Guy Thomas replies here to my earlier posts on distribution. As usual, he comes up with objections that are both ingenious and (in my view) weirdly mistaken. Today’s topic is the prospectively assumed distribution.
Guy refers many times to ‘prospective assumptions’ about the future path of a price. For example, quoting verbatim from his post, he writes:
- Suppose that as in my interpretation of the original post, the (95, 96, 95, 96…) price series is a prospectively assumed distribution. In this case, I say that a put option at 90 is always worth zero
- Suppose, alternatively, as in the interpretation put forward in the follow-up post, the (95, 96, 95, 96…) price series is a retrospectively observed single path. In this case, I say that we have not departed from the classical assumptions of Black-Scholes; we have not made a prospective assumption of mean reversion.
- If we then value an option using the Black-Scholes formula at every time t, we are implicitly making a prospective assumption of geometric Brownian motion at every time t. I agree that hedging allows us to correct for the retrospectively observed single path up to time t; but it says nothing about the validity of Black-Scholes valuation at time t if we were to make a prospective assumption of mean reversion at that time.
- In short, the original post does not show what it claims to show: that the Black-Scholes formula gives a good valuation of an option under a prospective assumption of mean reversion.
But what exactly is such an assumption? ‘Prospective’ comes from the Latin prospectus meaning that which is looked forward to or seen far away, sometimes it connotes foresight or knowledge of the future. The English meaning is ‘likely to come about’, implying foresight, but also ‘expected’, implying hope.
So is a prospective assumption one that is informed by foresight about the future? The problem is whether such foresight exists. We can of course predict eclipses, the reaction of hydrogen and oxygen to a flame, the acceleration due to gravity etc, but science hasn’t found a way to predict the path of market prices. The problem is that the market price of an asset itself involves a forecast, by the market, of future cashflows, so in trying to predict where the market will be in a year’s time, we are trying to forecast a forecast. It’s like, instead of trying to predict the result of the next election, we are trying to predict what the Times will predict it to be. Also, either the market is the best forecast, or it is not. If the former, we can’t improve on it. If the latter, we have to forecast what the bad forecast will be in a year’s time. You see the problem.
Or is a prospective assumption simply an expectation (in the psychological sense) of the future path? So when Guy says that if the 95, 96, 95, 96… price series is a prospectively assumed distribution, then the option is worth zero, he really means that it is worth zero to him, in the sense that he fully expects its payoff to be zero. Well fine, but that’s not the same as its market value. Different people have different views on the value of things, and the market price is a sort of equilibrium of those expectations.
None of this justifies marking the value of any asset to anything other than the market price. It may be that the market is completely wrong. Clearly it is, if the market price is changing all the time. But if I mark an asset on a firm’s books higher on the grounds that I have perfect foresight, or better judgment than the market, I am defrauding prospective shareholders of the firm. If I mark it lower, because my excellent judgment values it at less, I am defrauding existing shareholders.
If God were an accountant, He would not value an asset differently from its market value, despite being omniscient, for God is also perfectly good, and so would not get involved in false accounting.