arXiv subjects

Todor Bilarev

Publications and source records attributed to Todor Bilarev.

6 recordsLinked to original sources

arXiv2023

Hedging with physical or cash settlement under transient multiplicative price impact

We solve the superhedging problem for European options in an illiquid extension of the Black-Scholes model, in which transactions have transient price impact and the costs and the strategies for hedging are affected by physical or cash settlement requirements at maturity. Our analysis is based on a convenient choice of reduced effective coordinates of magnitudes at liquidation for geometric dynamic programming. The price impact is transient over time and multiplicative, ensuring non-negativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black-Scholes model with relative price impact being proportional to the volume of shares traded, where the transience for impact on log-prices is being modelled like in Obizhaeva-Wang \cite{ObizhaevaWang13} for nominal prices. More generally, we allow for non-linear price impact and resilience functions. The viscosity solutions describing the minimal superhedging price are governed by the transient character of the price impact and by the physical or cash settlement specifications. Pricing equations under illiquidity extend no-arbitrage pricing a la Black-Scholes for complete markets in a non-paradoxical way (cf.\ {Ç}etin, Soner and Touzi \cite{CetinSonerTouzi10}) even without additional frictions, and can recover it in base cases.

q-fin.PR↗

arXiv2018

Approximating diffusion reflections at elastic boundaries

We show a probabilistic functional limit result for one-dimensional diffusion processes that are reflected at an elastic boundary which is a function of the reflection local time. Such processes are constructed as limits of a sequence of diffusions which are discretely reflected by small jumps at an elastic boundary, with reflection local times being approximated by $\varepsilon$-step processes. The construction yields the Laplace transform of the inverse local time for reflection. Processes and approximations of this type play a role in finite fuel problems of singular stochastic control.

math.PR↗

arXiv2018

Stability for gains from large investors' strategies in M1/J1 topologies

We prove continuity of a controlled SDE solution in Skorokhod's $M_1$ and $J_1$ topologies and also uniformly, in probability, as a non-linear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that $M_1$-continuity is the key to ensure that proceeds and wealth processes from (self-financing) càdlàg trading strategies are determined as the continuous extensions for those from continuous strategies. We demonstrate by examples how continuity properties are useful to solve different stochastic control problems on optimal liquidation and to identify asymptotically realizable proceeds.

q-fin.MF↗

arXiv2017

Optimal Liquidation under Stochastic Liquidity

We solve explicitly a two-dimensional singular control problem of finite fuel type for infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determines the inter-temporal resilience of the market in spirit of Predoiu, Shaikhet and Shreve (2011), is taken to be stochastic, being driven by own random noise. The optimal control is obtained as the local time of a diffusion process reflected at a non-constant free boundary. To solve the HJB variational inequality and prove optimality, we need a combination of probabilistic arguments and calculus of variations methods, involving Laplace transforms of inverse local times for diffusions reflected at elastic boundaries.

math.PR↗

arXiv2017

Optimal Asset Liquidation with Multiplicative Transient Price Impact

We study a multiplicative transient price impact model for an illiquid financial market, where trading causes price impact which is multiplicative in relation to the current price, transient over time with finite rate of resilience, and non-linear in the order size. We construct explicit solutions for the optimal control and the value function of singular optimal control problems to maximize expected discounted proceeds from liquidating a given asset position. A free boundary problem, describing the optimal control, is solved for two variants of the problem where admissible controls are monotone or of bounded variation.

math.OC↗

arXiv2014

On the speed of convergence of Newton's method for complex polynomials

We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\mathcal{S}_d$ of $3.33d\log^2 d(1 + o(1))$ points with the following universal property: for every normalized polynomial of degree $d$ there are $d$ starting points in $\mathcal{S}_d$ whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least $1-2/d$ the number of iterations for these $d$ starting points to reach all roots with precision $\varepsilon$ is $O(d^2\log^4 d + d\log|\log \varepsilon|)$. This is an improvement of an earlier result in \cite{Schleicher}, where the number of iterations is shown to be $O(d^4\log^2 d + d^3\log^2d|\log \varepsilon|)$ in the worst case (allowing multiple roots) and $O(d^3\log^2 d(\log d + \log δ) + d\log|\log \varepsilon|)$ for well-separated (so-called $δ$-separated) roots. Our result is almost optimal for this kind of starting points in the sense that the number of iterations can never be smaller than $O(d^2)$ for fixed $\varepsilon$.

math.DS↗