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.