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Black-Scholes: a potentially dangerous assumption?

Discussion in 'Business & Economics' started by Onion Eater, Apr 13, 2009.

  1. Onion Eater

    Onion Eater Member

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    Black-Scholes: a potentially dangerous assumption?

    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.

    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:

    1) One's value scale is totally (linearly) ordered:

    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

    2) Marginal (diminishing) utility, u(s), is such that:

    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.

    3) First-unit demand conforms to proportionate effect:

    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
    value.

    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.
     
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