Black-Scholes: the formula at the origin of Wall Street

This year marks the 50th anniversary of the pub­lic­a­tion of a land­mark paper: “The Pri­cing of Options and Cor­por­ate Liab­il­it­ies” by Fisc­her Black and Myron Scholes. This paper describes a meth­od to determ­ine the price of a call option, a fin­an­cial con­tract which gives its hold­er the right (but not the oblig­a­tion) to pur­chase a fin­an­cial asset, called the under­ly­ing asset, at a pre­de­ter­mined price, at a pre­de­ter­mined future date. For all its import­ance, the for­mula itself is not the key con­tri­bu­tion of the paper, as some ver­sions of it were known well before Black and Scholes, not­ably from the 1900 thes­is of Louis Bacheli­er “Théor­ie de la Spécu­la­tion”.  The primary con­tri­bu­tion is the meth­od Black and Scholes used to prove that the for­mula is true.

To grasp their idea, think about a call option. Its price should clearly depend on the price of the under­ly­ing asset: when the price of the asset is high, the price of the asso­ci­ated option should also be high, and sim­il­arly, when the price of the asset is low, the option price should also be low. As time goes on and the asset price fluc­tu­ates, the option price will also fluc­tu­ate. It should then be pos­sible, by pur­chas­ing the assets dynam­ic­ally, to build a port­fo­lio whose value will fluc­tu­ate in exactly the same way as the price of the option. There­fore, if a trader has sold the option and holds this dynam­ic port­fo­lio, their pos­i­tion will be unaf­fected by mar­ket fluc­tu­ations, mak­ing it essen­tially risk-free.

As Black-Scholes the­ory became more wide­spread, options could be traded with great­er cer­tainty, without tak­ing on too much risk.

This is where the pivotal idea of Black and Scholes comes into play: if the pos­i­tion is risk-free, the return on this pos­i­tion should be equal to the return of the risk-free asset, such as an interest-bear­ing bond. The concept behind this idea is called ‘absence of arbit­rage’. If the return of the trader’s risk-free pos­i­tion were dif­fer­ent from the interest rate, the trader could earn money without tak­ing any risk and get very rich very quickly. Recog­niz­ing that the return of the hedged port­fo­lio is equal to the rate of interest, Black and Scholes then derived an equa­tion for the option’s price, whose solu­tion is giv­en by the Black-Scholes formula.

The key sig­ni­fic­ance of the Black and Scholes approach is that behind their for­mula there is a strategy for hedging the option: a trader selling an option at the price giv­en by the BS for­mula can imme­di­ately put in place a strategy, allow­ing to min­im­ize, if not com­pletely elim­in­ate, the risk asso­ci­ated to this pos­i­tion. Pri­or to Black and Scholes, such dynam­ic hedging strategies could not be com­puted in a sys­tem­at­ic way, which slowed down the devel­op­ment of deriv­at­ives markets.

As Black and Scholes the­ory became widely known, options could be traded with great­er secur­ity, without tak­ing too much risk. This led to expan­sion of option trad­ing, and the estab­lish­ment of option mar­kets, includ­ing the Chica­go Board of Options Exchange (1973), Marché des Options Négo­ci­ables de Par­is (1987) and others.

The Black-Scholes for­mula had a tur­bu­lent youth. The first wake-up call came with the 1987 fin­an­cial crisis. One key assump­tion behind the for­mula is that the asset price fol­lows a « con­tinu­ous time ran­dom walk ». This implies that the like­li­hood of hav­ing a large move over a short peri­od of time, such as as single day, is very small. Nev­er­the­less, on Monday, Octo­ber 19, 1987, fam­ously known as the Black Monday, Dow Jones Indus­tri­al Aver­age (the main index of the Amer­ic­an eco­nomy at that time) fell 22.6 per cent. Sellers of put options, designed to offer pro­tec­tion against such crashes, suffered heavy losses. It became clear that while the Black-Scholes for­mula per­formed well in nor­mal mar­ket con­di­tions, it failed to account for extreme events such as the Black Monday.

The answer of the fin­an­cial mar­kets was to adjust the para­met­ers of the for­mula: the options offer­ing pro­tec­tion against mar­ket crashes were now priced with high­er volat­il­ity para­met­er than the options cap­tur­ing small every­day mar­ket moves. This effect became known as the ‘volat­il­ity smile’ because of the smile-like shape the volat­il­ity graph has on traders’ screens. Since then, ever more com­plex exten­sions of the Black-Scholes for­mula were developed: loc­al volat­il­ity, stochast­ic volat­il­ity, rough volat­il­ity, etc.

The Black-Scholes paradigm was ques­tioned by sev­er­al authors who argue that rad­ic­ally dif­fer­ent mod­els are needed for bet­ter risk man­age­ment, such as the ones based on fractals intro­duced by Ben­oit Man­del­brot. How­ever, these mod­els nev­er took hold in the fin­an­cial industry because they do not allow for effect­ive hedging. Risk man­age­ment in options mar­kets is still based on the dynam­ic hedging prin­ciple pion­eered by Black and Scholes, and their for­mula, although rarely used dir­ectly, still provides the traders with a com­mon lan­guage to express more com­plex ideas.

The Black-Scholes for­mula res­ults from an equa­tion, remin­is­cent of the so called « heat equa­tion » in phys­ics, which describes the propaga­tion of heat in a sol­id body. It is there­fore no sur­prise that the first « quants » came from the phys­ics back­ground. How­ever, math­em­aticians soon real­ized that they, and not the phys­i­cists, had the per­fect tools to devel­op the the­ory of option pri­cing. With the pub­lic­a­tion of two land­mark papers by Har­ris­on and Kreps in 1979 and Har­ris­on and Pliska in 1982, it became clear, that the the­ory of stochast­ic cal­cu­lus is tail­or-made for describ­ing the notions of arbit­rage, dynam­ic hedging, and ulti­mately, option pri­cing. Stochast­ic cal­cu­lus was inven­ted by Japan­ese math­em­atician Kyosi Ito, and fur­ther developed by the French school of prob­ab­il­ity in Par­is and in Stras­bourg. No won­der then that many math­em­aticians found in the new fin­an­cial for­mu­las a per­fect applic­a­tion ter­rain with stim­u­lat­ing research ques­tions, curi­ous stu­dents, and sup­port­ive indus­tri­al part­ners. As a res­ult, a pro­duct­ive and endur­ing part­ner­ship formed between some parts of the math­em­at­ic­al com­munity and the fin­an­cial sec­tor. Not only math­em­aticians helped the traders to eval­u­ate options, but the fin­an­cial sec­tor was an import­ant source of ideas, which led to the emer­gence of new branches of probability.

A pro­duct­ive and last­ing part­ner­ship has been formed between part of the math­em­at­ic­al com­munity and the fin­an­cial sector.

Unfor­tu­nately, this last­ing rela­tion­ship led some traders to believe that the math­em­at­ics enabled them to per­fectly price and hedge any kind of option, how­ever soph­ist­ic­ated it may be. When the glob­al fin­an­cial crisis struck, some thought that the math­em­aticians were to blame, and that the math­em­at­ic­al mod­els were the « weapons of mass destruc­tion » that pre­cip­it­ated the crisis. In truth how­ever, the crisis was not caused by too much math­em­at­ic­al research, but too little of it. The for­mula used by the banks to price Col­lat­er­al­ized Debt Oblig­a­tions, a fin­an­cial deriv­at­ive largely respons­ible for the crisis, was too simple for this pur­pose, and failed to account for many risks asso­ci­ated with these com­plex products.

The crisis brought about pro­found changes, not only in the fin­an­cial industry but also in the fin­an­cial math­em­at­ics. Instead of devel­op­ing com­plex mod­els for option pri­cing, the focus of research shif­ted to more robust approaches and to the man­age­ment of new types of risk such as the risk of sys­tem­ic fail­ures of the fin­an­cial system.

Dur­ing the late 1980s, Par­is emerged as a prom­in­ent fin­an­cial cen­ter with numer­ous banks and a fledging option mar­ket. It was also the home of some of the world’s lead­ing experts in prob­ab­il­ity, stochast­ic cal­cu­lus and stochast­ic con­trol. On the oth­er hand, the French high­er edu­ca­tion sys­tem with its Grandes Ecoles had a stong emphas­is on com­pre­hens­ive train­ing in math­em­at­ics, and many stu­dents were keen to learn about new applic­a­tions of this sci­entif­ic discipline.

Par­is in the late 1980s was there­fore a fer­tile ground for fur­ther advance­ment of fin­an­cial math­em­at­ics, cre­ation of teach­ing pro­grams in quant­it­at­ive fin­ance, and part­ner­ships between uni­ver­sit­ies and fin­an­cial insti­tu­tions. This new domain attrac­ted the interest of the lead­ing French prob­ab­il­ists, among them Nicole El Karoui, Hély­ette Geman, Nic­olas Bouleau, Dami­en Lam­ber­ton and Bern­ard Lapeyre.

In 1990, a fin­an­cial math­em­at­ics track was cre­ated in the main mas­ter pro­gram in prob­ab­il­ity at Jussieu (now Sor­bonne Uni­versité). This pro­gram primar­ily attrac­ted stu­dents from the lead­ing engin­eer­ing schools such as Ecole Poly­tech­nique and Ecole des Ponts, who were taught the Black-Scholes the­ory with a dis­tinct French fla­vor of stochast­ic cal­cu­lus. Around the same time, a course in fin­an­cial math­em­at­ics was intro­duced in Ecole des Ponts, lead­ing to the pub­lic­a­tion, in 1992, of « Cal­cul stochastique appli­quée à la fin­ance » by D. Lam­ber­ton and B. Lapeyre, the first book on this top­ic in France and among the earli­est in the world. In 1997, Nicole El Karoui became pro­fess­or at Ecole Poly­tech­nique, and cre­ated the course « Méthodes stochastiques en fin­ance » in the applied math­em­at­ics major.

In the 10 years before the subprime crisis, the num­ber of stu­dents in these and oth­er pro­grams skyrock­eted, to the point that in 2006 Le Monde repor­ted that « one out of three quants in the world is French ». In the wake of the fin­an­cial crisis, stu­dent enroll­ment declined to some extent, due to a tem­por­ary decrease in hir­ing by banks. Moreover, the focus of the teach­ing pro­grams shif­ted from option pri­cing to risk man­age­ment and reg­u­la­tion. Cur­rently, the flow of French quants con­tin­ues at a more mod­er­ate pace. Non­ethe­less, the pro­gram at Poly­tech­nique and the his­tor­ic­al mas­ter pro­gram in prob­ab­il­ity and fin­ance, now jointly man­aged by Poly­tech­nique and Sor­bonne Uni­versité, still rep­res­ent a mark of excel­lence in the field.