21-270: Things to Keep in Mind #3

Many people have asked about purchasing fractional amount of securities. From page 18 of the notes:

Unless stated otherwise, we assume that all securities can be purchased and sold
short in any amounts we choose (including fractional numbers of shares of stock) and
we ignore transaction costs, including the bid-ask spread. Moreover, we assume the
interest rate for investing is the same as the rate for borrowing (although we do allow
the interest rate to depend on the duration of the loan or investment). We shall
also assume that the price per share of a security is independent of the number of
shares involved in the transaction. We shall also ignore the effects of taxation. A
discussion of the validity of these assumptions from a practical point of view is given
in Appendix II. For assets other than securities, we shall always indicate whether or
not short sales are allowed.

This is as good of a time as any for me to remark that I believe that the lecture notes for 21-270 are by far the best written source for the material covered in this course. If you haven't already, you should begin reading the notes as we go through the corresponding material in class.

Dr. Kramkov is one of the few individuals in the world in possession of extraordinarily deep knowledge of both the theoretical and practical aspects of financial mathematics. He is one of the fields leading researchers and has been responsible for some of its biggest advancements in the last 20 years. Moreover, he served as head of research of one of the world's largest banks. As if that were not enough, he and Dr. Hrusa have the amazing gift of being able to present ideas in a friendly, but thourough manner.

Although the lectures are self-contained, you will not get the most out of this class if you ignore the excellent text.

21-270: Things to Keep in Mind #2

Another common early mistake is to take the future value of a portfolio by netting out the initial cost. When we speak of value, we do not include any concept of profit; the value of a portfolio at any given time, t, is simply the sum of the time t values (prices) of the component securities. Note that short positions have values opposite in sign to their corresponding long position.

For example, suppose there exists a $4 European call and a $1.50 European put, both struck at $50 on a single share of the same stock initially trading at $50 and both expiring in one year. If we create a portfolio consisting in one call and a short position in two puts, the initial value of this portfolio is the sum of the values of the calls and the shorted puts: $4 + 2 x (-$1.50) = $1.

If, after a year, the stock goes up in price to $60 the puts are worthless and the call is worth $10 (why?). Many students will then calculate the time 1 value of the portfolio to be $10 + 2 x (- $0) - $1 = $9. This is incorrect; the value of the portfolio is simply $10 = $10 + 2 x (- $0).

Likewise. if the stock falls to $45, the portfolio will have value -$10, not -$11.


In general, absolute profit is not as useful of a number to think about as you might first believe. For one thing, it neglects the time value of money. Similarly, it's difficult to analyse without knowledge of alternative investments or a quantification of its associated risk.

21-270: Things to Keep in Mind #1

An early common mistake made by people learning mathematical finance is to think that the value of an option can always be found by using it's pricing formula at expiration.

For example, consider a European call option, currently selling for $8, struck at $50 with expiration T = 1 on a stock with current price S₀ = $60. As you know, if the stock price has fallen to S₁ = $55 in one year, we can calculate the terminal value of the call as:
C₁ = C₁(S₁) = (S₁ - K)⁺ = ($55 - $50)⁺ = $5
When asked for the time 0 value of the call, C₀, many students will argue that:
C₀ = C₀(S₀) = (S₀ - K)⁺ = ($60 - $50)⁺ = $10
This is incorrect. By the value of an option, we simply mean its price; in this example, C₀ = $8. For our purposes price and value are synonymous; we leave any distinction in the meaning of these terms to the Economists.

Valuing (or pricing) the call between time 0 and time T is a substantially more difficult problem and is one of the aims of modern mathematical finance. These valuations are covered in 21-370 and 21-420.