# Class Problem Solutions (1-timestep Delta Hedging and Portfolio Replication)

Here is a 3-1/2 page PDF file in which I provide the solutions.pdf for yesterday’s class problem in which we solved one time-step “arbitrage-free” call and put option prices using the Delta Hedging and Portfolio replication trading strategies:

One Timestep Delta Hedging and Portfolio Replication Class Problem (Solutions)

# On the origins of the binomial option pricing model

In my previous posting entitled “Historical context for the Black-Scholes-Merton option pricing model,” I provide links to the papers in which Black-Scholes and Merton presented the so-called “continuous time” version of the option pricing formula. Both of these papers were published in 1973 and eventually won their authors (with the exception of Fischer Black) Nobel prizes in 1997 (Black was not cited because he passed away in 1995 and Nobel prizes cannot be awarded posthumously).

Six years after the Black-Scholes and Merton papers were published, Cox, Ross, and Rubinstein (CRR) published a paper entitled “Option Pricing: A Simplified Approach”. This paper is historically significant because it presents (as per its title) a much simpler method for pricing options which contains (as a special limiting case) the Black-Scholes-Merton formula. The reason why we began our analysis of options by first studying CRR’s binomial model is because pedagogically, this makes the economics of option pricing much easier to comprehend. Furthermore, such an approach removes much (if not most of) the mystery and complexity of Black-Scholes-Merton and also makes that model much easier to comprehend.

# Historical context for the Black-Scholes-Merton option pricing model

Although we won’t get into the “gory” details on the famous Black-Scholes-Merton option pricing model until sometime later this semester when we cover Hull’s chapter entitled “The Black-Scholes-Merton Model” and my teaching note entitled “Derivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Equations“, I’d like to call your attention to the fact that the original papers by Black-Scholes and Merton are available on the web:

The Black-Scholes paper originally appeared in the Journal of Political Economy (Vol. 81, No. 3 (May – Jun., 1973), pp. 637-654). The Merton paper appeared at around the same time in The Bell Journal of Economics and Management Science (now called The Rand Journal). Coincidentally, the publication dates for these articles on pricing options roughly coincide with the founding of the Chicago Board Options Exchange, which was the first marketplace established for the purpose of trading listed options.

Apparently neither Black and Scholes nor Merton ever gave serious consideration to publishing their famous option pricing articles in a finance journal, instead choosing two top economics journals; specifically, the Journal of Political Economy and The Bell Journal of Economics and Management Science. Mehrling (2005) notes that Black and Scholes:

“… could have tried finance journals, but the kind of finance they were doing was outside the rubric of finance as it was then organized. There was a reason for the economist’s low opinion of finance, and that reason was the low analytical level of most of the work being done in the field. Finance was at that time substantially a descriptive field, involved mainly with recording the range of real-world practice and summarizing it in rules of thumb rather than analytical principles and models.”

Another interesting anecdote about Black-Scholes is the difficulty that they experienced in getting their paper published in the first place. Devlin (1997) notes: “So revolutionary was the very idea that you could use mathematics to price derivatives that initially Black and Scholes had difficulty publishing their work. When they first tried in 1970, Chicago University’s Journal of Political Economy and Harvard’s Review of Economics and Statistics both rejected the paper without even bothering to have it refereed. It was only in 1973, after some influential members of the Chicago faculty put pressure on the journal editors, that the Journal of Political Economy published the paper.”

References

Devlin, K., 1997, “A Nobel Formula”.

Mehrling, P., 2005, Fischer Black and the Revolutionary Idea of Finance (Hoboken, NJ: John Wiley & Sons, Inc.).

# Update of course website and dates for upcoming readings, problems sets, and exams

I have updated the Finance 4366 course website to realistically reflect dates for upcoming readings, problems sets, and exams. Next Tuesday, we begin class with a quiz based upon Hull’s Binomial Options chapter and also my teaching note entitled Binomial Option Pricing Model (single-period). During the next couple of weeks, we’ll focus attention in Finance 4366 upon various aspects of binomial option pricing. After spring break, we’ll delve into pricing American options and segue from the “discrete time” framework (upon which the binomial model is based) over to the “continuous time” framework (upon which the Black-Scholes-Merton model is based).

# Midterm Exam 1 Booklet and Solutions

Due to an important medical issue affecting a close member of my family, class and office hours tomorrow (Thursday, February 21) are canceled. See y’all next Tuesday; the assignments due tomorrow will now be due then.

# Information about tomorrow’s midterm exam in Finance 4366

Tomorrow’s midterm exam in Finance 4366 consists of 2 sections. The first section is required of all students and consists 4 multiple choice problems 8 points each. The second section is worth 64 points; it consists of 3 problems in which only two are required. At your option, you may complete all three problems, in which case I will count the two highest scoring problems (worth 32 points each). The maximum possible score on this exam is 100 (96 points possible for sections 1 and 2 and 4 points for including your name on the exam booklet :-)). Furthermore, I just posted the formula sheet for Midterm Exam 1, which will be included as part of the exam booklet. I recommend that y’all familiarize yourselves with this document sometime prior to tomorrow’s exam.

I’d like to make an important point about the formulas provided on the formula sheet for the replicating portfolio approach to option pricing. There, I list replicating portfolio values at inception and at expiration. Keep in mind that this approach involves determining a weighting scheme such that one replicates call or put payoffs by appropriately choosing stock exposure (delta) and bond exposure (beta). At this point in the course, we only allow for two possible option payoffs at expiration; thus, there actually are two equations for the replicating portfolio at expiration. These equations reflect the fact that the value of the underlying asset is different in each possible terminal state. However, since she value of the underlying asset is known at the inception of the option, there is only one equation for the replicating portfolio at that point in time.

# Options class problems from the February 5th FIN 4366 class meeting

I just realized that I had forgotten to post this information – better late then never, right?