90 research outputs found
Execution and block trade pricing with optimal constant rate of participation
When executing their orders, investors are proposed different strategies by
brokers and investment banks. Most orders are executed using VWAP algorithms.
Other basic execution strategies include POV (also called PVol) -- for
percentage of volume --, IS -- implementation shortfall -- or Target Close. In
this article dedicated to POV strategies, we develop a liquidation model in
which a trader is constrained to liquidate a portfolio with a constant
participation rate to the market. Considering the functional forms commonly
used by practitioners for market impact functions, we obtain a closed-form
expression for the optimal participation rate. Also, we develop a microfounded
risk-liquidity premium that permits to better assess the costs and risks of
execution processes and to give a price to a large block of shares. We also
provide a thorough comparison between IS strategies and POV strategies in terms
of risk-liquidity premium
Optimal market making
Market makers provide liquidity to other market participants: they propose
prices at which they stand ready to buy and sell a wide variety of assets. They
face a complex optimization problem with both static and dynamic components.
They need indeed to propose bid and offer/ask prices in an optimal way for
making money out of the difference between these two prices (their bid-ask
spread). Since they seldom buy and sell simultaneously, and therefore hold long
and/or short inventories, they also need to mitigate the risk associated with
price changes, and subsequently skew their quotes dynamically. In this paper,
(i) we propose a general modeling framework which generalizes (and reconciles)
the various modeling approaches proposed in the literature since the
publication of the seminal paper "High-frequency trading in a limit order book"
by Avellaneda and Stoikov, (ii) we prove new general results on the existence
and the characterization of optimal market making strategies, (iii) we obtain
new closed-form approximations for the optimal quotes, (iv) we extend the
modeling framework to the case of multi-asset market making and we obtain
general closed-form approximations for the optimal quotes of a multi-asset
market maker, and (v) we show how the model can be used in practice in the
specific (and original) case of two credit indices
A reference case for mean field games models.
Nous présentons un exemple archétypal de jeu à champ moyen. Cet exemple est important à deux égards. Tout d'abord, il est suffisamment simple pour permettre l'obtention de solutions explicites : les fonctions de Bellman sont quadratiques, les mesures stationnaires gaussiennes et l'étude de la stabilité peut se faire explicitement en utilisant les polynômes d'Hermite. Aussi, et malgré la simplicité du problème, l'exemple que nous présentons est suffisamment riche pour être transposé mutatis mutandis à d'autres domaines d'application plus complexes.In this article, we present a reference case of mean field games. This case can be seen as a reference for two main reasons. First, the case is simple enough to allow for explicit resolution: Bellman functions are quadratic, stationary measures are normal and stability can be dealt with explicitly using Hermite polynomials. Second, in spite of its simplicity, the case is rich enough in terms of mathematics to be generalized and to inspire the study of more complex models that may not be as tractable as this one.Partial differential equations; Mean field games; Control theory; Numerical methods;
Mean field games equations with quadratic Hamiltonian: a specific approach
Mean field games models describing the limit of a large class of stochastic
differential games, as the number of players goes to , have been
introduced by J.-M. Lasry and P.-L. Lions. We use a change of variables to
transform the mean field games (MFG) equations into a system of simpler coupled
partial differential equations, in the case of a quadratic Hamiltonian. This
system is then used to exhibit a monotonic scheme to build solutions of the MFG
equations. Effective numerical methods based on this constructive scheme are
presented and numerical experiments are carried out.Comment: Submitted in June 201
- …