Fall 2004 Finance 4366 Weblog

Risk Neutral Valuation, RADR, CEQ, etc.

As we go through the mathematical details concerning the derivation of the Black-Scholes model, I think it is important to step back for a moment and think about the rationale/motivation behind "risk neutral valuation" and how this concept is related to very simple valuation concepts which were (hopefully) covered in the basic principles of finance course (i.e., Finance 3310).

Risk Neutral Valuation

Perhaps the most important insight in the theory of derivatives pricing involves the notion that a "risk neutral valuation relationship" exists between the price of a derivative security and its underlying asset.  With some algebra, the Black-Scholes nonstochastic partial differential equation (see equation (8) on page 28 of http://129.62.162.249/ofod/fall2004/lecture14.ppt) can be solved for the current call option price ("C").  By inspection, the current option price depends upon the current price of the underlying stock along with other (deterministic) factors such as the rate of interest, the volatility of the underlying stock, and values of three (math) derivatives; specifically, the derivative of the call price with respect to time and the first and second derivatives of the call price with respect to the price of the underlying stock.

You may be thinking "so what?"  The answer is that this result is huge.  The Black-Scholes nonstochastic partial differential equation implies that for a given price of the underlying asset, a call option written against that asset must trade at the same price in a risk neutral economy as it would in a risk averse or risk loving economy.  Since it is easiest to price the option as if investors are risk neutral (since this allows us to be completely agnostic concerning the nature of risk preferences), this is what we do.

Finance 3310 revisited (the "RADR" approach)

Most students' first exposure to pricing of risk comes in the Finance 3310 course, where they learn about the Capital Asset pricing model.  Once this concept is introduced, students are taught to use the CAPM in order to determine the appropriate rate at which to discount the expected value of a future risky cash flow.  This approach to pricing risk is commonly known as the risk adjusted discount rate, or RADR approach.   

The CEQ approach

An alternative approach to pricing risk which was probably not covered in Finance 3310 involves the certainty equivalent, or CEQ approach.  This is a concept which I cover in some detail in the courses that I teach (i.e., FIN/RMI 4335 and FIN 4366).  The CEQ approach involves discounting the certainty equivalent value of the risky cash flow at the risk free rate of interest.  The certainty equivalent is determined by deducting the dollar value of the risk premium from the expected cash flow, and as we showed in class, can also be accomplished by summing the products of state-contingent cash flows multiplied by their corresponding "risk neutral" probabilities.

In summary, the two approaches to pricing risky assets are as follows:

• RADR approach: discount the expected value of the cash flow at a risk adjusted discount rate; e.g., as indicated by CAPM; and
• CEQ approach: discount the certainty equivalent value of the cash flow at the risk free rate.

Risk Neutral Valuation and the Integration approach to computing the value of an option

Derivatives pricing is better understood once one goes through some of the mathematical details.  I realize that for most typical undergraduate (or even master's level) finance students, stochastic calculus is not a particularly familiar topic.  Having said that, I (as well as your textbook author) believe that a basic introduction to concepts such as geometric brownian motion and Ito's Lemma make the underlying economics of the option pricing problem much more transparent; specifically, the notion of dynamic hedging.  However, once we have established this concept, I believe that the integration approach to computing the Black-Scholes option price is easier to grasp than the differential equations approach, since it makes the link between risk neutral valuation and the CEQ approach to pricing risky assets much more transparent. 

Conceptually, the integration approach to option pricing theory represents a calculus-based implementation of the CEQ approach.  Specifically, we compute the certainty equivalent value of the expected payoff on the option, and then discount it back to the present time.  The details of this are laid out in my paper entitled "Derivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Equations," as well as in the lecture note entitled "The Black-Scholes Model".

Some changes to the problem set dates/assignments

I have changed problem set 7 so that only questions 11.12, 11.13, and 11.16 will be due.  Time permitting, I might try tackling 11.14 and 11.15 in class - no promises though!

Also, the new due date for the final problem set (#8) will be Tuesday, November 30.  Problem set #8 consists of questions 12.24, 12.25, 12.26, 12.27, 12.28 from page 261 of Hull.

Against grade inflation - How to counter declining rigor in US university courses.

I am attaching a copy of a very interesting editorial about grade inflation at private universities.  This appeared recently in Nature (October 14, 2004 issue). 

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Call it Moore's law of US higher education: the quantity and quality of work that undergraduates must do to get top grades halves every decade. This is an exaggeration, of course, but many readers will recognize the sentiment. Is it just the jaded perception of cynical academics? On the contrary: the evidence suggests that there is a real problem of grade inflation in degree courses, especially at private universities. And the assessment of teachers by students, as well as parents' demands that they get what they think they've paid for, are making the problem worse.

Course evaluations were intended to give the instructor feedback about how well he or she was doing. But they rapidly became a favored tool of deans, tenure and promotion committees because they were quantifiable. Now there is an implicit understanding that if instructors give good grades, they will not be judged too severely by students. New faculty often grade more harshly than other members of the department, only to be 'punished' by students. Deans who believe that this doesn't happen are deluding themselves.

Also worrying is the idea - particularly evident at costly private universities - that students and their parents believe they are paying for a degree that will lead to a good job, rather than for a good education that will help them to think independently. The pressure on teachers to appease demanding students and parents by awarding high grades is obvious.

The consequences are all too clear. Anecdotal evidence suggests that there is a general unwarranted upward creep in grades (http://ctl.stanford.edu/Tomprof/postings/444.html). More objectively, the fact that graduate schools rely for admission criteria almost exclusively on the results of standardized tests, rather than on universities' individual grading, points to a systematic failure to ensure that grading standards are being maintained.

What to do? More universities should focus seriously on improving the instructional abilities of their faculty in programs - mandatory for new instructors - to videotape classes and analyze them with the faculty member to highlight strengths and weaknesses. And evaluations should take note of thoughtful individual comments by students, rather than relying on scores, or be abandoned.

What's Behind Edward C. Prescott's Nobel Prize?

Last month Edward C. Prescott and Finn E. Kydland won the 2004 Nobel Prize in Economic Sciences for two important papers they coauthored that advanced the field of "dynamic macroeconomics". As the Royal Swedish Academy of Sciences put it, Prescott and Kydland's work "has not only transformed economic research, but has also profoundly influenced the practice of economic policy in general, and monetary policy in particular." The Knowledge@Wharton website has an excellent article which explains the importance of Prescott's research to issues that confront our society today.  The title of the article is: "What's Behind Edward C. Prescott's Nobel Prize?"

Pension Guarantor's Deficit Doubles

"Struggling under a cascade of bankruptcy filings in the airline and steel industries, the government's pension insurance agency (aka the Pension Benefit Guaranty Corporation, or PBGC) said yesterday that its deficit has more than doubled in the past year -- to $23.3 billion," according to The Washington Post. In "How to Reduce the Cost of Federal Pension Insurance," Richard A. Ippolito, former chief economist at the PBGC, warns that the agency is poised for a taxpayer bailout similar to the 1980s savings and loan crisis. He recommends transforming the PBGC into a private insurance program that sets premiums according to the amount of risk plan sponsors add to the program.

Midterm 2 Grades and Current Grade Distribution for FIN 4366

The second midterm scores are in.  Here are the descriptive statistics:

The course grade distribution after the first exam is as follows:

If letter grades had to be assigned today, based upon this distribution I would use the following curve (which would generate a class GPA of 3.17):

due date for problem set #7 will be Thursday, November 18

The due date for problem set #7 will be Thursday, November 18 rather than tomorrow.  I figured that y'all might be interested in knowing this before unduly torturing yourselves this evening.