Black-Scholes: a potentially dangerous assumption?Submitted by Shaka on Sat, 03/21/2009 - 22:25
The Black-Scholes model of the market for an equity makes the following explicit assumptions:
1) It is possible to borrow and lend cash at a known constant risk-free interest rate.
2) The price follows a geometric Brownian motion with constant drift and volatility.
Also, Black and Scholes make the simplifying assumptions that all securities are perfectly divisible, there are no transaction costs or dividends and there are no restrictions on short selling. These are just simplifications that later, more complicated versions, have worked around. So there are really only two essential assumptions made by quantitative analysts - quants.
Let us begin by considering Black and Scholes’ first axiom.
Suzy at Debate Politics writes:
"I am an economist by training and spent 25 years in the banking industry. I have personally sat in loan committee meetings and had bank examiners demand that we 'not discriminate against low and moderate income borrowers.' Never mind that the reason they are low and moderate income in the first place is their inability to make good financial decisions... like establishing a steady work history, paying bills on time, living within their means, obeying the law, buying insurance to guard against catastrophic illness or property loss, (women) having multiple children with multiple men. And on, and on.
"The CRA was well intentioned, and lawmakers from both sides of the aisle rightly noted the positive effects that homeownership can have on a society. The problem was, these middle and upper class lawmakers made the erroneous assumption that if you put poor people in houses, they would suddenly start behaving like financially responsible middle class people. All of a sudden lawnmowers would replace lottery tickets and backyard barbeques would take the place of drive by shootings. Alas, these hopes for change were empty promises as they always are, and borrowers who had to get their down payments from “third party non-profit agencies” (by the way, someone always makes a profit, otherwise why are they in it) had nothing to fall back on when the hot water heater broke or the roof leaked. The houses fell into disrepair and by the time the foreclosure papers were posted, the occupants and hopes of any recovery by the lender long gone. But the originating lender didn't care... the loan had been sold, not their problem any more!
"There is more blame to go around... mortgage companies and builders soon realized there was money being printed and there sprung up companies that specialized in getting subprime borrowers into low cost (and low quality) housing. You probably heard them advertising on the radio and saw the ads in the Sunday paper. Did you ever wonder what kind of people would need a no-doc loan? And who would be stupid enough to make such a loan? I think we all know the answer to that now... the ultimate lender was of course Fannie Mae or Freddie Mac, the original loan having long since been sold by the originating bank. All that bad paper, and Franklin Raines out the back door with his suitcase full of money..."
Here, Suzy is referring to the Community Reinvestment Act (CRA) that was passed in 1977, four years after Black and Scholes introduced their groundbreaking axiomatic system.
Clearly, the CRA flies in the face of Black and Scholes’ first axiom by systematically discriminating against segments of the population in the distribution of credit. Contra Black and Scholes, it is NOT possible to borrow and lend cash at a known constant risk-free interest rate. Instead, loans are made on the basis of ethnicity and other non-economic factors, in spite of their known risks.
I argue that the failure of Black and Scholes to anticipate that the CRA would up-end their first axiom is the principle cause of our current financial crisis. Who among you would deny this?
Contra Milton Friedman, assumptions DO matter!
Instead of Black-Scholes, I recommend the following axiomatic system. Notice that this system does NOT make any ridiculous assumptions about credit being distributed in a fair and even-handed manner. Also, notice that my third axiom is compatible with Black and Scholes' second axiom, that price follows a geometric Brownian motion with constant drift and volatility. I have no argument with Black and Scholes’ second axiom.
My assumptions are three:
i) Transitive; p ≤ q and q ≤ r imply p ≤ r
ii) Reflexive; p ≤ p
iii) Anti-Symmetric; p ≤ q and q ≤ p imply p = q
iv) Total; p ≤ q or q ≤ p
i) It is independent of first-unit demand.
ii) It is negative monotonic; that is, u'(s) < 0.
iii) The integral of u(s) from zero to infinity is finite.
i) Value changes each day by a proportion (called 1+εj, with j denoting
the day), of the previous day's value.
ii) In the long run, the εj's may be considered random as they are not
directly related to each other nor are they uniquely a function of
iii) The εj's are taken from an unspecified distribution with a finite
mean and a non-zero, finite variance.
Read my Simplified Exposition of Axiomatic Economics for a more detailed, but still undergraduate-level, discussion of my economic theory. This paper requires knowledge of multi-variable calculus, but omits the real analysis that plagues readers of my 1999 book.