Peter Carr left us, saddened and shocked, on the morning of March 1, 2022, after an earlier devastating internal hemorrhage. I spoke with Peter on January 29 asking about his response to my remarks on a discussion that took place at a virtual seminar on January 25, the second day of the Spring semester of 2022. He replied that he had not been feeling well and would get back to me when he was better. This was the first time, and now also the last time, that Peter responded in such a manner. I have never known him to be in such a state. Something was clearly amiss.
Undoubtedly, Peter leaves us with many wonderful memories of our interactions with him and here I try and relate some of them.
I first met Peter in the summer of 1991 at a Cornell conference that initiated the journal Mathematical Finance. At the time I had just arrived back to the US from twelve years in Australia and was an unknown personality in the US community of mathematicians with an interest in Finance. The meeting was brief, but we got to appreciate each other. We got to know each other much better in the Fall of 1995 when I spent the Fall semester on sabbatical at Cornell. Many discussions followed on the theory of no-arbitrage pricing.
By the summer of 1996, Peter had moved from Cornell to essentially head the quant group at Morgan Stanley. He may not have been the head, but was recognized by his brilliance, intellect, and energy as such by those around him. His natural inclination was to question, debate, and challenge what is being discussed from multiple directions in the friendliest way possible. Recognizing this the rest of the team just fell in place.
Peter arranged to have me hired as a consultant at Morgan Stanley in the summer of 1996 and I have since spent two to three days a week there. The spring of 1995 saw the birth of the variance gamma model with skewness added into it. Trader overrides were posing a strain on capital requirements supporting the options trading business. Marking to the variance gamma model was proposed, tried out on a small set of underliers, and then adopted firm-wide. The marking activity needed fast computation of variance gamma prices for thousands of underliers at market close. It must have been Thanksgiving of 1996, when my family and I were making our annual visit to my brother-in-law in Connecticut, where instead of joining the families to go shopping, the following Friday, I met Peter at Morgan Stanley and worked on developing the fast Fourier transform method for pricing options. This is what made the marking possible. Neither Peter nor I were interested in shopping. For many years the Friday after Thanksgiving was spent this way. These two papers are among our most cited papers. Many asked Peter to take charge of the model and its implementation. But Peter’s active mind had to move on to other pastures. It cannot stand in the light of any past glories.
Since then, we have worked in many ways and places. When it came to financial mathematics Peter did not take breaks. We worked in Sydney, Budapest, Paris, and London on numerous occasions. Once on the barrier reef in Australia when most of the conference party was taking dives or snorkeling, Peter could be seen on deck, laptop in hand, interviewing colleagues on matters of mathematical detail. He embraced early on the idea of proof in mathematics and loved it all the way. Directing donations to the math museum is most appropriate.
Peter moved on to revising the VIX computation and developing hedges for the variance and volatility swaps. I interacted with him on hedging insurance risks. We worked on the implications of the absence of static arbitrage. This led us to the study of additive processes of which the Sato process was another contribution. My conversation with Peter between January 25 and 29 of this year was on additive processes and has led to a new paper, now without him, that we could have co-authored. Peter, prolific with both ideas and the personalities he engaged with has published with many other co-authors on numerous aspects of the subject and we may note in this regard the extensive work with Liuren Wu, Roger Lee, Gurdip Bakshi, Andrey Itkin, Johannes Ruf, Travis Fisher, Matthew Lorig, Lorrenzo Torricelli, Sergey Nadtochiy to mention a few.
Apart from the papers and investigations, Peter gave freely of his time in delivering courses at conferences during the years he was with the financial industry. I participated in a few of these. One of these was a weeklong course, in January 2012, at the Tata Institute of Fundamental Research in Bombay, my hometown. Peter could not take financial remunerations for these and diverted the funds to be received to various causes. One of which was funding my FestSchrift conference in 2006. In 2016 Peter, Philip Protter, Rene Carmona, and myself worked in bringing the World Congress of the Bachelier Finance Society to New York.
Let me close by noting that there is much to fondly remember Peter by. I remember a night in Paris when we went to dinner at a restaurant with no menu, but you got what the house had prepared that day. The line to enter was too long and at Peter’s suggestion, we returned the next night to be first in line. Another time in the financial district of London we went to a pub and on some rare occasions Peter would take an alcoholic beverage, standard fare for me, but to my surprise, he took to a cherry flavored beer. A bit courageous for my taste but Peter delved into the new easily. Bought an electric vehicle easily and really enjoyed driving it. Would go out of the way to extend the ride. In Sydney, we enjoyed the early morning swim at Manly beach followed by Tasmanian scallops, one of my favorites there. He bought me a gigantic wine glass that holds a bottle of wine so I could say I just had one glass of wine at night! On another occasion, Peter stayed at my house overnight for a joint presentation in Baltimore the next morning. Peter was using the shower in my daughter’s bedroom when there were just the two of us in the house. Neither of us, with three PhDs between us, could figure out how to turn the shower on. We gave up and he finally showered in my bedroom shower.
His spirit will definitely live on in our memories and our lives. Many thanks, Peter for uplifting all the lives you touched so deeply.