Damir Kinzebulatov

Optimal Execution with a Price Limiter

Agents often wish to limit the price they pay for an asset. If they are acquiring a large number of shares, they must balance the risk of trading slowly (to limit price impact) with the risk of future uncertainty in prices. Here, we address the optimal acquisition problem for an agent who is unwilling to pay more than a specified price for an asset while they are subject to market impact and price uncertainty. The problem is posed as an optimal stochastic control and we provide an analytical closed form solution for the perpetual case as well as a dimensionally reduced PDE for the general case. The optimal speed of trading is found to no longer be deterministic and instead slows when maturity and the price barrier approaches. Moreover, we demonstrate that a price limiter constraint significantly reduces the conditional tail expectation of the securities costs.

Optimal accelerated share repurchases

An accelerated share repurchase (ASR) allows a firm to repurchase a significant portion of its shares immediately, while shifting the burden of reducing the impact and uncertainty in the trade to an intermediary. The intermediary must then purchase the shares from the market over several days, weeks, or as much as several months. Some contracts allow the intermediary to specify when the repurchase ends, at which point the corporation and the intermediary exchange the difference between the arrival price and the TWAP over the trading period plus a spread. Hence, the intermediary effectively has an American option embedded within an optimal execution problem. As a result, the firm receives a discounted spread relative to the no early exercise case. In this work, we address the intermediarys optimal execution and exit strategy taking into account the impact that trading has on the market. We demonstrate that it is optimal to exercise when the TWAP exceeds Ζ(t) St where St is the fundamental price of the asset and Ζ(t) is deterministic. Moreover, we develop a dimensional reduction of the stochastic control and stopping problem and implement an efficient numerical scheme to compute the optimal trading and exit strategies.

Dynamical generalized functions and the multiplication problem

1. In this paper, we consider the problem ofmultiplication of a generalized function by a discontinuous function. The need in such an operation appears in the study of ordinary differential equations with generalized functions [1–8]; it can be viewed as a particular case of a more general operation of multiplication of two generalized functions [9–11] the latter operation arises, in particular, in applications to the problems of quantum mechanics [12, 13]. As is known [14, 15], in the classical space D′, it is impossible to define a continuous operation of multiplication of generalized functions. It is also impossible to define a continuous partial operation of multiplication of a generalized function from D′ by a discontinuous function [14, 15]. A continuous operation of elements of D′ is defined in the algebra of Colombeau generalized functions [9], which contains D′ as a subspace. In the algebra of Colombeau generalized functions, there exists a product of any two elements of D′, although, in the general case, this product is a Colombeau generalized function and does not belong to D′ [9]. In particular, the product of the unit function θτ discontinuous at τ and the delta-function δτ does not belong to D′. This leads to a series of inconveniences in the study of differential equations with generalized functions containing the product of a generalized function and a discontinuous function. The main inconvenience is in the fact that the solution is not an ordinary discontinuous function but a Colombeau generalized function, which makes it difficult to give its physical interpretation [9, 16]. In this paper, we construct a space of dynamical generalized functions T ′ in which a continuous and associative operation of multiplication of a generalized function by a discontinuous function is defined. The definitions of the product of the unit function θτ and the classical delta-function δτ ∈ D′ given in [4, 11, 17, 18] (which, generally speaking, are neither associative, nor continuous) are particular cases of the multiplication in the space T ′.

Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds. II

We develop complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds. This, in particular, includes the results on holomorphic extension from complex submanifolds, corona type theorems, properties of divisors, holomorphic analogs of the Peter-Weyl approximation theorem, Hartogs type theorems, characterization of uniqueness sets. The model examples of these algebras are: (1) Bohrs algebra of holomorphic almost periodic functions on tube domains; (2) algebra of all fibrewise bounded holomorphic functions (e.g., arising in the corona problem for

Algorithmic Trading with Learning

H^\infty

On linear perturbations of the Ricker model

). Our approach is based on an extension of the classical Oka-Cartan theory to coherent-type sheaves on the maximal ideal spaces of these algebras -- topological spaces having some features of complex manifolds.