High Frequency Asymptotics for the Limit Order Book
Peter Lakner (New York University, USA)
Joint work with Joshua Reed and Sasha Stoikov

Wednesday June 4, 16:30-17:00 | session 6.8 | Stochastic Analysis | room 1+2

We study the one-sided limit order book for sell (or buy) orders and model it as a measure-valued process. Limit sell (or buy) orders are offers to sell (or buy) an equity at a price determined by the seller (or buyer). Market buy (or sell) orders are orders to buy (or sell) an equity at the best, that is, least expensive (most expensive, in case of sell market orders) price offered by previous limit sell (or buy) orders. Limit orders arrive to the book according to a Poisson process and are placed on the book according to a distribution which varies depending on the current best price. Market orders to buy (or sell) periodically arrive to the book according to a second, independent Poisson process and remove from the book the order corresponding to the current best price. We consider the above described order book in a high frequency regime in which the rate of incoming limit and market orders is large and traders place their limit sell orders close to the current best price. Our first set of results provide weak limits for the price process and the properly scaled measure-valued order book process in the high frequency regime. In particular, we characterize the limiting measure-valued order book process as the solution to a measure-valued stochastic differential equation. We then provide an analysis of both the transient and long-run behavior of the limiting order book process.

Markov Chain BSDEs and risk averse networks
Samuel Cohen (University of Oxford, UK)

Wednesday June 4, 17:00-17:30 | session 6.8 | Stochastic Analysis | room 1+2

When studying a financial network, one is often interested in the importance of a particular node. This can be measured in various ways, for example, by the ergodic probabilities of an associated Markov chain. We consider ergodic BSDEs based on countable state Markov chains, and use these to derive nonlinear, risk-averse versions of these probabilities and similar quantities. With this machinery, one can also consider various problems in ergodic stochastic control, and can incorporate model or statistical uncertainties into the assessment of the importance of different nodes and groups of nodes. We apply these techniques to interbank liability networks, and see that the stability of a bank with respect to the liability network can be directly calculated.

Market capitalizations, Poisson-Dirichlet subordinators and Fractional Calculus
Sergey Sosnovskiy (Frankfurt School of Finance and Management, Germany)

Wednesday June 4, 17:30-18:00 | session 6.8 | Stochastic Analysis | room 1+2

Power law appears in finance in distributions of ranked data such as stock market capitalizations, volatility clusters, market returns, volumes, etc. In particular, ranked market shares, known as capital distribution curves are known to possess certain shape, which is stable over periods of time. Despite different natures of these phenomena it is possible to look at them as random division of a constrained resource under some symmetric measure.
In the first part of this paper we propose to model distribution of ranked market capitalizations by means of the two-parameter Poisson-Dirichlet measure, since it provides greater flexibility over Dirichlet and stable distributions. Definition of the Poisson-Dirichlet process by subordination according to propositions 14 and 21 in Pitman, Yor [1997] generalizes Kingman's ranked jumps analysis of random measures.
Second part of this paper is devoted to application of fractional operator calculus for analytic representation of subordinators. Fractional derivative operator is related to generalized Laplace transform and allows simplified treatment of special functions expressed in power series, such as Bessel and Mittag-Leffler functions, as well as divergent series. Particularly, we show that cumulative density functions of stable and tempered stable subordinators admit simple closed form expressions in terms of fractional derivative operator applied to unity. Consequently, Levy measure appears naturally as probability distribution of jumps over infinitesimal time increments and also has fractional calculus representation. This also allows to obtain Levy measure of composition of two subordinators.
Stochastic volatility can be also considered as random division of some finite activity in limited period of time, where non-normalized two-parameter Poisson-Dirichlet process provides subordinator, which sits between gamma subordinator (VG model) and tempered stable one (CGMY model). The model can be generalized by consideration of alternative subordinators and by application to other types of financial data exhibiting power laws.