Celebratio Mathematica

The pa­per is or­gan­ized in three parts: the first part, writ­ten by Hely­ette Ge­man, dis­cusses the first en­coun­ters of Marc Yor with the field of math­em­at­ic­al fin­ance and a num­ber of an­ec­dotes re­lated to his life in the field. The second part, writ­ten by Dilip Madan, starts in 1997, when Dilip began vis­it­ing Par­is every Janu­ary for a month. The Ap­pendix is the testi­mony of Marc At­lan, a former PhD stu­dent of Marc Yor.

My trib­ute to Marc will be a col­lec­tion of short stor­ies, since I got the hon­or of shar­ing with him not only sci­entif­ic mo­ments but also mo­ments of fun, after which I would al­ways tell him that these would be pub­lished in my book of “Short Stor­ies”, but he would be al­lowed to proofread. The last piece is not go­ing to hap­pen, un­for­tu­nately.

One morn­ing at the end of the year 1989, I gathered my au­da­city and went to see the great Marc Yor with a pro­ject: I needed his (cru­cial) help to solve the prob­lem of eval­u­at­ing op­tions writ­ten on the arith­met­ic av­er­age of the stock price (so called Asi­an when they ap­peared in 1989, prob­ably fol­low­ing the names of European and Amer­ic­an op­tions) in the ref­er­ence Black–Scholes–Mer­ton set­ting, namely a stock price driv­en by a geo­met­ric Browni­an mo­tion.

I was wel­come by Marc’s usu­al kind­ness and ex­plained the chal­len­ging prob­lem: the sum of two geo­met­ric Browni­an mo­tions (today, it would be the sum of the ex­po­nen­tials of a Lévy pro­cess) is not a geo­met­ric Browni­an mo­tion, and many of us had un­der­stood at the time, us­ing sev­er­al av­en­ues and not ne­ces­sar­ily the ‘par­tial dif­fer­en­tial equa­tions’ ap­proach of the found­ing pa­pers, why this as­sump­tion in the Black–Scholes–Mer­ton mod­el made the valu­ation so ‘simple’. I also men­tioned the ex­ist­ing lit­er­at­ure at the time, and the battles ra­ging in the journ­al RISK — then read by prac­ti­tion­ers and aca­dem­ics in fin­ance — between those who were writ­ing that an “Av­er­age In­tel­li­gence” was needed to solve the prob­lem; and those who thought that the dif­fi­culty went “Bey­ond Av­er­age In­tel­li­gence”. Marc listened with amuse­ment; he al­ways smiled at my fin­ance stor­ies, agreed to look at the prob­lem but did not prom­ise to solve it or to write a pa­per on the sub­ject. I in­sisted on the fact that we should find the ex­act solu­tion of the ex­act prob­lem: not re­place the arith­met­ic av­er­age by a geo­met­ric one, not find a nu­mer­ic­al ap­prox­im­a­tion. Even though this was his first en­counter, to my best know­ledge, with fin­ance, Marc un­der­stood right away that an ex­act solu­tion would provide us with the Greeks, hence the hedging and trad­ing strategies.

Off we went on the pro­ject: Marc was a vis­it­ing pro­fess­or at ETH at the time. He would travel from Zurich by the night train on Fri­day and be in his of­fice by 7 am in the morn­ing, time at which we would meet. At 7h00 on Sat­urday, we did not get dis­turbed by too many people in the Labor­atoire de Prob­ab­ilités and I could ask again and again my ques­tions about pos­it­ive pro­cesses stable un­der ad­dit­iv­ity. In the mean time, we com­puted all mo­ments of the arith­met­ic av­er­age that we pub­lished in a Note aux Comptes Ren­dus de l’Académie des Sci­ences; a Ca­na­dian ac­tu­ary, Daniel Du­fresne, was also work­ing on the prob­lem.

One Sat­urday morn­ing, it happened: Marc said to me “What about us­ing the prop­erty that a geo­met­ric Browni­an mo­tion can be writ­ten as a time-changed squared Bessel pro­cess?” (Lamperti (1972), Wil­li­ams and Ro­gers (1987)). From that minute, stochast­ic time changes ap­plied to fin­ance provided us with their beau­ties: their ‘ob­vi­ous’ abil­ity to rep­res­ent stochast­ic volat­il­ity (ac­cel­er­ate the time when volat­il­ity is high­er, re­duce the speed when it is lower, and you re­cov­er the con­stant volat­il­ity of the Black–Scholes–Mer­ton mod­el). In the years 1991, 1992, the sub­ject of stochast­ic volat­il­ity was get­ting wide­spread on both sides of the ocean (see Dupire (1993), Der­man & Kani (1993), He­ston (1993)).

We ded­ic­ated a sec­tion of our pa­per to the use of stochast­ic time changes to ad­dress stochast­ic volat­il­ity; con­versely, we kept in mind that an easy way to in­tro­duce stochast­ic volat­il­ity in the stock price pro­cess is to change the time, as we did in Carr–Ge­man–Madan–Yor (2003). Moreover, the choice of the in­creas­ing pro­cess for the time change could in­cor­por­ate the prop­er­ties we per­ceived in volat­il­ity, such as clus­ter­ing for in­stance. In fact, I was so thrilled by the many ap­plic­a­tions of time changes to fin­ance that, for a while, the title of the pa­per was “Time Changes in Math­em­at­ic­al Fin­ance”, a title that should have been kept if it was not for the buzz around Asi­an op­tions…

The fi­nal ver­sion of the pa­per was typ­ic­ally French: very long, ad­dress­ing three big top­ics, the third one be­ing the valu­ation of per­petu­it­ies in the so-called Cox–In­ger­soll–Ross (1985) mod­el where the short-term rate is driv­en by a squared Bessel pro­cess. Be­cause of the square root present in the stochast­ic dif­fer­en­tial equa­tion sat­is­fied by the pro­cess, the CIR mod­el is also usu­ally re­ferred to as the square-root pro­cess. Des­pite its “dens­ity”, we re­ceived in 1994 the first prize of the Mer­rill Lynch awards for it and were happy… I went on and wrote in 1995 with Alex Ey­de­land a solu­tion to the in­ver­sion of the Laplace trans­form which ex­ten­ded the prob­lem to the com­plex plane.

To fin­ish on the sub­ject, which is an im­port­ant one to me since it gave me the op­por­tun­ity to build over three years a sol­id friend­ship with Marc, be­came a re­newed sub­ject of at­ten­tion be­cause of its gi­gant­ic use in the world of crude oil and oth­er com­mod­it­ies and was the theme of the last con­fer­ence we at­ten­ded to­geth­er in Ju­ly 2013, I will men­tion a fi­nal story. In March 1992 took place at the Uni­versity of War­wick a con­fer­ence on “Com­plex Op­tions”. Ton Vorst, from Erasmus Uni­versity, was go­ing to dis­cuss his clev­er ap­prox­im­a­tion of the arith­met­ic av­er­age by the geo­met­ric av­er­age, hence pre­serving the geo­met­ric Browni­an mo­tion. Prac­ti­tion­ers were present­ing vari­ous ap­prox­im­a­tion res­ults and my own slides were quite heavy: one stochast­ic time change plus two meas­ure changes to ‘only’ ob­tain the Laplace trans­form of the price with re­spect to time to ma­tur­ity. So, in or­der to in­tro­duce some num­bers, I de­cided to present an ex­ample based on the para­met­ers used in the valu­ation of an Asi­an op­tion writ­ten on a cur­rency dis­cussed in a pub­lished pa­per. I took a num­ber of strikes \( k \) for the op­tion, in­clud­ing \( k=0 \), and com­puted in par­al­lel the price of the plain-vanilla European call op­tion with same strike and ma­tur­ity. The Asi­an call op­tion turned out to be more ex­pens­ive in my com­pu­ta­tions and I say to my­self: this is im­possible, since the volat­il­ity is lower than the one of the European call. It is 8.30 pm in War­wick, 9.30 pm in St Chéron (with no un­cer­tainty!); I man­aged to find coins, a phone booth and called Marc about the situ­ation. After fif­teen minutes, I had con­sumed my coins, Marc had to go to bed after one of his usu­al long days and I was alone with my so-called mis­take or para­dox. Late at night, I un­der­stood: un­der the pri­cing meas­ure, the drift of the cur­rency in the nu­mer­ic­al ex­ample was neg­at­ive (as it may be the case also for di­vidend pay­ing stocks or com­mod­it­ies) and in­deed, the Asi­an call was more ex­pens­ive. I had — and oth­ers at the time — been fooled by the slo­gan “the price of the op­tion is the price of the volat­il­ity”; but the risk-ad­jus­ted drift mat­ters, ob­vi­ously…

We used again the meth­od of the com­pu­ta­tion of the Laplace trans­form of the op­tion price three years later in the valu­ation of double- bar­ri­er op­tions, a prob­lem that could also be solved as a triple in­teg­ral of the dens­ity of the triple (Browni­an mo­tion \( (t) \), Max­im­um \( (t) \), Inf \( (t) \)) provided by Louis Bacheli­er in 1941.

So much was hap­pen­ing at that time in the field! In par­al­lel to op­tion pri­cing, I con­tin­ued to work on the prob­lem of stochast­ic time changes with one of my PhD stu­dents as of 1994. At that time, there was a first large move of the fin­an­cial mar­kets in­to what is called today “High fre­quency trad­ing”; in the work with Ané, we were already look­ing at time in­ter­vals of one minute versus 30 minutes. We dis­covered the pa­per by Clark (1973) who had re­mark­ably con­jec­tured that rep­res­ent­ing the stock re­turn as a sub­or­din­ated Browni­an mo­tion was a way to ad­dress the non-nor­mal­ity of as­set re­turns without re­sort­ing to pro­cesses with in­fin­ite vari­ance, as pro­posed by Man­del­brot in 1966. The beau­ti­ful Clark’s rep­res­ent­a­tion of the stock as a sub­or­din­ated pro­cess was ob­tained through the ap­proach of Func­tion­al Ana­lys­is. The stochast­ic time-change prob­ab­il­ist­ic ap­proach does not re­quire the time-change to be a sub­or­din­at­or. Dur­ing Sum­mer of 1997, Marc was on the West Coast as usu­al and I was at the Uni­versity of Mas­sachu­setts. I had dis­cus­sions on the top­ic of time change with the prob­ab­il­ist Joe Horow­itz; Joe in­dic­ated to me the pa­per writ­ten by Mon­roe in 1978 about the prop­erty that “Any se­mi­martin­gale can be ex­pressed as a time-changed Browni­an mo­tion”. So con­sequently, the circle was closed: in Fin­ance, un­der No Ar­bit­rage, the (log) price pro­cess has to be a semi mar­tin­gale un­der the real prob­ab­il­ity meas­ure (since a mar­tin­gale un­der an equi­val­ent mar­tin­gale meas­ure). Itrel Mon­roe (1978) in his beau­ti­ful pa­per said that any se­mi­martin­gale could be em­bed­ded/writ­ten as a time-changed Browni­an mo­tion. Hence, writ­ing the log stock price as a time-changed Browni­an mo­tion was not a con­jec­ture any more (but Clark “could” not know it in 1973!). I was so ex­cited that I called Marc on the West Coast that very day (his land­lord was cer­tainly amazed at the num­ber of groupies this quiet math­em­atician had!). And Marc, with his usu­al hu­mil­ity, told me “I should have thought about this pa­per by Mon­roe !”. I went on and wrote the res­ult in my pa­per with Ané pub­lished in 2000 — now, we only needed to identi­fy em­pir­ic­ally and/or math­em­at­ic­ally the stochast­ic clock provid­ing nor­mal­ity, and em­pir­ic­ally, the ob­vi­ous can­did­ates to drive the clock were the volume or num­ber of trades. We also wrote with Marc and Dilip in 2001 our first pro­pos­al for the in­tro­duc­tion of jumps in stock prices un­der the title “As­set Prices are Browni­an mo­tion: Only in Busi­ness Time”: the time change had to be dis­con­tinu­ous in or­der to al­low for the loc­al un­cer­tainty con­tained in the driver of the clock, namely the news/trad­ing activ­ity.

One re­cur­rent fea­ture of my time with Marc was the pre­val­ence of joy: I would make jokes about the fin­an­cial com­munity and Marc — who nev­er made a penny from his deep in­tu­ition of the field and mas­ter­ing of the maths around — would smile and be totally in­dul­gent, so smart in un­der­stand­ing a jar­gon he had nev­er really learnt. His in­tel­li­gence was fly­ing at so many levels!

A high mo­ment of amuse­ment was a con­fer­ence on De­riv­at­ives at Bo­ston Uni­versity in June 1998. The two guest speak­ers were Robert Mer­ton, No­bel Prize win­ner and part­ner of the then alive hedge fund Long Term cap­it­al man­age­ment; and Jef­frey Skilling, Chief Ex­ec­ut­ive Of­ficer of the then ma­gic com­pany En­ron. Robert Mer­ton made an aca­dem­ic present­a­tion about de­riv­at­ives. Jef­frey Skilling had not bothered to pre­pare any talk; he told the audi­ence he could not dis­close the amaz­ing trad­ing strategies En­ron was im­ple­ment­ing in their Hou­s­ton headquar­ters to gen­er­ate hun­dreds of mil­lions, but was just go­ing to give us one tip: “Turn on your wash­ing ma­chine at night, when elec­tri­city is priced at the off peak re­gime”. Marc was un­im­pressed and said to me “I can do that; it is the time in the day when I an­swer the dozens of faxes I have re­ceived”. We nev­er dis­cussed the sub­ject any more and I was so sur­prised three years later, after En­ron col­lapsed with a lot of dam­age to its em­ploy­ees and Jef­frey Skilling was sent to jail for wrong­do­ings, to hear Marc say to me on the phone “Is the per­son in jail the one wear­ing these in­cred­ibly poin­ted Tex­an boots? I knew his dis­closed strategies were too simple to be so prof­it­able!”

Dur­ing the First Bacheli­er World Con­gress I had the hon­or to or­gan­ize in 2000 at Collège de France and In­sti­tut Henri Poin­caré, Marc was a Key­note Speak­er, of course. He was so happy to meet Henry McK­ean, an­oth­er amaz­ingly humble char­ac­ter and an­oth­er math­em­atician able to cov­er the twelve boards of the Am­phi­théatre Mar­guer­ite de Nav­arre with an im­pec­cable hand­writ­ing of beau­ti­ful for­mu­las. McK­ean had writ­ten in 1965 the ex­act for­mula for the valu­ation of an Amer­ic­an op­tion in the geo­met­ric Browni­an mo­tion set­ting, in the case of an in­fin­ite ma­tur­ity. The prob­lem in the case of a fi­nite ma­tur­ity of the op­tion is still open today — -but there are many re­mark­able pa­pers on the sub­ject, of course.

An­oth­er joy­ful memory was the time of the first ses­sion on Math­em­at­ic­al Fin­ance Marc or­gan­ized at the Académie des Sci­ences, in the early years 2000. The day be­fore, a ma­jor strike had stopped all sub­ways and trains, and Marc had stayed overnight at the Hôtel de Sen­lis with Dilip. When I ar­rived to pick them up on the way to the Académie, they had already checked out of their rooms. Marc calmly went be­hind an arm­chair to change in­to the fresh clothes I had brought from my place. I was laugh­ing stu­pidly and re­peat­ing that I would write one day about this in­cred­ibly fam­ous sci­ent­ist get­ting dressed so un­com­fort­ably on his way to the Académie des Sci­ences…

Like every­one else, my greatest memory of Marc is his in­cred­ible gen­er­os­ity: gen­er­os­ity with his time, gen­er­os­ity for the way he would thank us for bring­ing for­ward in­ter­est­ing prob­lems, gen­er­os­ity with the amount of in­cred­ible ef­forts he put in­to the suc­cess­ful open­ing of Wolfgang Doeblin’s “Pli Cacheté”.

My last con­fer­ence with Marc took place in Ju­ly 2013 at the Uni­versity of St An­drews, in Scot­land. The sub­ject was “Asi­an Op­tions and Com­mod­it­ies”. The loc­a­tion was spec­tac­u­lar, the group very friendly and Marc, tired at the start, looked in­creas­ingly happy. An­drew Ly­as­soff, from Bo­ston Uni­versity, Marc and I were stay­ing at the same hotel. The second night, the three of us left our keys on the key hold­er at the en­trance, and we came back to find the hotel locked. So, the two math­em­aticians had to break in­to the place, and just when they had suc­ceeded, a po­lice car stopped by. I ex­plained to the po­lice­men that they should not worry be­cause these two men were “prob­ab­il­ists”, hence harm­less; they looked puzzled and left. Marc then told me that Scot­land should not leave the United King­dom, and once again, I was so im­pressed with how much he was in touch with the real world while fly­ing so high sci­en­tific­ally…

Marc Yor was a bril­liant math­em­atician whose love for math­em­at­ics led him to work tire­lessly to help any­one try­ing to use it to do whatever. We came from fin­ance to seek his help and ad­vice to find an en­gage­ment of time, ef­fort and en­cour­age­ment that at first we did not un­der­stand. But he just loved the sub­ject and all who tried to do something with it. With this in mind we re­min­isce here about our in­ter­ac­tions and dis­cus­sions that led to our joint math­em­at­ic­al con­tri­bu­tions to fin­ance. There are of course many oth­er con­tri­bu­tions of a schol­ar with his stature and abil­ity, but we let oth­ers speak about them. Giv­en Marc’s avail­able time, these en­deav­ours en­cour­aged Madan to spend every Janu­ary for some ten years in Par­is. Marc al­ways greeted Madan on these trips by ask­ing what was not work­ing and this set the agenda for the month.

The dis­cus­sions that led to the work usu­ally took place at Café Souf­flot in the month of Janu­ary, start­ing at 9 in the morn­ing on a Sat­urday, and go­ing through till 11 when we moved to some oth­er Café for a bit, fol­lowed by lunch after which Marc took the train back home. These week­end meet­ings set the sched­ule for the work week when Marc was of course also en­gaged in many oth­er mat­ters at the Uni­versity.

The early work centered around dis­cus­sions about how as­set prices should be modeled. We re­cog­nized that quite gen­er­ally they could be seen as Browni­an mo­tion un­der a time change. It was the re­cog­ni­tion that all in­creas­ing pro­cesses that are can­did­ates for time changes are dis­con­tinu­ous if they have some loc­al un­cer­tainty that led us to fo­cus on dis­con­tinu­ous pro­cesses. Be­sides the fact that for prac­tic­al pur­poses they are flex­ible enough to al­ter both skew­ness and ex­cess kur­tos­is loc­ally. We were already deal­ing with such a time change giv­en by the gamma pro­cess as em­bed­ded in the vari­ance gamma pro­cess. Marc main­tained that both Browni­an mo­tion and the gamma pro­cess were fun­da­ment­al, one for real val­ued pro­cesses and the oth­er for pos­it­ive in­creas­ing pro­cesses.

At one these meet­ings Madan, who was a con­sult­ant at Mor­gan Stan­ley, com­plained that the vari­ance gamma mod­el be­ing used at the bank to mark op­tion books reg­u­larly at mar­ket close world wide had to have dif­fer­ent para­met­ers for each op­tion ma­tur­ity. The mod­el was defined for all ma­tur­it­ies but the para­met­ers had to be made ma­tur­ity spe­cif­ic. It was a suc­cess that they were not strike spe­cif­ic and it would be nice to have a mod­el that worked not only across op­tion strikes but ma­tur­it­ies as well. These mod­els are con­struc­ted by de­fin­ing a func­tion de­scrib­ing the rate at which price moves of dif­fer­ent sizes are oc­cur­ring. The vari­ance gamma had such a rate func­tion, and the first at­tempt was to al­ter this rate func­tion, make it more gen­er­al with some more para­met­ers. Marc had worked with Ver­shik on com­bin­ing the rate func­tion for the gamma pro­cess with that for the stable \( \alpha \) laws. We de­cided to in­vest­ig­ate this rate func­tion com­bin­a­tion as a new rate func­tion for moves in the log­ar­ithm of the stock price. The vari­ance gamma mod­el already had three para­met­ers and the \( \alpha \) para­met­er of the stable \( \alpha \) laws was a fourth para­met­er. Re­mem­ber­ing Samuel­son’s story, at the Bacheli­er con­gress 2000 in Par­is, of how he chose the name “Amer­ic­an Op­tions” for such op­tions, by em­phas­iz­ing the power au­thors have in nam­ing things, we de­noted each of the para­met­ers by the last name ini­tials of the four au­thors. This led to the \( CGMY \) mod­el. The para­met­er \( Y \) was the stable \( \alpha \) para­met­er and ad­dressed the fine struc­ture of the pro­cess go­ing from fi­nite activ­ity to in­fin­ite vari­ations as \( Y \) ranged between neg­at­ive to pos­it­ive unity. It was ap­pro­pri­ately labeled after Marc.

Un­for­tu­nately the \( CGMY \) mod­el did not solve the prob­lem of find­ing a parsi­mo­ni­ously para­met­er­ized mod­el con­sist­ent with some 300 op­tion prices across strike and ma­tur­ity at mar­ket close on a single day. All the four para­met­ers had to be made ma­tur­ity spe­cif­ic as they had to be for the vari­ance gamma mod­el. In fact we learned from this ex­per­i­ence that no rate func­tion for log­ar­ithmic price moves that was in­de­pend­ent of the time of the move could ever fit op­tion prices across ma­tur­ity. This was be­cause all time in­de­pend­ent rate func­tions were as­so­ci­ated with skew­ness of log prices at ma­tur­ity \( t \) fall­ing like the re­cip­roc­al of the square root of the ma­tur­ity \( t \), while ex­cess kur­tos­is fell like the re­cip­roc­al of \( t \). In the op­tion data one could check that both skew­ness and ex­cess kur­tos­is ten­ded to be con­stant or slightly rising.

At an­oth­er Café Souf­flot meet­ing Marc offered yet an­oth­er re­mark­able and beau­ti­ful solu­tion to the prob­lem. He ob­served that both Lévy and Kh­intchine had stud­ied in 1937–1938 the ques­tion of all the lim­it laws that may pos­sibly ex­ist on tak­ing lim­its of centered and scaled se­quences of ran­dom vari­ables. They had been clas­si­fied as the set of self-de­com­pos­able laws all which were a sub­set of laws with the time in­de­pend­ent rate func­tions we had been work­ing with. These lim­it laws had just been sit­ting there re­ceiv­ing little at­ten­tion from any­one. Marc com­men­ted that Sato had re­cently shown how to as­so­ci­ate with each self-de­com­pos­able or lim­it law a pro­cess with a time de­pend­ent rate func­tion called an ad­dit­ive pro­cess. It turned out that the vari­ance gamma law was self-de­com­pos­able and one could build the as­so­ci­ated ad­dit­ive pro­cess that we termed the Sato pro­cess. This was suc­cess­ful and re­mains to date the only parsi­mo­ni­ous, in fact four para­met­er, mod­el con­sist­ent with op­tion prices across strike and ma­tur­ity at mar­ket close on a single stock. The mod­el syn­thes­izes some 300 op­tion prices in just four para­met­ers. We pub­lished the res­ults with some re­ques­ted delay in 2007.

Marc was of course also in­ter­ested not just in the Sato pro­cess but in all oth­er pro­cesses that may have the same con­sist­ency with op­tion prices across strikes and ma­tur­it­ies. This led to the con­struc­tion of nu­mer­ous such ex­amples and the pa­per titled “Mak­ing Markov Mar­tin­gales Meet Mar­gin­als”. He began with the work on Skoro­hod em­bed­ding and gen­er­al­ized from there­on. It was won­der­ful to watch all the vari­ous con­struc­tions come in­to ex­ist­ence.

In the mean­time the mar­ket had be­gun trad­ing the volat­il­ity of stock prices dir­ectly along with op­tions on this volat­il­ity. Mod­els with a con­stant rate func­tion or even one with a fixed time de­pend­ence as in the Sato pro­cess would see no volat­il­ity of volat­il­ity and must price op­tions on volat­il­ity at the lar­ger strikes at zero. Real­ized vari­ance however was a ran­dom vari­able and our first piece in this dir­ec­tion ex­pli­citly priced op­tions on real­ized in such struc­tures. The ac­cess was via ex­pli­cit con­struc­tions of the Laplace trans­form of the quad­rat­ic vari­ation for such a pro­cess.

Later we for­mu­lated mod­els with our suc­cess­ful rate func­tions that ex­pli­citly al­lowed volat­il­ity to be volat­ile as op­posed to be­ing con­stant. We bor­rowed from the lit­er­at­ure on stochast­ic volat­il­ity to change the clock of our con­stant rate func­tion mod­els by run­ning them on clocks that ran­domly speed up or slow down but re­vert to­wards some con­stant speed in the longer term. We came up with two classes of mod­els with the re­quired volat­il­ity of volat­il­ity. In the first, the ex­pec­ted fu­ture stock price ap­pro­pri­ately dis­coun­ted equalled the ini­tial stock price. In the second the ex­pec­ted fu­ture stock price com­puted at any in­ter­me­di­ate time ap­pro­pri­ately dis­coun­ted to the in­ter­me­di­ate time equalled the stock price at this in­ter­me­di­ate time. The the­ory of no dy­nam­ic ar­bit­rage in mar­kets ar­gued that the second mod­el was the right one. However, the first mod­el fit the data much bet­ter. This out­come posed a quandary to be un­der­stood and cla­ri­fied.

The an­swer came in dif­fer­en­ti­at­ing dif­fer­ent ar­bit­rage op­por­tun­it­ies or op­por­tun­it­ies to make money at no risk. One of these in­volves put­ting on a risk­less prof­it­able trade at today’s prices and then hold­ing po­s­i­tions to ma­tur­ity with no fur­ther trad­ing. Such ar­bit­rages are called stat­ic ar­bit­rages and their ab­sence is termed no stat­ic ar­bit­rage. Oth­er dy­nam­ic ar­bit­rages present in mod­els in­volve trad­ing in the fu­ture as well at prices pre­vail­ing in the mod­el at the fu­ture dates. A trader who sees a stat­ic ar­bit­rage will put it on, but one who sees a dy­nam­ic ar­bit­rage in a mod­el may be reti­cent in put­ting it on as he or she does not be­lieve that the re­quired fu­ture trades can be con­duc­ted at the prices be­ing pro­posed or as­sumed in the mod­el. The first mod­el that fit the data was con­sist­ent with no stat­ic ar­bit­rage. The second that did not fit the data was con­sist­ent with no dy­nam­ic ar­bit­rage but as these ar­bit­rages were mod­el de­pend­ent and they could be ig­nored in ac­tu­al mar­kets. Un­der­stand­ing these dif­fer­ences in the po­ten­tial in­form­a­tion used by mar­kets and those made avail­able in mod­els was crit­ic­al in resolv­ing the quandary. Marc showed how the ab­sence of stat­ic ar­bit­rage was con­sist­ent with no dy­nam­ic ar­bit­rage in some dif­fer­ent mod­el for how in­form­a­tion may be evolving and the mar­ket may pay at­ten­tion to this dif­fer­ence. Dy­nam­ic ar­bit­rage in some spe­cif­ic mod­el is just that and un­less you have com­plete mod­el con­fid­ence it may be mar­ket ig­nored.

The ab­sence of stat­ic ar­bit­rage was tied to dis­tri­bu­tions for the stock price at each ma­tur­ity be­ing in­creas­ing in the con­vex or­der. This just means that the price of con­vex pay­offs prom­ised later are worth more. The same ques­tion could be asked about dis­tri­bu­tions of volat­il­ity in the mar­kets for op­tions on volat­il­ity. Though one could prove they had to be in­creas­ing in the con­vex or­der we ob­served that once scaled to a com­mon ex­pect­a­tion, they ten­ded to be in­creas­ing in the con­vex or­der. These in­vest­ig­a­tions led Marc to de­vel­op the prop­er­ties of what he even­tu­ally called pea­cocks and lyre­birds.

Leav­ing aside parsi­mo­ni­ous para­met­ric mod­els, the in­dustry of­ten uses non­para­met­ric mod­els whereby the volat­il­ity of the stock is a modeled as a de­term­in­ist­ic func­tion of the price and cal­en­dar time. For rate func­tions this re­quires mak­ing the speed of the clock a de­term­in­ist­ic func­tion of the price and cal­en­dar time. The ques­tion then arises on how to re­cov­er such clock de­pend­ence from quoted op­tions prices. The cor­res­pond­ing ques­tion for volat­il­ity in the case of Browni­an mo­tion driv­ing the stock was answered in the Dupire loc­al volat­il­ity mod­el and for­mula re­lat­ing loc­al volat­il­ity to the op­tion prices. Marc showed via an ap­plic­a­tion of the Mey­er–Tana­ka–Itô for­mula in the con­text of rate func­tions how one may ex­tract a de­term­in­ist­ic clock speed func­tion from op­tion prices. This work has been ex­ten­ded to in­clud­ing a sep­ar­ate stochast­ic com­pon­ent to the volat­il­ity or the clock speed but now re­cov­ery from op­tion prices is no longer pos­sible and one has to look to oth­er sources. At­ten­tion is be­ing paid to the joint in­form­a­tion that may be re­covered from op­tions on the stock and op­tions on the volat­il­ity of the stock.

Apart from Par­is, Marc reg­u­larly vis­ited the United States and Berke­ley in par­tic­u­lar. On one of these oc­ca­sions he vis­ited Madan at Mor­gan Stan­ley, on his way back to Par­is, and spent one day at the bank. He had a desk in a line of com­puters with no use for one as he worked only with a pen and pa­per. The head trader on struc­tured equity asked the head quant what the ef­fect of stochast­ic volat­il­ity would be on the price of swap that pays the sum of ab­so­lute price changes in the geo­met­ric Browni­an mo­tion mod­el. The head quant came over to Madan with the ques­tion and it was clear that with Marc in the room, where the ques­tion was to go. So we asked Marc and he im­me­di­ately got to work with the pre­dic­a­ment that stochast­ic volat­il­ity was stochast­ic time and time ap­peared in the Black–Scholes for­mula in many places. He then trans­formed the Black–Scholes for­mula till when time ap­peared in only place and then pro­claimed that he knew this for­mula as it was the for­mula for the last time Browni­an mo­tion was at a pre­spe­cified level. He thus saw the op­tion price as a prob­ab­il­ity for the first time and this led to the book, a year or two later en­titled Op­tion Prices as Prob­ab­il­it­ies.

On an­oth­er trip of Madan to Mor­gan Stan­ley, he asked Marc with as­sist­ing bet­ter sim­u­la­tions of the \( CGMY \) mod­el by writ­ing it as a time-changed Browni­an mo­tion. Marc al­ways be­lieved this was pos­sible but one had to get down to the de­tails and show how to con­struct the re­quired time change. We star­ted with the sym­met­ric case and fi­nally wrote this as a time change by a shaved stable sub­or­din­at­or where one threw out some of the jumps in this sub­or­din­at­or. The asym­met­ric pro­cess fol­lowed on ex­po­nen­tial tilt­ing the sym­met­ric one. The same meth­od worked on oth­er pro­cesses like the Meixn­er pro­cess.

In the math­em­at­ics of un­cer­tainty all events have a prob­ab­il­ity. In the fin­an­cial ana­lys­is of un­cer­tainty all events have a price. In gen­er­al the two are not the same. The price of an event is like a prob­ab­il­ity and in gen­er­al will ex­ceed prob­ab­il­ity when people are ad­versely af­fected by the event and wish to in­sure against it. Ex­amples be­ing the pur­chase of in­sur­ance of many kinds. Oth­er events have prob­ab­il­it­ies high­er than price re­flect­ing a pos­it­ive ex­pec­ted re­turn to com­pensate for the risk be­ing un­der­taken. Ex­amples in­clude the loan­ing of mon­ies to fund busi­ness activ­ity. The prob­ab­il­ity of suc­cess must be sub­stan­tial when com­pared to the price. We asked the ques­tion of wheth­er it was pos­sible that all risks were priced and there were no events for which the price equalled the prob­ab­il­ity. We came close to con­struct­ing such a situ­ation but not quite there. In any case it is pos­sible that many events see a change between price and prob­ab­il­ity.

More re­cently mar­kets are trad­ing equity linked notes with long ma­tur­it­ies of 15 to 20 years with claims re­turn­ing prin­cip­al but pay­ing sub­stan­tial coupons provided stock in­dices lie above a pre­spe­cified bar­ri­er. The mod­els be­ing used to price such claims have the stock price go­ing to zero with prob­ab­il­ity one at in­fin­ity and Madan was con­cerned that the prob­ab­il­ity of stay­ing above the bar­ri­er was be­ing un­der­es­tim­ated and the coupon over­stated. He asked Marc about hav­ing mod­els where the dis­coun­ted stock is well defined at in­fin­ity. Marc sug­ges­ted the use of Wien­er gamma in­teg­rals for the con­struc­tion of such pro­cesses. They con­firm the coupon over­state­ment and lead to mod­els that al­low on to dis­tin­guish booms from busts in mar­kets based on wheth­er it is up or down move­ments that are poorly modeled. Madan and Yor gone on to use these meth­ods to de­vel­op a pro­gram to de­tect mar­ket mal­func­tion by the loss of a prop­er lim­it­ing dis­coun­ted stock price and the near fu­ture will see the ap­plic­a­tion of such meth­ods to ac­tu­al mar­kets, but now without Marc’s as­sist­ance.

With a view to en­han­cing the ana­lys­is of hy­brid products writ­ten on two or more un­der­ly­ing un­cer­tain­ties we de­veloped a simple hy­brid mod­el for in­terest rate and equity hy­brids. Mod­el­ing the in­terest rate as mean re­vert­ing and driv­en by pos­it­ive gamma shocks with the rate moves dir­ectly im­pact­ing the stock price we for­mu­lated the joint char­ac­ter­ist­ic func­tion for the rate, its in­teg­ral and the log­ar­ithm of the stock. This gave us a sim­u­la­tion en­gine for the ana­lys­is of many hy­brid products.

In still more re­cent work, sub­mit­ted a day after Marc passed away, we linked the the­ory of two price eco­nom­ies in con­tinu­ous time to the the­ory of non­lin­ear \( G \) ex­pect­a­tions, via mod­el­ing the non­lin­ear­ity in pri­cing op­er­at­ors by meas­ure dis­tor­tions of the rate func­tions men­tioned earli­er. Madan has gone on to show that the re­quired non­lin­ear­it­ies oc­cur in both meas­ure changes and the dis­count­ing func­tions with pos­sible in­ter­ac­tions between them. These ad­vances are de­liv­er­ing non­lin­ear valu­ation meth­od­o­lo­gies for cor­por­ate de­cision, bring­ing new in­sights in­to old and of­ten in­tract­able prob­lems blocked by the use of lin­ear­ity in valu­ation. We an­ti­cip­ate much fur­ther de­vel­op­ments in this non­lin­ear link­age between rate func­tions de­scrib­ing un­cer­tainty at the in­stant­an­eous time evol­u­tion, and the valu­ation op­er­at­ors guid­ing in­vest­ment, risk man­age­ment and fin­an­cial de­cision mak­ing more broadly. Un­for­tu­nately, we shall have to go without the bril­liant in­sights of Marc at our side.

I had the hon­our and the priv­ilege to be a PhD stu­dent of Marc Yor between 2002 and 2006. Hence I had the chance to in­ter­act with him on a daily basis with re­gards to math­em­at­ics and, in fact, many oth­er top­ics. In do­ing so I was ex­posed to his in­cred­ible know­ledge and tal­ent, kind­ness, hu­mil­ity and help. I will now try to provide the read­ers with an in­sight in­to his qual­it­ies by shar­ing some of my per­son­al ex­per­i­ences with him.

In this sec­tion, I will try to present in my opin­ion some es­sen­tial skills and val­ues Marc Yor taught and in­stilled in his doc­tor­al stu­dents.