Why We Have Never Ud the Black-Scholes-Merton Option Pricing Formula1
Espen Gaarder Haug & Nassim Nicholas Taleb
February 2009 – Fifth Version
Abstract: Options traders u a 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 contradiction with it). However we have historical
evidence that 1) Black, Scholes and Merton did not invent any formula, just
found an argument to make a well known (and ud) formula compatible with
the economics establishment, by removing the “risk” parameter through
“dynamic hedging”, 2) Option traders u (and evidently have ud since 1902)
2012灾难片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 by using put-call parity. 3) Option
traders did not u formulas after 1973 but continued their bottom-up heuristics.shooin
The Bachelier-Thorp approach is more robust (among other things) 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
calling the formula by the wrong name.
1 Thanks to Russ Arbuthnot for uful comments.
B REAKING THE
C HAIN OF T RANSMISSION
For us, practitioners, theories should ari from practice2. This explains our concern with the “scientific” notion that practice should fit theory. Option hedging, pricing, and trading is neither philosophy nor mathematics. It is a rich craft with traders learning from traders (or traders copying other traders) and tricks developing under evolution pressures, in a bottom-up manner. It is technë, not ëpistemë. Had it been a science it would not have survived – for the empirical and scientific fitness of the pricing and hedging theories offered are, we will e, at best, defective and unscientific (and, at the worst, the hedging methods create more risks than they reduce). Our approach in this paper is to ferret out historical evidence of technëshowing how option traders went about their business in the past.
Options, we will show, have been extremely active in the pre-modern finance world. Tricks and heuristically derived methodologies in option trading and risk management of derivatives books have been developed over the past century, and ud quite effectively by operators. In parallel, many derivations were produced by mathematical rearchers. The economics literature, however, did not recognize the contributions, substituting the rediscoveries or subquent reformulations done by (some) economists. There is evidence of an attribution problem with Black-Scholes-Merton option “fo
rmula”, which was developed, ud, and adapted in a robust way by a long tradition of rearchers and ud heuristically by option book runners. Furthermore, in a ca of scientific puzzle, the exact formula called “Black-Sholes-Merton” was written down (and ud) by Edward Thorp which, paradoxically, while being robust and realistic, has been considered unrigorous. This rais the following: 1) The Black Scholes Merton was just a neoclassical finance argument, no more than a thought experiment3, 2) We are not aware of traders using their argument or their version of the formula.
It is high time to give credit where it belongs.
2 For us, in this discussion, a practitioner is deemed to be someone involved in repeated decisions about option hedging, not a support quant who writes pricing software or an academic who provides “consulting” advice.
3 Here we question the notion of confusing thought experiments in a hypothetical world, of no predictive power, with either science or practice. The fact that the Black-Scholes-Merton argument works in a Platonic world and appears to be “elegant” does not mean anything since one can always produce a Platonic world in which a certain equation works, or in which a “rigorous” proof can be pro
vided, a process called rever-engineering. T HE B LACK-S CHOLES-M ERTON “F ORMULA” WAS AN
A RGUMENT
Option traders call the formula they u the “Black-Scholes-Merton” formula without being aware that by some irony, of all the possible options formulas that have been produced in the past century, what is called the Black-Scholes-Merton “formula” (after Black and Scholes, 1973, and Merton, 1973) is the one the furthest away from what they are using. In fact of the formulas written down in a long history it is the only formula that is fragile to jumps and tail events.
First, something ems to have been lost in translation: Black and Scholes (1973) and Merton (1973) actually never came up with a new option formula, but only an theoretical economic argument built on a new way of “deriving”, rather rederiving, an already existing –and well known –formula. The argument, we will e, is extremely fragile to assumptions. The foundations of option hedging and pricing were already far more firmly laid down before them. The Black-Scholes-Merton argument, simply, is that an option can be hedged using a certain methodology called “dynamic hedging” and then turned into a risk-free instrument, as the portfolio would no longer be stochastic. I
ndeed what Black, Scholes and Merton did was “marketing”, finding a way to make a well-known formula palatable to the economics establishment of the time, little el, and in fact distorting its esnce.
Such argument requires strange far-fetched assumptions: some liquidity at the level of transactions, knowledge of the probabilities of future events (in a neoclassical Arrow-Debreu style)4, and, more critically, a certain mathematical structure that requires “thin-tails”, or mild randomness, on which, later. The entire argument is indeed, quite strange and rather inapplicable for someone clinically and obrvation-driven standing outside conventional neoclassical economics. Simply, the dynamic hedging argument is dangerous in practice as it subjects you to blowups; it makes no n unless you are concerned with neoclassical economic theory. The Black-Scholes-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 e, are highly invalid mathematically, the main one being the ability to cut the risks using continuous trading which only works
4 Of all the misplaced assumptions of Black Scholes that cau it to be a mere thought experiment, though an extremely elegant one, a flaw shared with modern portfolio theory, is the certain knowledg
e of future delivered variance for the random variable (or, equivalently, all the future probabilities). This is what makes it clash with practice –the rectification by the market fattening the tails is a negation of the Black-Scholes thought experiment.
英语话题作文编程java培训班in the very narrowly special ca of thin-tailed distributions. But it is not just the flaws that make it inapplicable: option traders do not “buy theories”, particularly speculative general equilibrium ones, which they find too risky for them and extremely lacking in standards of reliability. A normative theory is, simply, not good for decision-making under uncertainty (particularly if it is in chronic disagreement with empirical evidence). People may take decisions bad on speculative theories, but avoid the fragility of theories in running their risks.
Yet professional traders, including the authors (and, alas, the Swedish Academy of Science) have operated under the illusion that it was the Black-Scholes-Merton “formula” they actually ud –we were told so. This myth has been progressively reinforced in the literature and in business schools, as the original sources have been lost or frowned upon as “anecdotal” (Merton,
1992).
Figure 1 The typical "risk reduction" performed by the Black-Scholes-Merton argument. The are the variations of a dynamically hedged portfolio. BSM indeed "smoothes" out risks but expos the operator to massive tail events –reminiscent of such blowups as LTCM. Other option formulas are robust to the rare event and make no such claims.
This discussion will prent our real-world, ecological understanding of option pricing and hedging b
ad on what option traders actually do and did for more than a hundred years.
This is a very general problem. As we said, option traders develop a chain of transmission of technë, like many professions. But the problem is that the “chain” is often broken as universities do not store the acquired skills by operators. Effectively plenty of robust heuristically derived implementations have been developed over the years, but the economics establishment has refud to quote them or acknowledge them. This makes traders need to relearn matters periodically. Failure of dynamic hedging in 1987, by such firm as Leland O’Brien Rubinstein, for
instance, does not em to appear in the academic literature published after the event 5 (Merton, 1992, Rubinstein, 1998, Ross, 2005); to the contrary dynamic hedging is held to be a standard operation.
There are central elements of the real world that can escape them –academic rearch without feedback from practice (in a practical and applied field) can cau the diversions we witness between laboratory and ecological frameworks. This explains why some many finance academics have had the tendency to make smooth returns, then blow up using their own theories 6. We started the other way around, first by years of option trading doing million of hedges and thousands of option
trades. This in combination with investigating the forgotten and ignored ancient knowledge in option pricing and trading we will explain some common myths about option pricing and hedging. There are indeed two myths:
•
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 incread our risks and t 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 “u” the Black-Scholes-Merton options “pricing formula”. We, simply don’t.
In our discussion of the myth we will focus on the bottom-up literature on option theory that has been hidden in the dark recess of libraries. And that address only recorded matters –not the actual practice of option trading that has been lost. M YTH 1: P EOPLE DID NOT PROPERLY “PRICE ” OPTIONS BEFORE THE B LACK -S CHOLES -M ERTON THEORY
It is assumed that the Black-Scholes-Merton theory is what made it possible for option traders to calculate their delta hedge (against the underlying) and to price options. This argument is highly debatable, both historically and analytically.
5
For instance –how mistakes never resurface into the consciousness, Mark Rubinstein was awarded in 1995 the Financial Engineer of the Year award by the International Association of Financial Engineers. There was no mention of portfolio insurance and the failure of dynamic hedging.
6 For a standard reaction to a rare event, e the following: "Wednesday is the type of day people will remember in quant-land for a very long time," said Mr. Rothman, a University of Chicago Ph.D. who ran a quantitative fund before joining Lehman Brothers. "Events that models only predicted would happen once in 10,000 years happened every day for three days." One 'Quant' Sees Shakeout For the Ages -- '10,000 Years' By Kaja Whitehou, August 11, 2007; Page B3.
Options were actively trading at least already in the 1600 as described by Joph De La Vega –implying some form of technë, a heuristic method to price them
and deal with their exposure. De La Vega describes option trading in the Netherlands, indicating that operators had some experti in option pricing and hedging. He diffuly points to the put-call parity, and his book was not even meant to teach people about the technicalities in option trading. Our insistence on the u of Put-Call parity is critical for the following reason: The Black-Scholes-Merton’s claim to fame is removing the necessity of a risk-bad drift from the underlying curity –to make the trade “risk-neutral”. But one does not need dynamic hedging for that: simple put call parity can suffice (Derman and Taleb, 2005), as we will discuss later. And it is this central removal of the “risk-premium” that apparently was behind the decision by the Nobel committee to grant Merton and Scholes the (then called) Bank of Sweden Prize in Honor of Alfred Nobel: “Black, Merton and Scholes made a vital contribution by showing that it is in fact not necessary to u any risk premium when valuing an option. This does not mean that the risk premium disappears; instead it is already included in the stock price.”7 It is for having removed the effect of the drift on the value of the option, using a thought experiment, that their work was originally cited, something that was mechanically prent by any form of trading and converting using far simpler techniques.
Options have a much richer history than shown in the conventional literature. Forward contracts ems to date all the way back to Mesopotamian clay tablets dating all the way back to 1750 B.C. G
elderblom and Jonker (2003) show that Amsterdam grain dealers had ud options and forwards already in 1550.
In the late 1800 and the early 1900 there were active option markets in London and New York as well as in Paris and veral other European exchanges. Markets it ems, were active and extremely sophisticated option markets in 1870. Kairys and Valerio (1997) discuss the market for equity options in USA in the 1870s, indirectly showing that traders were sophisticated enough to price for tail events8.
7 e www.Nobel.
8 The historical description of the market is informative until Kairys and Valerio try to gauge whether options in the 1870s were underpriced or overpriced (using Black-Scholes-Merton style methods). There was one tail-event in this period, the great panic of September 1873. Kairys and Valerio find that holding puts was profitable, but deem that the market panic was just a one-time event :
“However, the put contracts benefit from the “financial panic” that hit the market in September, 1873. Viewing this as a “one-time” event, we repeat the analysis for puts excluding any unexpired contracts written before the stock market panic.”
Using references to the economic literature that also conclude that options in general were overpriced in the 1950s 1960s and 1970s they conclude: "Our analysis shows that option There was even active option arbitrage trading taking
place between some of the markets. There is a long list of missing treatis on option trading: we traced at研究生面试英语自我介绍
least ten German treatis on options written between the late 1800s and the hyperinflation episode9.ddp
One informative extant source, Nelson (1904), speaks volumes: An option trader and arbitrageur, S.A. Nelson published a book “The A B C of Options and Arbitrage” bad on his obrvations around the turn of the twentieth century. According to Nelson (1904) up to 500 messages per hour and typically 2000 to 3000 messages per day were nt between the London and the New York market through the cable companies. Each message was transmitted over the wire system in less than a minute. In a heuristic method that was repeated in Dynamic Hedging by one of the authors, (Taleb,1997), Nelson, describe in a theory-free way many rigorously clinical aspects of his arbitrage business: the cost of shipping shares, the cost of insuring shares, interest expens, the possibilities
to switch shares directly between someone being long curities in New York and short in London and in this way saving shipping and insurance costs, as well as many more tricks etc.
The formal financial economics canon does not include historical sources from outside economics, a mechanism discusd in Taleb (2007a). The put-call parity was according to the formal option literature first fully described by Stoll (1969), but neither he nor others in the field even mention Nelson. Not only was the put-call parity argument fully understood and described in detail by Nelson (1904), but he, in turn, makes frequent reference to Higgins (1902). Just as an example Nelson (1904) referring to Higgins (1902) writes:
contracts were generally overpriced and were unattractive for retail investors to purcha”. They add: ”Empirically we find that both put and call options were regularly overpriced relative to a theoretical valuation model."
The results are contradicted by the practitioner Nelson (1904): “…the majority of the great option dealers who have found by experience that it is the givers, and not the takers, of option money who have gained the advantage in the long run”.
9 Here is a partial list:
Bielschowsky, R (1892): Ueber die rechtliche Natur der Prämiengeschäfte, Bresl. Genoss.-Buchdr
Granichstaedten-Czerva, R (1917): Die Prämiengeschäfte an der Wiener Bör, Frankfurt am Main
Holz, L. (1905) Die Prämiengeschäfte, Thesis (doctoral)--Universität Rostock
Kitzing, C. (1925): Prämiengeschäfte : Vorprämien-, Rückprämien-, Stellagen- u. Nochgeschäfte ; Die solidesten Spekulationsgeschäfte mit Versicherg auf Kursverlust, Berlin Ler, E, (1875): Zur Geschichte der Prämiengeschäfte
Szkolny, I. (1883): Theorie und praxis der prämiengeschäfte nach einer originalen methode dargestellt., Frankfurt am Main
Author Unknown (1925): Das Wen der Prämiengeschäfte, Berlin : Eugen Bab & Co., Bankgeschäft
“It may be worthy of remark that ‘calls’ are more
often dealt than ‘puts’ the reason probably being
that the majority of ‘punters’ in stocks and shares
are more inclined to look at the bright side of things,
哈佛大学入学条件and therefore more often ‘e’ a ri than a fall in
prices.
This special inclination to buy ‘calls’ and to leave the
‘puts’ verely alone does not, however, tend to
make ‘calls’ dear and ‘puts’ cheap, for it can be
shown that the adroit dealer in options can convert
a ‘put’ into a ‘call,’ a ‘call’ into a ‘put’, a ‘call o’ more’
into a ‘put- and-call,’ in fact any option into another,
by dealing against it in the stock. We may thereforeproducer
assume, with tolerable accuracy, that the ‘call’ of a
stock at any moment costs the same as the ‘put’ of
that stock, and half as much as the Put-and-Call.” The Put-and-Call was simply a put plus a call with the same strike and maturity, what we today would call a straddle. Nelson describes the put-call parity over many pages in full detail. Static market neutral delta hedging was also known at that time, in his book Nelson for example writes:
“Sellers of options in London as a result of long
experience, if they ll a Call, straightway buy half
the stock against which the Call is sold; or if a Put is
sold; they ll half the stock immediately.”
We must interpret the value of this statement in the light that standard options in London at that time were issued at-the-money (as explicitly pointed out by Nelson); furthermore, all standard options in London were European style. In London in- or out-of-the-money options were only traded occasionally and were known as “fancy options”. It is quite clear from this and the rest of Nelson’s book that the option dealers were well aware that the delta for at-the-money options was approximat
ely 50%. As a matter of fact, at-the-money options trading in London at that time were adjusted to be struck to be at-the-money forward, in order to make puts and calls of the same price. We know today that options that are at-the-money forward and do not have very long time to maturity have a delta very clo to 50% (naturally minus 50% for puts). The options in London at that time typically had one month to maturity when issued.
Nelson also diffuly points to dynamic delta hedging, and that it worked better in theory than practice (e Haug, 2007). It is clear from all the details described by Nelson that options in the early 1900 traded actively and that option traders at that time in no way felt helpless in either pricing or in hedging them.
Herbert Filer was another option trader that was involved in option trading from 1919 to the 1960s. Filer(1959) describes what must be considered a reasonable active option market in New York and Europe in the early 1920s and 1930s. Filer mentions however that due to World War II there was no trading on the European Exchanges, for they were clod. Further, he mentions that London option trading did not resume before 1958. In the early 1900’s, option traders
in London were considered to be the most sophisticated, according to Nelson. It could well be that
World War II and the subquent shutdown of option trading for many years was the reason known robust arbitrage principles about options were forgotten and almost lost, to be partly re-discovered by finance professors such as Stoll (1969).
Earlier, in 1908, Vinzenz Bronzin published a book
deriving veral option pricing formulas, and a formula very similar to what today is known as the Black-Scholes-Merton formula, e also Hafner and Zimmermann (2007). Bronzin bad his risk-neutral option valuation on robust arbitrage principles such as the put-call parity and the link between the forward price and call and put options –in a way that was rediscovered by Derman and Taleb (2005)10. Indeed, the put-call parity restriction is sufficient to remove the need to incorporate a future return in the underlying curity –it forces the lining up of options to the forward price11.
Again, 1910 Henry Deutsch describes put-call parity but
in less detail than Higgins and Nelson. In 1961 Reinach again described the put-call parity in quite some detail (another text typically ignored by academics). Traders at New York stock exchange specializing in using the put-call parity to convert puts into calls or calls into puts was at that time known as Converters. Reinach (1961):
一对一真人外教“Although I have no figures to substantiate my
claim, I estimate that over 60 per cent of all Calls
are made possible by the existence of Converters.” In other words the converters(dealers) who basically operated as market makers were able to operate and hedge most of their risk by “statically” hedging options with options. Reinach wrote that he was an option trader (Converter) and gave examples on how he and his colleagues tended to hedge and arbitrage options
10 The argument Derman Taleb(2005) was prent in Taleb (1997) but remained unnoticed.
11 Ruffino and Treussard (2006) accept that one could have solved the risk-premium by happenstance, not realizing that put-call parity was so extensively ud in history. But they find it insufficient. Indeed the argument may not be sufficient for someone who subquently complicated the reprentation of the world with some implements of modern finance such as “stochastic discount rates” –while simplifying it at the same time to make it limited to the Gaussian and allowing dynamic hedging. They write that “the u of a non-stochastic discount rate common to both the call and the put options is inconsistent with modern equilibrium capital ast pricing theory.” Given that we have never en a practitioner u “stochastic discount rate”, we, like our option trading predecewt
ssors, feel that put-call parity is sufficient & does the job.
The situation is akin to that of scientists lecturing birds on how to fly, and taking credit for their subquent performance –except that here it would be lecturing them the wrong way.
