arXiv subjects
Publications and source records attributed to Yan Dolinsky.
We study exponential-utility maximization for high-frequency trading in a discretized fractional Brownian motion model. Using spectral methods for stationary Gaussian sequences, we derive the asymptotic growth rate of the optimal certainty equivalent. We also show that the suitably rescaled optimal positions converge in finite-dimensional distributions to a Gaussian white-noise-type field.
In this note, we derive a large-deviation-type scaling limit for a discretization of the Martingale Representation Theorem. Somewhat surprisingly, and to the best of our knowledge, this result has not been previously obtained in the literature.
We study scaled trinomial models converging to the Black--Scholes model, and analyze exponential certainty-equivalent prices for path-dependent European options. As the number of trading dates $n$ tends to infinity and the risk aversion is scaled as $nl$ for a fixed constant $l>0$, we derive a nontrivial scaling limit. Our analysis is purely probabilistic. Using a duality argument for the certainty equivalent, together with martingale and weak-convergence techniques, we show that the limiting problem takes the form of a volatility control problem with a specific penalty. For European options with Markovian payoffs, we analyze the optimal control problem and show that the corresponding delta-hedging strategy is asymptotically optimal for the primal problem.
We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the matrix being inverted is required to lie in the orthogonal complement of that subspace with respect to the trace inner product. Under a sharp nondegeneracy condition on the subspace, we show that every positive definite matrix admits a unique decomposition of this form. This decomposition admits a variational characterization as the unique minimizer of a strictly convex log-determinant optimization problem, which in turn yields a natural dual formulation that can be efficiently exploited computationally. We derive several properties, including the stability of the decomposition. We further develop feasibility-preserving Newton-type algorithms with provable convergence guarantees and analyze their per-iteration complexity in terms of algebraic properties of the decomposed matrix and the underlying subspace. Finally, we present two applications: one in mathematical finance, where the decomposition arises naturally in exponential utility maximization, and one in linear algebra, where our main theorem characterizes a new bilinear triangular decomposition for positive definite matrices, obtained together with a direct elimination algorithm.
In this work we study a continuous time exponential utility maximization problem in the presence of a linear temporary price impact. More precisely, for the case where the risky asset is given by the Ornstein-Uhlenbeck diffusion process we compute the optimal portfolio strategy and the corresponding value. Our method of solution relies on duality, and it is purely probabilistic.
We study an optimal execution problem in the infinite horizon setup. Our financial market is given by the Black-Scholes model with a linear price impact. The main novelty of the current note is that we study the constrained case where the number of shares and the selling rate are non-negative processes. For this case we give a complete characterization of the value and the optimal control via a solution of a non-linear ordinary differential equation (ODE). Furthermore, we provide an example where the non-linear ODE can be solved explicitly. Our approach is purely probabilistic.
In this note we consider the maximization of the expected terminal wealth for the setup of quadratic transaction costs. First, we provide a very simple probabilistic solution to the problem. Although the problem was largely studied, as far as we know up to date this simple and probabilistic form of the solution has not appeared in the literature. Next, we apply the general result for the numerical study of the case where the risky asset is given by a fractional Brownian Motion and the information flow of the investor can be diversified.
In this work we study the continuous time exponential utility maximization problem in the framework of an investor who is informed about the price changes with a delay. This leads to a non-Markovian stochastic control problem. In the case where the risky asset is given by a Gaussian process (with some additional properties) we establish a solution for the optimal control and the corresponding value. Our approach is purely probabilistic and is based on the theory for Radon-Nikodym derivatives of Gaussian measures developed by Shepp [6] and Hitsuda [5].
In this work we consider the exponential utility maximization problem in the framework of semistatic hedging.
The aim of this short note is to present a solution to the discrete time exponential utility maximization problem in a case where the underlying asset has a multivariate normal distribution. In addition to the usual setting considered in Mathematical Finance, we also consider an investor who is informed about the risky asset's price changes with a delay. Our method of solution is based on the theory developed in [4] and guessing the optimal portfolio.
We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock's price fluctuations. With linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be not only well posed, but it even allows for a closed-form solution. We describe this solution and the resulting problem value for this stochastic control problem with partial observation by solving its convex-analytic dual problem.
We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options in the case where the investor is required to liquidate her position. Our main result is establishing a non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. We compute the limit of the corresponding utility indifference prices and find explicitly a family of portfolios which are asymptotically optimal.
In this paper, we obtain a duality result for the exponential utility maximization problem where trading is subject to quadratic transaction costs and the investor is required to liquidate her position at the maturity date. As an application of the duality, we treat utility-based hedging in the Bachelier model. For European contingent claims with a quadratic payoff, we compute explicitly the optimal trading strategy.
We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options and we compute their non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. Moreover, we find explicitly a family of portfolios which are asymptotically optimal.
In this paper we study optimal investment when the investor can peek some time units into the future, but cannot fully take advantage of this knowledge because of quadratic transaction costs. In the Bachelier setting with exponential utility, we give an explicit solution to this control problem with intrinsically infinite-dimensional memory. This is made possible by solving the dual problem where we make use of the theory of Gaussian Volterra integral equations.
The aim of this short note is to establish a limit theorem for the optimal trading strategies in the setup of the utility maximization problem with proportional transaction costs. This limit theorem resolves the open question from [4]. The main idea of our proof is to establish a uniqueness result for the optimal strategy. The proof of the uniqueness is heavily based on the dual approach which was developed recently in [6,7,8].
We consider the discretized Bachelier model where hedging is done on an equidistant set of times. Exponential utility indifference prices are studied for path-dependent European options and we compute their non-trivial scaling limit for a large number of trading times $n$ and when risk aversion is scaled like $n\ell$ for some constant $\ell>0$. Our analysis is purely probabilistic. We first use a duality argument to transform the problem into an optimal drift control problem with a penalty term. We further use martingale techniques and strong invariance principles and get that the limiting problem takes the form of a volatility control problem.
We consider the Bachelier model with information delay where investment decisions can be based only on observations from $H>0$ time units before. Utility indifference prices are studied for vanilla options and we compute their non-trivial scaling limit for vanishing delay when risk aversion is scaled liked $A/H$ for some constant $A$. Using techniques from [7], we develop discrete-time duality for this setting and show how the relaxed form of martingale property introduced by [9] results in the scaling limit taking the form of a volatility control problem with quadratic penalty.