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.

Copié !

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.

Copié !

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.