## Black-Scholes: my "Buffett mistakes"

A confession: I am the actuary who disagreed with this post by Dean Buckner, a former PRA official, which asserts that the Black-Scholes formula gives a good valuation of an option under the assumption of mean reversion in prices. In a follow-up post he said my critique was “ingenious” but “wrong” and that it was similar to an earlier “Buffett mistake”. ( I am also the “the firm-friendly friend” who previously drew his attention to Buffett’s views on the pricing of long-term options as described there.) I would very happy go on making “Buffett mistakes” for the rest of my life. But despite further discussion with Dean, I remain in disagreement with both the original post and the follow-up. This post explicates my view.

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 (except for the chance that our (95, 96, 95,96…) assumption may be wrong, which has nothing to do with Black-Scholes).

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. (The 95,96, 95,96..) is merely one possible observed path amongst many under geometric Brownian motion. 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. Its criticisms of the Institute and Faculty of Actuaries (like much else on the Eumaeus website) are entertaining as slapstick, but also intemperate and wrong. And its claim that *“The family of Black pricing models are amongst the most practical and robust models of reality that science possesses”* is simply absurd.

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(*) We may be able to fudge Black-Scholes by adjusting the volatility input to allow for mean reversion, for example as in this paper. But fudged Black-Scholes is not the same as Black-Scholes!

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