Category Archives: Helpful Hints

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.

Problem Set 2 helpful hints

Problem Set 2 is now available from the course website at http://fin4366.garven.com/spring2019/ps2.pdf; its due date is Tuesday, January 29.

Problem Set 2 consists of two problems. The first problem requires calculating expected value, standard deviation, and correlation, and using this information as inputs into determining expected returns and standard deviations for 2-asset portfolios; see pp. 17-22 of the http://fin4366.garven.com/spring2019/lecture3.pdf lecture note for coverage of this topic. The second problem involves using the standard normal probability distribution to calculate the probabilities of earning various levels of return by investing in risky securities and portfolios.

Visualizing Taylor polynomial approximations

In his video lesson entitled “Visualizing Taylor polynomial approximations“, Sal Kahn essentially replicates the tail end of last Thursday’s Finance 4366 class meeting in which we approximated y = eˣ with a Taylor polynomial centered at x=0.  Sal approximates y = eˣ with a Taylor polynomial centered at x=3 instead of x=0, but the same insight obtains in both cases, which is that one can approximate functions using Taylor polynomials, and the accuracy of the approximation increases as the order of the polynomial increases (see pp. 19-25 in my Mathematics Tutorial lecture note if you wish to review what we did in class last Thursday).

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.

Problem Set 1 hint…

Problem Set 1 is due at the beginning of class on Tuesday, January 22. Here is a hint for solving the 4th question on problem set 1.

The objective is to determine how big a hospital must be so that the cost per patient-day is minimized. We are not interested in minimizing total cost; if this were the case, there would be no hospital because marginal costs are positive, which implies that total cost is positively related to the number of patient-days.

The cost equation C = 4,700,000 + 0.00013X2 tells you the total cost as a function of the number of patient-days. This is why you are asked in part “a” of the 4th question to derive a formula for the relationship between cost per patient-day and the number of patient days. Once you have that equation, then that is what you minimize, and you’ll be able to answer the question concerning optimal hospital size.

Khan Academy “Finance and capital markets” videos

Not only are the Khan Academy Calculus and Statistics videos that I referenced in a previous posting quite useful; I am also a big fan of the Khan Academy “Finance and capital markets” videos which are located at https://www.khanacademy.org/economics-finance-domain/core-finance; these videos do a great job of effectively presenting many of the most important concepts which are typically covered in undergraduate and MBA level finance curricula (indeed, the content provided by the “Options, swaps, futures, MBSs, CDOs, and other derivatives” subsection of the “Finance and capital markets” page effectively subsumes most of the Finance 4366 course content!