Convergence of approximation schemes for fully nonlinear second order equations

We propose a framework to study optimal trading policies in a one-tick pro-rata limit order book, as typically arises in short-term interest rate futures contracts. The high-frequency trader has the choice to trade via market orders or limit orders, which are represented respectively by impulse controls and regular controls. We model and discuss the consequences of the two main features of this particular microstructure: first, the limit orders sent by the high frequency trader are only partially executed, and therefore she has no control on the executed quantity. For this purpose, cumulative executed volumes are modelled by compound Poisson processes. Second, the high frequency trader faces the overtrading risk, which is the risk of brutal variations in her inventory. The consequences of this risk are investigated in the context of optimal liquidation.The optimal trading problem is studied by stochastic control and dynamic programming methods, which lead to a characterization of the value function in terms of an integro quasi-variational inequality. We then provide the associated numerical resolution procedure, and convergence of this computational scheme is proved. Next, we examine several situations where we can on one hand simplify the numerical procedure by reducing the number of state variables, and on the other hand focus on specific cases of practical interest. We examine both a market making problem and a best execution problem in the case where the mid-price process is a martingale. We also detail a high frequency trading strategy in the case where a (predictive) directional information on the mid-price is available. Each of the resulting strategies are illustrated by numerical tests.Keywords: Market making, limit order book, pro-rata microstructure, inventory risk, marked point process, stochastic control. 1 IntroductionIn most of modern public security markets, the price formation process, or price discovery, results from competition between several market agents that take part in a public auction. In particular, day trading sessions, which are also called continuous trading phases, consist of continuous double auctions. In these situations, liquidity providers 1 continuously set bid and ask prices for the considered security, and the marketplace publicly displays a (possibly partial) information about these bid and ask prices, along with transactions prices. The action of continuously providing bid and ask quotes during day trading sessions is called market making, and this role was tradionnally performed by specialist firms. However, due to the recent increased availability of electronic trading technologies, as well as regulatory changes, a large range of investors are now able to implement such market making strategies. These strategies are part of the broader category of high frequency trading (HFT) strategies, which are characterized by the fact that they facilitate a larger number of orders being sent to the market per unit of time. HFT takes place in the continuous trading phase, and ...

Section: Convergence Of the Numerical Schemementioning

confidence: 98%

Optimal High‐frequency Trading in a Pro Rata Microstructure With Predictive Information

Guilbaud

Pham

2013

“…Following Barles and Souganidis (1990) methodology, we know that there exists a sequence ( n , t n , q n ) n such that r n → 0, (t n , q n ) → (t * , q * ), r θ c n (t n , q n ) → θ (t * , q * ), r θ c n − φ has a global maximum at (t n , q n ). Now, because of the definition of θ c n , if (t * , q * ) ∈ [0, T) × (0, q 0 ], we can always suppose that q n ≥ n and t n = T and then by definition of θ n :…”

Section: Lemma 53 H Converges Locally Uniformly Toward H Whenmentioning

confidence: 99%

General Intensity Shapes in Optimal Liquidation

Guéant

Lehalle

2013

The classical literature on optimal liquidation, rooted in Almgren–Chriss models, tackles the optimal liquidation problem using a trade‐off between market impact and price risk. It answers the general question of optimal scheduling but the very question of the actual way to proceed with liquidation is rarely dealt with. Our model, which incorporates both price risk and nonexecution risk, is an attempt to tackle this question using limit orders. The very general framework we propose to model liquidation with limit orders generalizes existing ones in two ways. We consider a risk‐averse agent, whereas the model of Bayraktar and Ludkovski only tackles the case of a risk‐neutral one. We consider very general functional forms for the execution process intensity, whereas Guéant, Lehalle and Fernandez‐Tapia are restricted to exponential intensity. Eventually, we link the execution cost function of Almgren–Chriss models to the intensity function in our model, providing then a way to see Almgren–Chriss models as a limit of ours.

“…Let us first recall the definition of viscosity solutions. Consider a nonlinear elliptic second-order partial differential equation of the form Ishii (1985), Barles and Souganidis (1991), and Barles (1994), we define the notion of viscosity solutions as follows.…”

Section: Viscosity Solutionsmentioning

confidence: 99%

“…remark 6.3. In Barles and Souganidis (1991) and Barles (1994), the set D is supposed to be either an open set or the closure of an open set. Since the notion of viscosity solution is a "local" property, conditions on D need only to be local.…”

Section: Viscosity Solutionsmentioning

confidence: 99%

“…The function U h is defined on h and interpolated linearly with respect to each coordinate in [0, 1] n . If the discrete operators C h a satisfy the discrete maximum principle (DMP), then (9.19) is monotone in the sense of Barles and Souganidis (1991). Moreover (9.19) can then be interpreted as a Dynamic Programming Equation for an ergodic discrete control problem.…”

Section: Numerical Solution Of the Variational Inequalitymentioning

confidence: 99%

See 1 more Smart Citation

Dynamic Optimization of Long‐Term Growth Rate for a Portfolio with Transaction Costs and Logarithmic Utility

Akian

Sulem

Taksar

2001

We study the optimal investment policy for an investor who has available one bank account and n risky assets modeled by log-normal diffusions. The objective is to maximize the long-run average growth of wealth for a logarithmic utility function in the presence of proportional transaction costs. This problem is formulated as an ergodic singular stochastic control problem and interpreted as the limit of a discounted control problem for vanishing discount factor. The variational inequalities for the discounted control problem and the limiting ergodic problem are established in the viscosity sense. The ergodic variational inequality is solved by using a numerical algorithm based on policy iterations and multigrid methods. A numerical example is displayed for two risky assets.