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(Very) Sophisticated Heuristics, Never the Black– Scholes–Merton Formula1

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(Very) Sophisticated Heuristics, Never the Black– Scholes–Merton Formula1 Option traders use (very) sophisticated heuristics, never the Black– Scholes–Merton formula1 Espen Gaarder Haug & Nassim Nicholas Taleb Version November 2010 Abstract: Option traders use a heuristically derived pricing formula which they adapt by fudging and changing the tails and skewness by varying one parameter, the standard deviation of a Gaussian. Such formula is popularly called “Black– Scholes–Merton” owing to an attributed eponymous discovery (though changing the standard deviation parameter is in contra- diction with it). However, we have historical evidence that: (1) the said Black, Scholes and Merton did not invent any formula, just found an argument to make a well known (and used) formula compatible with the economics establishment, by removing the “risk” parameter through “dynamic hedging”, (2) option traders use (and evidently have used since 1902) sophisticated heuristics and tricks more compatible with the previous versions of the formula of Louis Bachelier and Edward O. Thorp (that allow a broad choice of probability distributions) and removed the risk parameter using put-call parity, (3) option traders did not use the Black–Scholes–Merton formula or similar formulas after 1973 but continued their bottom-up heuristics more robust to the high impact rare event. The paper draws on historical trading methods and 19th and early 20th century references ignored by the finance literature. It is time to stop using the wrong designation for option pricing. 1 Thanks to Russ Arbuthnot for useful comments. Electronic copy available at: http://ssrn.com/abstract=1012075 THE BLACK-SCHOLES-MERTON “FORMULA” WAS AN BREAKING THE CHAIN OF TRANSMISSION ARGUMENT For us, practitioners, theories should arise from Option traders call the formula they use the “Black- practice2. This explains our concern with the “scientific” Scholes-Merton” formula without being aware that by notion that practice should fit theory. Option hedging, some irony, of all the possible options formulas that pricing, and trading is neither philosophy nor have been produced in the past century, what is called mathematics. It is a rich craft with traders learning the Black-Scholes-Merton “formula” (after Black and from traders (or traders copying other traders) and Scholes, 1973, and Merton, 1973) is the one the tricks developing under evolution pressures, in a furthest away from what they are using. In fact of the bottom-up manner. It is technë, not ëpistemë. Had it formulas written down in a long history it is the only been a science it would not have survived – for the formula that is fragile to jumps and tail events. empirical and scientific fitness of the pricing and First, something seems to have been lost in translation: hedging theories offered are, we will see, at best, Black and Scholes (1973) and Merton (1973) actually defective and unscientific (and, at the worst, the never came up with a new option formula, but only an hedging methods create more risks than they reduce). theoretical economic argument built on a new way of Our approach in this paper is to ferret out historical “deriving”, rather re-deriving, an already existing –and evidence of technë showing how option traders went well known –formula. The argument, we will see, is about their business in the past. extremely fragile to assumptions. The foundations of Options, we will show, have been extremely active in option hedging and pricing were already far more firmly the pre-modern finance world. Tricks and heuristically laid down before them. The Black-Scholes-Merton derived methodologies in option trading and risk argument, simply, is that an option can be hedged management of derivatives books have been developed using a certain methodology called “dynamic hedging” over the past century, and used quite effectively by and then turned into a risk-free instrument, as the operators. In parallel, many derivations were produced portfolio would no longer be stochastic. Indeed what by mathematical researchers. The economics literature, Black, Scholes and Merton did was “marketing”, finding however, did not recognize these contributions, a way to make a well-known formula palatable to the substituting the rediscoveries or subsequent economics establishment of the time, little else, and in reformulations done by (some) economists. There is fact distorting its essence. evidence of an attribution problem with Black-Scholes- Such argument requires strange far-fetched Merton option “formula”, which was developed, used, assumptions: some liquidity at the level of transactions, and adapted in a robust way by a long tradition of knowledge of the probabilities of future events (in a researchers and used heuristically by option book neoclassical Arrow-Debreu style)4, and, more critically, runners. Furthermore, in a case of scientific puzzle, the a certain mathematical structure that requires “thin- exact formula called “Black-Sholes-Merton” was written tails”, or mild randomness, on which, later. The entire down (and used) by Edward Thorp which, argument is indeed, quite strange and rather paradoxically, while being robust and realistic, has been inapplicable for someone clinically and observation- considered unrigorous. This raises the following: 1) The driven standing outside conventional neoclassical Black-Scholes-Merton was just a neoclassical finance economics. Simply, the dynamic hedging argument is 3 argument, no more than a thought experiment , 2) We dangerous in practice as it subjects you to blowups; it are not aware of traders using their argument or their makes no sense unless you are concerned with version of the formula. neoclassical economic theory. The Black-Scholes- It is high time to give credit where it belongs. Merton argument and equation flow a top-down general equilibrium theory, built upon the assumptions of operators working in full knowledge of the probability distribution of future outcomes –in addition to a collection of assumptions that, we will see, are highly invalid mathematically, the main one being the ability to 2 For us, in this discussion, a practitioner is deemed to be cut the risks using continuous trading which only works someone involved in repeated decisions about option hedging, not a support quant who writes pricing software or an academic who provides “consulting” advice. 4 Of all the misplaced assumptions of Black Scholes that 3 Here we question the notion of confusing thought cause it to be a mere thought experiment, though an extremely experiments in a hypothetical world, of no predictive power, elegant one, a flaw shared with modern portfolio theory, is the with either science or practice. The fact that the Black-Scholes- certain knowledge of future delivered variance for the random Merton argument works in a Platonic world and appears to be variable (or, equivalently, all the future probabilities). This is “elegant” does not mean anything since one can always what makes it clash with practice –the rectification by the produce a Platonic world in which a certain equation works, or market fattening the tails is a negation of the Black-Scholes in which a “rigorous” proof can be provided, a process called thought experiment. reverse-engineering. Electronic copy available at: http://ssrn.com/abstract=1012075 in the very narrowly special case of thin-tailed instance, does not seem to appear in the academic distributions. But it is not just these flaws that make it literature published after the event 5 (Merton, 1992, inapplicable: option traders do not “buy theories”, Rubinstein, 1998, Ross, 2005); to the contrary dynamic particularly speculative general equilibrium ones, which hedging is held to be a standard operation. they find too risky for them and extremely lacking in There are central elements of the real world that can standards of reliability. A normative theory is, simply, escape them –academic research without feedback not good for decision-making under uncertainty from practice (in a practical and applied field) can cause (particularly if it is in chronic disagreement with the diversions we witness between laboratory and empirical evidence). People may take decisions based ecological frameworks. This explains why some many on speculative theories, but avoid the fragility of finance academics have had the tendency to make theories in running their risks. smooth returns, then blow up using their own theories6. Yet professional traders, including the authors (and, We started the other way around, first by years of alas, the Swedish Academy of Science) have operated option trading doing million of hedges and thousands of under the illusion that it was the Black-Scholes-Merton option trades. This in combination with investigating “formula” they actually used –we were told so. This the forgotten and ignored ancient knowledge in option myth has been progressively reinforced in the literature pricing and trading we will explain some common and in business schools, as the original sources have myths about option pricing and hedging. been lost or frowned upon as “anecdotal” (Merton, There are indeed two myths: 1992). • That we had to wait for the Black-Scholes- Merton options formula to trade the product, price options, and manage option books. In fact the introduction of the Black, Scholes and Merton argument increased our risks and set us back in risk management. More generally, it is a myth that traders rely on theories, even less a general equilibrium theory, to price options. • That we “use” the Black-Scholes-Merton options “pricing formula”. We, simply don’t. In our discussion of these myth we will focus on the bottom-up literature on option theory that has been Figure 1 The typical "risk reduction" performed hidden in the dark recesses of libraries. And that by the Black-Scholes-Merton argument. These addresses only recorded matters –not the actual are the variations of a dynamically hedged practice of option trading that has been lost. portfolio. BSM indeed "smoothes" out risks but exposes the operator to massive tail events – reminiscent of such blowups as LTCM. Other MYTH 1: PEOPLE DID NOT PROPERLY “PRICE” OPTIONS option formulas are robust to the rare event and BEFORE THE BLACK-SCHOLES-MERTON THEORY make no such claims.
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