arXiv subjects
Publications and source records attributed to Mikhail Urusov.
In this paper we propose a novel dual regression-based approach for pricing American options. This approach reduces the complexity of the nested Monte Carlo method and has especially simple form for time discretised diffusion processes. We analyse the complexity of the proposed approach both in the case of fixed and increasing number of exercise dates. The method is illustrated by several numerical examples.
In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm ($\varepsilon^{-3}$ in the case of a first order scheme and $\varepsilon^{-2.5}$ in the case of a second order scheme) can be reduced down to $\varepsilon^{-2+δ}$ for any $δ\in [0,0.25)$ with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.
In this paper we present an enhancement of the regression-based variance reduction approaches recently proposed in Belomestny et al. This enhancement is based on a truncation of the control variate and allows for a significant reduction of the computing time, while the complexity stays of the same order. The performances of the proposed truncated algorithms are illustrated by a numerical example.
In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm ($\varepsilon^{-3}$) can be reduced down to $\varepsilon^{-2}\sqrt{\left|\log(\varepsilon)\right|}$ in case of the Euler scheme with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.
In this paper we suggest a modification of the regression-based variance reduction approach recently proposed in Belomestny et al. This modification is based on the stratification technique and allows for a further significant variance reduction. The performance of the proposed approach is illustrated by several numerical examples.
We consider the decreasing and the increasing $r$-excessive functions $φ_r$ and $ψ_r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints $α< β$. We prove that the $r$-excessive local martingale $\bigl( e^{-r (t \wedge T_α)} φ_r (X_{t \wedge T_α}) \bigr)$ $\bigl($resp., $\bigl( e^{-r (t \wedge T_β)} ψ_r (X_{t \wedge T_β}) \bigr) \bigr)$ is a strict local martingale if the boundary point $α$ (resp., $β$) is inaccessible and entrance, and a martingale otherwise.
Let $X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^N_{k+1} = Y^N_{k} + a_N(Y^N_k) X_{k+1}$, where $a_N: \mathbb R \to \mathbb R_+$. We show, under mild assumptions on the law of $X_i$, that one can choose the scale factor $a_N$ in such a way that the process $(Y^N_{\lfloor N t \rfloor})_{t \in \mathbb R_+}$ converges in distribution to a given diffusion $(M_t)_{t \in \mathbb R_+}$ solving a stochastic differential equation with possibly irregular coefficients, as $N \to \infty$. To this end we embed the scaled random walks into the diffusion $M$ with a sequence of stopping times with expected time step $1/N$.
The main result is a counterpart of the theorem of Monroe [\emph{Ann. Probability} \textbf{6} (1978) 42--56] for a geometric Brownian motion: A process is equivalent to a time change of a geometric Brownian motion if and only if it is a nonnegative supermartingale. We also provide a link between our main result and Monroe [\emph{Ann. Math. Statist.} \textbf{43} (1972) 1293--1311]. This is based on the concept of a \emph{minimal} stopping time, which is characterised in Monroe [\emph{Ann. Math. Statist.} \textbf{43} (1972) 1293--1311] and Cox and Hobson [\emph{Probab. Theory Related Fields} \textbf{135} (2006) 395--414] in the Brownian case. We finally suggest a sufficient condition for minimality (for the processes other than a Brownian motion) complementing the discussion in the aforementioned papers.
This paper studies the loss of the semimartingale property of the process $g(Y)$ at the time a one-dimensional diffusion $Y$ hits a level, where $g$ is a difference of two convex functions. We show that the process $g(Y)$ can fail to be a semimartingale in two ways only, which leads to a natural definition of non-semimartingales of the \textit{first} and \textit{second kind}. We give a deterministic if and only if condition (in terms of $g$ and the coefficients of $Y$) for $g(Y)$ to fall into one of the two classes of processes, which yields a characterisation for the loss of the semimartingale property. A number of applications of the results in the theory of stochastic processes and real analysis are given: e.g. we construct an adapted diffusion $Y$ on $[0,\infty)$ and a \emph{predictable} finite stopping time $ζ$, such that $Y$ is a semimartingale on the stochastic interval $[0,ζ)$, continuous at $ζ$ and constant after $ζ$, but is \emph{not} a semimartingale on $[0,\infty)$.
In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. In this article we consider a limit order book model that allows for time-dependent, deterministic depth and resilience of the book and determine optimal portfolio liquidation strategies. In a first model variant, we propose a trading dependent spread that increases when market orders are matched against the order book. In this model no price manipulation occurs and the optimal strategy is of the wait region - buy region type often encountered in singular control problems. In a second model, we assume that there is no spread in the order book. Under this assumption we find that price manipulation can occur, depending on the model parameters. Even in the absence of classical price manipulation there may be transaction triggered price manipulation. In specific cases, we can state the optimal strategy in closed form.
In this expository paper we describe the pathwise behaviour of the integral functional $\int_0^t f(Y_u)\,\dd u$ for any $t\in[0,ζ]$, where $ζ$ is (a possibly infinite) exit time of a one-dimensional diffusion process $Y$ from its state space, $f$ is a nonnegative Borel measurable function and the coefficients of the SDE solved by $Y$ are only required to satisfy weak local integrability conditions. Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where either the Williams theorem and the theory of Bessel processes or the first Ray-Knight theorem can be applied to prove the characterisation.
In this note we re-examine the analysis of the paper "On the martingale property of stochastic exponentials" by B. Wong and C.C. Heyde, Journal of Applied Probability, 41(3):654-664, 2004. Some counterexamples are presented and alternative formulations are discussed.
Modelling stock prices via jump processes is common in financial markets. In practice, to hedge a contingent claim one typically uses the so-called delta-hedging strategy. This strategy stems from the Black--Merton--Scholes model where it perfectly replicates contingent claims. From the theoretical viewpoint, there is no reason for this to hold in models with jumps. However in practice the delta-hedging strategy is widely used and its potential shortcoming in models with jumps is disregarded since such models are typically incomplete and hence most contingent claims are non-attainable. In this note we investigate a complete model with jumps where the delta-hedging strategy is well-defined for regular payoff functions and is uniquely determined via the risk-neutral measure. In this setting we give examples of (admissible) delta-hedging strategies with bounded discounted value processes, which nevertheless fail to replicate the respective bounded contingent claims. This demonstrates that the deficiency of the delta-hedging strategy in the presence of jumps is not due to the incompleteness of the model but is inherent in the discontinuity of the trajectories.
The stochastic exponential $Z_t=\exp\{M_t-M_0-(1/2) _t\}$ of a continuous local martingale $M$ is itself a continuous local martingale. We give a necessary and sufficient condition for the process $Z$ to be a true martingale in the case where $M_t=\int_0^t b(Y_u)\,dW_u$ and $Y$ is a one-dimensional diffusion driven by a Brownian motion $W$. Furthermore, we provide a necessary and sufficient condition for $Z$ to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function $b$ and the drift and diffusion coefficients of $Y$. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.
For a real Borel measurable function b, which satisfies certain integrability conditions, it is possible to define a stochastic integral of the process b(Y) with respect to a Brownian motion W, where Y is a diffusion driven by W. It is well know that the stochastic exponential of this stochastic integral is a local martingale. In this paper we consider the case of an arbitrary Borel measurable function b where it may not be possible to define the stochastic integral of b(Y) directly. However the notion of the stochastic exponential can be generalized. We define a non-negative process Z, called generalized stochastic exponential, which is not necessarily a local martingale. Our main result gives deterministic necessary and sufficient conditions for Z to be a local, true or uniformly integrable martingale.
We obtain a deterministic characterisation of the \emph{no free lunch with vanishing risk}, the \emph{no generalised arbitrage} and the \emph{no relative arbitrage} conditions in the one-dimensional diffusion setting and examine how these notions of no-arbitrage relate to each other.