What is Optimal Execution With Quadratic Variation Inventories?
Education
The first half of the paper is devoted to description and implementation of statistical tests arguing for the presence of a Brownian component in the inventories and wealth processes of individual traders.
Leal & Carmona use intra-day data from the Toronto Stock Exchange to provide empirical evidence of this claim.
In addition, work with regularly spaced time intervals, as well as with asynchronously observed data. Moreover, their tests reveal with high significance the presence of a non-zero Brownian motion component.
Brownian motion, also known as Brownian movement, is the random motion of microscopic particles suspended in a fluid or gas, caused by collisions with molecules of the surrounding medium. The phenomenon became named after the British botanist Robert Brown, who first observed it in 1827 while studying pollen grains suspended in water.
Brownian motion occurs due to the thermal energy of the medium, which causes the particles to move in a zigzag or erratic pattern. The motion is stochastic, meaning that it is random and unpredictable, and the direction and speed of the particle’s movement change continuously.
The mathematical description of Brownian motion became first formulated by the French mathematician Louis Bachelier in 1900.
And it has since become used in various fields! Including physics, chemistry, finance, and biology, to describe phenomena such as diffusion, particle transport, and stock price movements.
Furthermore, the second half of the paper becomes concerned with the analysis of trader behaviors throughout the day.
Leal & Caromona “extend the theoretical analysis of an existing optimal execution model to accommodate the presence of Itô inventory processes. And compare empirically the optimal behavior of traders in such fitted models, to their actual behavior as inferred from the data.”
High-frequency trading is a computerized practice based on algorithms that allow firms to trade stocks. And other financial instruments in microseconds, or even smaller fractions of a second.
Over the past two decades, the use of statistical and econometric methods to analyze high-frequency financial data has seen a huge growth.
Mathematical finance has long become involved in modeling price processes, which amounts to specifying their dynamics.
The key in this modeling is to take into account the randomness of the quantities of interest.
If randomness were not involved, Leal & Carmona; “would only need to specify a differential equation concerning the time evolution of the process.”
“Accounting for randomness in price processes has evolved. Moreover, from the use of standard Itô processes. To include jumps through the introduction of Poisson and Lévy processes. To general Markov processes and semi-martingales, and to processes which are neither Markov nor semi-martingales.
With the introduction of terms driven by fractional Brownian motions!
Meanwhile, the dynamics of the inventory and wealth processes of the trading agents has gone largely ignored.”
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Written by René Carmona and Laura Leal
Laura Leal
Masters student in Operations Research and Financial Engineering
Rio de Janeiro, Brazil
B.S., Brazilian School of Economics and Finance; M.S., Graduate School of Economics
“I am a Princeton engineer because I joyfully follow curiosity wherever she leads and make sure to follow up with execution.”
Laura Leal – Princeton University School of Engineering and Applied Science
René Carmona, Ph.D., is the Paul M. Wythes ’55 Professor of Engineering and Finance at Princeton University in the department of Operations Research and Financial Engineering.
Firstly, he is an associate member of the Department of Mathematics. In addition, a member of the Program in Applied and Computational Mathematics, and Director of Graduate Studies of the Bendheim Center for Finance where he oversees the Master in Finance program. In addition, he obtained a PhD in probability from Marseille university where he held his first academic job. After time spent at Cornell and a couple of stints at Princeton. He moved to the University of California at Irvine in 1981 and eventually Princeton University in 1995.
Dr Carmona is a Fellow of the Institute of Mathematical Statistics (IMS) since 1984, of the Society for Industrial and Applied Mathematics (SIAM) since 2009 and of the American Mathematical Society (AMS) since 2020. Furthermore, he is the founding chair of the SIAM Activity Group on Financial Mathematics and Engineering. A founding editor of the Electronic Journal & Communications in Probability, and the SIAM Journal on Financial Mathematics. He is on the editorial board of several peer-reviewed journals and book series. He was/is on the scientific board of several research institutes. More recently, the NSF Institute for Mathematical and Statistical Innovation (IMSI) in Chicago.
Moreover, his publications include over one hundred fifty articles and eleven books in probability, statistics, mathematical physics, signal analysis and financial mathematics.
He also developed computer programs for teaching and research. He has worked on the commodity and energy markets as well as the credit markets, and he is recognized as a leading researcher and consultant in these areas. Over the last decade his research focused on the development of a probabilistic approach to Mean Field Games and Mean Field Control. His two-volume book on the subject, co-authored with F. Delarue, was the recipient of the J.L. Doob Prize awarded every three years by the American Mathematical Society.
Lastly, in 2020 he was awarded a competitive ARPA-E grant under the Performance-based Energy Resource Feedback, Optimization and Risk Management (PERFORM) program, and together with colleagues from Princeton University, UC Santa Barbara, and Scoville Risk Partners, leads thewebpush research team Operational Risk Financialization of Electricity Under Stochasticity (ORFEUS).
