Category Archives: Helpful Hints

Midterm Exam 2 information

Midterm 2 will be given during class on Thursday, April 4. This test consists of 4 problems. You are only required to complete 3 problems. At your option, you may complete all 4 problems, in which case I will throw out the problem on which you receive the lowest score.

The questions involve topics which we have covered since the first midterm exam. I expect my Finance 4366 students to demonstrate mastery concerning how arbitrage-free prices for European and American call and put options obtain within the binomial framework. Irrespective of the framework you apply (e.g., delta hedging, replicating portfolio, or risk neutral valuation), prices obtained via these methods are arbitrage-free in the sense that if the market price is not equal to the arbitrage-free price, then you can earn riskless trading profits without having to commit any of your own capital. Other than the binomial model, there’s also a question pertaining to the “Wiener Processes and Ito’s Lemma” readings and related class discussions and problems.

By the way, I have posted the formula sheet that I plan to use on the exam at the following location:

This coming Tuesday’s will be devoted to a review session for midterm exam 2. If you haven’t already done so, I highly recommend that you review Problem Sets 6-8 and also try working the Sample Midterm 2 Exam prior to coming to class on Tuesday.

Some Problem Set 8 Hints

Here are some “helpful hints” pertaining to problem set 8, which is due at the beginning of class on Tuesday, April 2.

1. Problem 1 asks for probability distributions. Given that the company’s cash position follows a generalized Wiener process with a drift rate of .2 per month and a variance rate of .5 per month (with an initial cash position of $3 million), then the probability distributions for 1 month, 6 months, and 1 year are all normal, and it’s your job to determine what the parameter values are for the mean and variance of the company’s cash position.  Note that the generalized Wiener process implies that the expected value of the cash position will equal the initial value of the cash position ($3 million) plus \mu multiplied by the number of months (1, 6, and 12), and the variance of the cash position will be the monthly variance rate multiplied by the number of months.   The probability of a negative cash position in x months time is calculated by finding the z statistic, i.e., z(x) = (0 – (3 + .2(x))/(\sqrt {{\sigma ^2}x}), and then plugging the z statistic into Excel or relying upon the Standard Normal Distribution Function (“z”) Table.

Perhaps the most challenging aspect of problem 1 is part 3: “At what time is the probability of a negative cash position the greatest?” The trade-off here is that as time passes, the expected value of the cash position increases linearly, whereas the volatility increases in a non-linear fashion. To solve this problem, note that the probability of the cash position being negative is maximized when (3 + .2(x))/(\sqrt {{\sigma ^2}x})) is minimized (note that this will yield the most negative possible value for z(x)).

2. The key to solving problem 2 is to recognize that since the stock price is lognormally distributed, the natural logarithm of the stock price is normally distributed. Equation 8 from my “Applying Ito’s Lemma to determine the parameters of the probability distribution for the continuously compounded rate of return” reading shows the distribution in this case:

From this expression, it follows that the probability distribution for log returns is \ln ({S_T}/S) \sim N\left( {(\mu - .5{\sigma ^2})T,{\sigma ^2}T} \right). Thus, the “true” (i.e., actual as opposed to risk neutral) probability that the option will expire in the money is equal to 1-N(z), where

z = \left( {\ln (K/S) - (\mu - .5{\sigma ^2})T} \right)/\sigma \sqrt T.

Also, as we have previously mentioned in class, the risk neutral probability that the option will expire in the money is given by the N({d_2}) term which appears in the Black-Scholes-Merton call option pricing formula (see equation (30) of Teaching the Economics and Convergence of the Binomial and Black-Scholes Option Pricing Formulas, for a description of this pricing formula).

Discrete time to continuous time…

As we segue from discrete-time (binomial) to continuous time (Black-Scholes-Merton) pricing models in Finance 4366, you will surely find this topic to be quite challenging. But it is essential, since a basic understanding of the continuous time framework is, as Hull puts it, “central to the pricing of derivatives”. As you read Hull’s textbook chapter entitled “Wiener Processes and Ito’s Lemma” and my teaching notes entitled “Applying Ito’s Lemma to determine the parameters of the probability distribution for the continuously compounded rate of return” and “Geometric Brownian Motion Simulations“, keep the following exhortation by Hull in mind (this appears as the third paragraph on the first page of the textbook chapter assigned for tomorrow):

“Many people feel that continuous-time stochastic processes are so complicated that they should be left entirely to ‘‘rocket scientists.’’ This is not so. The biggest hurdle to understanding these processes is the notation. Here we present a step-by-step approach aimed at getting the reader over this hurdle. We also explain an important result known as Ito’s lemma that is central to the pricing of derivatives.”


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 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; 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 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; 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!