Category Archives: Assignments

Solution for part 1 of Class Problem 1 from today’s meeting of Finance 4335

Here is the solution for part 1 of Class Problem 1 which we worked on in class today.  Since we showed that it is possible to completely eliminate risk via delta hedging, the arbitrage-free price of virtually financial derivative must equal to the risk-neutral expected value of the time payoff, discounted at the riskless rate of interest.  Between now and Thursday, please try to take a stab on your own at part 2 of this problem, which asks you to confirm that the price determined in part 1, in fact, satisfies the Black-Scholes-Merton equation (shown below, and also appearing as equation (16) in the Black-Scholes-Merton Model textbook reading!

Delta hedging, replicating portfolio, and risk neutral valuation pricing in a multiple time-step setting

The delta hedging and replicating portfolio perspectives for pricing options are important in that they enable us to think carefully as well as deeply concerning the economics of option pricing. For example, the delta hedging approach illustrates that an appropriately hedged portfolio consisting of either long-short call-share positions or long-long put-share positions is riskless and consequently must produce a riskless rate of return. Similarly, the replicating portfolio approach reminds us that a call option represents a “synthetic” margined stock investment, whereas a put option represents a “synthetic” short sale of the share combined with lending money. As we saw in our study earlier this semester of forward contracts, if there is any difference between the value of a derivative and its replicating portfolio, then one can earn profits with zero net investment and no exposure to risk. Thus, the “arbitrage-free” price for the derivative (option or forward) corresponds to the value of its replicating portfolio.

An important result shown in the Binomial Option Pricing Model (single-period) reading which we will cover tomorrow is that the delta hedging and replicating portfolio approaches to pricing options both imply that a “risk-neutral” valuation relationship exists between the derivative and its underlying asset.  This insight provides a deceptively simple pricing equation which enables us to price options by discounting the risk-neutral expected value of the option payoff (i.e., its certainty-equivalent) at the riskless rate of interest.  During tomorrow’s class meeting, I also hope to extend the pricing model from a single timestep to a multiple timestep setting. To accomplish this task, we rely upon backward induction. Backward induction involves beginning at the very end of the binomial tree and working our way back to the beginning. We’ll continue with the numerical example introduced during class today and illustrate the backward induction method for the delta hedging, replicating portfolio and risk neutral valuation approaches. We’ll discover that multiple timesteps imply that the delta hedging and replicating portfolio methods imply “dynamic” trading strategies which require portfolio rebalancing as the price of the underlying asset changes over time (cf. the Dynamic Delta Hedging Numerical Example (calls and puts) and Dynamic Replicating Portfolio Numerical Example (calls and puts) readings).

Availability of Problem set 4, along with a helpful hint

I have posted Problem set 4 on the course website.  This problem set is based upon the “Properties of Stock Options” reading, and it consists of four problems.  It is due at the beginning of class on Tuesday, February 12.

The fourth problem in this problem set references an Excel spreadsheet template called “Derivagem” which you can download from http://fin4366.garven.com/spring2019/DG300.xls. Open DG300.xls up in Excel, and you’ll encounter a dialog box which looks something like this:

Click on “Enable Macros”, and once the  spreadsheet is open, select the “Equity_FX_Index_Futures_Options” worksheet. The upper left corner of the spreadsheet is where you input the data for the problem.  For this problem, “Underlying Type” is “Equity”, “Option Type” is “Black-Scholes – European”, a $.50 dividend is paid in 6 months, which corresponds to .5 year, the current stock price is $41, volatility is 35%, and the risk-free rate of interest is 6%, the option’s life is 1 year, and the strike or exercise price is $40:

Once you have input these data, select the call button for the call option price and the put button for the put option price.  Depending on whether you select “Put” or “Call”, when you click on “Calculate”, the spreadsheet will report back put and call option prices based on the Black-Scholes-Merton (BSM) option pricing formula.  For now,  this formula may be somewhat of a black box for many (if not most) of you, which is a “problem” we will surely rectify in the not-too-distant future in Finance 4366.

Reminder about tomorrow’s assignments in Finance 4366

Problem Set 1 is due at the beginning of class tomorrow (see the hint that I gave about this problem set in a previous blog posting.  Class will also begin tomorrow with a brief quiz based upon the assigned readings, which include “The New Religion of Risk Management”, by Peter Bernstein and “Normal and standard normal distribution”, by yours truly.

Going forward, I will typically not post reminders like this concerning Finance 4335 assignment deadlines; however, you’ll be “good to go” in Finance 4335 if you faithfully follow the guidelines listed in my “How to know whether you are on track with Finance 4366 assignments” posting.

Plans for next week’s Finance 4366 class meetings, along with a preview of future topics

Next week in Finance 4366 will be devoted to tutorials on probability and statistics. These tools are foundational for the theory of pricing and managing risk with financial derivatives, which is what this course is all about.  Next Tuesday’s class meeting will be devoted to introducing discrete and continuous probability distributions, calculating parameters such as expected value, variance, standard deviation, covariance and correlation, and applying these concepts to measuring expected returns and risks for portfolios consisting of risky assets. Next Thursday will provide a deeper dive into discrete and continuous probability distributions, in which the binomial and normal distributions are showcased.

Beginning Tuesday, January 29, we introduce financial derivatives, We will begin by defining financial derivatives and motivating their study with examples of forwards, futures, and options. Derivatives are so named because the prices of these instruments are derived from the prices of one or more underlying assets. The types of underlying assets upon which derivatives are based are often traded financial assets such as stocks, bonds, currencies, or other derivatives, but they can be pretty much anything. For example, the Chicago Mercantile Exchange (CME) offers exchange-traded weather futures contracts and options on such contracts (see “Market Futures: Introduction To Weather Derivatives“). There are also so-called “prediction” markets in which derivatives based upon the outcome of political events are actively traded (see “Prediction Markets“).

Besides introducing financial derivatives and discussing various institutional aspects of markets in which they are traded, we’ll consider various properties of forward and option contracts, since virtually all financial derivatives feature payoffs that are isomorphic to either or both of these schemes. For example, a futures contract is simply an exchange-traded version of a forward contract. Similarly, since swaps involve exchanges between counter-parties of payment streams over time, these instruments essentially represent a series of forward contracts. In the option space, besides traded stock options there are also embedded options in corporate securities; e.g., a convertible bond represents a combination of a non-convertible bond plus a call option on company stock. Similarly, when a company makes an investment, there may be embedded “real” options to expand or abandon the investment at some future date.

Perhaps the most important (pre-Midterm 1) idea that we’ll introduce is the concept of a so-called “arbitrage-free” price for a financial derivative. While details will follow, the basic idea is that one can replicate the payoffs on a forward or option by forming a portfolio consisting of the underlying asset and a riskless bond. This portfolio is called the “replicating” portfolio, since it is designed to perfectly replicate the payoffs on the forward or option. Since the forward or option and its replicating portfolio produce the same payoffs, then they must also have the same value. However, suppose the replicating portfolio (forward or option) is more expensive than the forward or option (replicating portfolio). It this occurs, then one can earn a riskless arbitrage profit by simply selling the replicating portfolio (forward or option) and buying the forward or option (replicating portfolio). However, competition will ensure that opportunities for riskless arbitrage profits vanish very quickly. Thus the forward or option will be priced such that one cannot earn arbitrage profit from playing this game.

Lagrangian Multipliers

There is a section in the assigned “Optimization” reading due Thursday, 1/17 on pp. 74-76 entitled “Lagrangian Multipliers” which (as noted in footnote 9 of that reading) 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 this coming Thursday’s meeting of Finance 4366, make sure that you fill out and email the student information form as a file attachment to options@garven.com prior to the beginning of tomorrow’s class. As I explained during today’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 obtain a Wall Street Journal subscription

A subscription to the Wall Street Journal is required for Finance 4366. In order to subscribe to the Wall Street Journal (WSJ), go to http://r.wsj.net/j73NM. Your WSJ subscription includes access to print, online, tablet and mobile editions, and only costs $1 for a 15 week subscription. At your option, you may choose to receive both the digital and paper versions of WSJ or only the WSJ digital version.

Throughout the semester, I will often reference specific WSJ articles in class and on the course blog. Finance 4366 topics (as well as topics in many of your other business school courses) come to life in the world outside the Baylor bubble when you make a habit of reading the WSJ on a regular basis. Furthermore, if you expect to interview for jobs or internships anytime soon, reading the WSJ will give you a leg up on the competition, since you will be better informed and have more compelling ideas and insights to share with recruiters.

In closing, the following (2 minute) video provides a helpful introduction to the WSJ, providing time-saving tips to help you get the most from WSJ and succeed not only in Finance 4366, but also your other courses and careers: