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The Limits of Theoretical Models



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Professor Succo,

Firstly, I wanted to congratulate you on (and thank you) for
your excellent coverage of the events of last Thursday. I thought it was right out of the top drawer.

Anyway, onto your discussion of portfolio insurance.

Though it may well be theoretically possible to synthetically replicate the curvilinear nature of options prices in a continuous time environment with perfect liquidity, or even on an incremental basis in a real market environment in the belly of the bell curve. Indeed this dynamic replication is the basis of the Black Scholes valuation model. But a model is by necessity a simplification of reality, if not, what is the purpose of the model? But the assumptions made by BS73 are quite clearly not representative of reality at all times, especially not at the tails of a leptokurtic distribution (like the events of '87).

There is also a game theoretic aspect there, in that while it might have made sense for individual market participants to engage in the dynamic replication of non-existent options, it did not when done on a systematic basis. The large scale of engagement in portfolio insurance exacerbated the differences between the model and reality. It, in a sense, contributed to its own failures, failures that would likely have been nowhere near as severe had it been done on a smaller scale.

In many ways I think you could say that portfolio insurance was the triumph of theory (and perhaps greed) over practical experience.

I think you're correct in pointing out the similarities between modern structured (principal protected) hedge fund products and portfolio insurance. Unless the firm offering the protected note has extensive real-time knowledge of the underlying positions of the fund (and the positions are in liquid instruments, something that is often not the case) in question, a hedge fund is in many ways almost the diametric opposite of a liquid instrument traded in continuous time.

Obviously the structurers by and large understand this and account for it in the implicit price of the option, but given the fat-tailed nature of most markets (and the skewness of many, implicitly if not explicitly, short-gamma) hedge fund strategies would you not say that the potential for discontinuities beyond those priced into the options exists?

With thanks,

Minyan Michael


The models used to describe markets, including the Black-Scholes, at least up to now, are all based on linear math. This linear math cannot incorporate the small permutations that exist in reality that over time add up and become manifest in clear results that these models cannot explain. They are at best approximations that describe "parts" of reality.

So Black-Scholes does a reasonable job of explaining the relationship between option prices and asset behavior day-to-day of only small deviations.

You eloquently describe the problem. By applying only theory, portfolio insurance or principal protection become schemes that do not work when markets behave non-linearly. Theory interrupts market mechanisms.

When someone buys a real option, the price for the next option goes up. This process eventually balances supply and demand and correctly prices the amount of gamma in that market. Portfolio insurance interrupted that pricing mechanism as the suppliers of the insurance never raised the price; this caused far too much gamma (leverage) to exist in the system.

So Scott Reamer's work delves into non-linear applications to try to better understand "the markets." He is attempting to describe the real nature of distributions by understanding how social and psychological factors add up and "move markets from the belly of the curve to the tails."

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