurer earns expected profit of $0 on A per high-risk type (because A is purchased exclusively by high-risk types) and expected profit of $18 on C (because C is purchased exclusively by low-risk types).  If Policy D is offered instead of Policy C, then high-risk types still prefer A (and yield expected profit per high-risk type of $0), whereas low-risk types prefer D (and yield expected profit per low-risk type of $3.15)