+ I had coffee the other day with an insightful fellow, the Dean of a business school. He said: well, you know, you seem to be telling me that we don’t value options. It then hit me that, when it comes to options, what we call “value” is not quite the same thing that a neoclassical economist calls “value”. By “value” they mean some top-down equilibrium induced number that is compatible with what one should expect for a series of Arrow-Debreu elementary claims of which it is built. For instance, when you value a stock you incorporate elements that YOU KNOW about the future distribution of its value, in line with the other junk ideas they have about the world, the “market price of risk” and some other NONOBSERVABLES.
+ That is not at all what option traders do (or what I did with 400,000 option trades and 2 million dynamic hedges). By “value” we mean the best guess price we can produce, knowing that we may change our mind later based on what information will be revealed in the future, and compatible with supply and demand of financial instruments and NOT INCOMPATIBLE with some elementary arbitrage relationships, i.e., consistent with other options in the market and matters of a STRICTLY OBSERVABLE character. We do not hold options to expiration, so we care about supply and demands of the part of the distribution that we deal with. Even if we held options to expiration, we only have a fuzzy idea of what the distribution looks like. I repeat: We do not “know” the Arrow-Debreu states that are not observed from other instruments. We do not know future distributions; we only have a vague idea of what’s going on –vague, but sufficient to produce a price.
+ Based on that point, Espen and I are looking at what people call in the vulgate the Black-Scholes-Merton “formula”. It is not a formula, just an argument to remove the risk thanks to dynamic hedging, hence price options as a risk-free instrument, and that ONLY works if you assume a Gaussian (i.e. ability to smooth out the square variations in continuous time).
+ So if you do not accept the Gaussian distribution (i.e. if you have some ethics) AND do not “value” options in a axiomatized top-down fashion (i.e. only price them as some informed and temporary guess), then YOU ARE NOT USING THE BLACK SCHOLES FORMULA, but one of the modifications of the one started by Bachelier (the latest contributer being Ed Thorp’s). They did not ground their formula in the Gaussian.
+ I do not understand something. If you either 1) Do not accept the hypothesis that you know the parameters of the future asset value with any precision (i.e. do not know the volatility along the path of the asset), or
2) Do not accept the Gaussian as an “approximation” of asset values, do not think that the P/L of an option position becomes “smooth” by dynamic hedging,
3) Do not think that LTCM blew up because it was “a highly remote incidence that should happen every 10^10 years”
4) Have some eyes to see that stocks have always jumped between open and close, during earnings, or upon some announcements and that such jump represents a sizeable portion of the volatility, and that such jump cannot be smooted out, etc.
5) Something called “equilibrium” is not observable, plain metaphysics
then you are accepting that the Black-Scholes-Merton is WRONG. But not the Sprenkle, Thorp, or Bachelier’s results. We are using some variation of the initial Bachelier equation.