I met Peter for the first time in February 1989 in New York, at a Colloquium of the American Stock Exchange. We were in the same session: Peter was presenting an essay of his dissertation dedicated to an exchange option where one party could choose the asset to deliver out of a defined set. I was presenting a paper on ‘Portfolios of Bonds and Futures on Bonds’ where the ‘Cheapest to Deliver’ was chosen by the seller of the Future among a basket of Treasuries. We were very happy to meet and soon after, Peter accepted my invitation to ESSEC, probably because he knew that France was a country with a long tradition in Probability – a subject he had not fully studied in his first education and fascinated him. He came for two weeks in the Finance Department, where my colleagues (all US Finance PhDs after a French Grande Ecole) were already quite familiar with the intricacies of complex Derivatives and greatly contributed to the construction of the ‘DEA Probabilities Finance’ and to the ‘school in Mathematical Finance’ which was growing in France, in particular after the long visit of Ioannis Karatzas invited by Nicole Elkaroui and the weekly Seminar started by Thierry Jeulin. Peter was carrying around the young and enthusiastic look he would keep all his life, as well as an immense kindness. His paper on American options (‘Alternative Characterizations of American Put Options’, Carr, Jarrow and Myneni, Mathematical Finance, 1992) contained so many rich results unknown at the time by many of us.
I met Dilip Madan in 1995 at a conference in Montreal and a three-way collaboration started after Dilip had joined Peter at Morgan Stanley. Our first joint paper discussed ‘Acceptability’ to address ‘Market Incompleteness’(CGM , Journal of Financial Economics, 2001). Dilip would come to Paris for two months a year; I naturally introduced him to Marc Yor and this is how the CGMY model came into existence. Marc Yor was bringing his ‘total’ knowledge of Brownian Motion and Levy Processes — I also salute his memory with emotion. Dilip had worked with Eugene Seneta on the ‘Variance Gamma Process (‘The Variance Gamma Process for Share Markets’, Madan and Seneta, Journal of Business, 1995) Peter had the perspective of a practitioner searching for a model better ‘adapted’ to the markets of the time. In my case, I had been very interested since 1995 in the ‘Subordination’ introduced in Finance by Clark, who offered to represent Cotton Futures prices by subordinated processes ( A Subordinated Stochastic with Finite Variance Model , Clark, Econometrica , 1973) This paper was remarkable for several reasons: firstly, Clark rejects the processes with infinite variance proposed by Mandelbrot in 1965 (‘Forecasts of Future Prices, Unbiased Markets and Martingale Models’, Mandelbrot, Journal of Business, 1965). Secondly, he introduces Subordination in Finance, which was a brilliant way to create new processes to address the non-normality of returns observed already in 1960 by Fama and many others after him. Thirdly, his paper first had an unfortunate fate because of the exclusive attention brought to the papers by Black & Scholes (1973) and Merton (1973) and time went by before the original contribution of Clark received deserved attention.
I read Clark’s paper with fascination, but the limits of subordination imposed by the mathematical constraints on the subordinator — when one wants precisely to account for the non-uniform arrival of the order flow in the markets and the resulting stochastic volatility — rapidly convinced me of the value of extending subordination to ‘stochastic time changes’ ( Geman, RISK, 1996). In Summer 1997, Joe Horowitz, a probabilist at the University of Massachusetts, brought to my attention the beautiful paper written by Monroe in 1978: ‘ Any semimartingale is a time-changed Brownian Motion’, (Itrel Monroe , ‘Processes that can be embedded in Brownian motion’, Annals of Probability, 1978). This paper was written after the Clark (1973) article (which Monroe certainly ignored) — and extended to semimartingales the theorem established for martingales by Dubins and Schwartz (1965). This brought the final piece …
By No Arbitrage, discounted stock prices are martingales under an equivalent probability measure Q (Harrison, Kreps, Pliska, Delbaen, Schachermayer). Consequently, stock prices have to be semimartingales under the real probability measure P, and, up to a change of filtration, they are ‘Time Changed Brownian Motion’.( G.& Ane, 2000, Journal of Finance). One can then go to market data to infer properties of the stochastic clock as we did with my Ph.D. student. Or one can decide to choose the stochastic time change that brings a ‘compound’ stochastic process with desirable mathematical properties, which was our approach in CGMY (Journal of Business, 2002), CGMY with Stochastic Volatility (Mathematical Finance, 2003), and other papers, for instance looking at Options on Variance. Peter used similar techniques in his work on FX options with his Ph.D. student Liuren Wu to study FX options (‘Stochastic Skew in Currency Options, Journal of Financial Economics, 2009).
So many other papers of Peter’s deserve to be mentioned and discussed, but I prefer to end on a joyful memory.
In 2008, the CGMY model received an Award from the Alma Mater Studiorum Institute of the University of Bologna (the oldest University in the world, and a remarkable place). The four of us had to each give a talk on the genesis of CGMY, and beyond. And we received an allowance’ of $500 per person that had to be spent exclusively in Bologna. I was the fastest one to fill the mission, obviously; the three others were walking all around the city where so many choices were offered. The result was that at departure, my three co-authors looked like elegant Italian gentlemen rather than usual mathematicians …
Since then, I have continued to meet Peter at conferences or at friendly dinners, mostly with Dilip and Master and doctoral students, sometimes with my Ph.D. student Nassim who had become his colleague at NYU Tandon.
Peter, we will miss your kindness and generosity. Your abrupt departure is another proof of the lack of Antifragility in human life.