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Patrick Cheridito

Publications and source records attributed to Patrick Cheridito.

46 records · Page 3Linked to original sources
arXiv2013

BSDEs with terminal conditions that have bounded Malliavin derivative

We show existence and uniqueness of solutions to BSDEs of the form $$ Y_t = ξ+ \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$$ in the case where the terminal condition $ξ$ has bounded Malliavin derivative. The driver $f(s,y,z)$ is assumed to be Lipschitz continuous in $y$ but only locally Lipschitz continuous in $z$. In particular, it can grow arbitrarily fast in $z$. If in addition to having bounded Malliavin derivative, $ξ$ is bounded, the driver needs only be locally Lipschitz continuous in $y$. In the special case where the BSDE is Markovian, we obtain existence and uniqueness results for semilinear parabolic PDEs with non-Lipschitz nonlinearities. We discuss the case where there is no lateral boundary as well as lateral boundary conditions of Dirichlet and Neumann type.

math.PR↗
arXiv2013

BSΔEs and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$Δ$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$Δ$Es and BSDEs are governed by drivers $f^N(t,ω,y,z)$ and $f(t,ω,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BS$Δ$Es are based on d-dimensional random walks $W^N$ approximating the d-dimensional Brownian motion W underlying the BSDE and that $f^N$ converges to f. Conditions are given under which for any bounded terminal condition $ξ$ for the BSDE, there exist bounded terminal conditions $ξ^N$ for the sequence of BS$Δ$Es converging to $ξ$, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when $f^N$ and f are convex in z. We show that in this situation, the solutions of the BS$Δ$Es converge to the solution of the BSDE for every uniformly bounded sequence $ξ^N$ converging to $ξ$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^N,ξ^N)$ is close to $(W,ξ)$ in distribution, then the solution of the Nth BS$Δ$E is close to the solution of the BSDE in distribution too.

math.PR↗
arXiv2011

Pricing and Hedging in Affine Models with Possibility of Default

We propose a general framework for the simultaneous modeling of equity, government bonds, corporate bonds and derivatives. Uncertainty is generated by a general affine Markov process. The setting allows for stochastic volatility, jumps, the possibility of default and correlation between different assets. We show how to calculate discounted complex moments by solving a coupled system of generalized Riccati equations. This yields an efficient method to compute prices of power payoffs. European calls and puts as well as binaries and asset-or-nothing options can be priced with the fast Fourier transform methods of Carr and Madan (1999) and Lee (2005). Other European payoffs can be approximated with a linear combination of government bonds, power payoffs and vanilla options. We show the results to be superior to using only government bonds and power payoffs or government bonds and vanilla options. We also give conditions for European continent claims in our framework to be replicable if enough financial instruments are liquidly tradable and study dynamic hedging strategies. As an example we discuss a Heston-type stochastic volatility model with possibility of default and stochastic interest rates.

q-fin.PR↗
arXiv2011

Existence, minimality and approximation of solutions to BSDEs with convex drivers

We study the existence of solutions to backward stochastic differential equations with drivers f(t,W,y,z) that are convex in z. We assume f to be Lipschitz in y and W but do not make growth assumptions with respect to z. We first show the existence of a unique solution (Y,Z) with bounded Z if the terminal condition is Lipschitz in W and that it can be approximated by the solutions to properly discretized equations. If the terminal condition is bounded and uniformly continuous in W, we show the existence of a minimal continuous supersolution by uniformly approximating the terminal condition with Lipschitz terminal conditions. Finally, we prove existence of a minimal RCLL supersolution for bounded lower semicontinuous terminal conditions by approximating the terminal condition pointwise from below with Lipschitz terminal conditions.

math.PR↗
arXiv2010

Optimal consumption and investment in incomplete markets with general constraints

We study an optimal consumption and investment problem in a possibly incomplete market with general, not necessarily convex, stochastic constraints. We give explicit solutions for investors with exponential, logarithmic and power utility. Our approach is based on martingale methods which rely on recent results on the existence and uniqueness of solutions to BSDEs with drivers of quadratic growth.

q-fin.PM↗
arXiv2009

Processes of class Sigma, last passage times and drawdowns

We propose a general framework to study last passage times, suprema and drawdowns of a large class of stochastic processes. A central role in our approach is played by processes of class Sigma. After investigating convergence properties and a family of transformations that leave processes of class Sigma invariant, we provide three general representation results. The first one allows to recover a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process hit a certain level or was equal to its running maximum. It also leads to a formula recently discovered by Madan, Roynette and Yor expressing put option prices in terms of last passage times. Our second representation result is a stochastic integral representation of certain functionals of processes of class Sigma, and the third one gives a formula for their conditional expectations. From the latter one can deduce the laws of a variety of interesting random variables such as running maxima, drawdowns and maximum drawdowns of suitably stopped processes. As an application we discuss the pricing and hedging of options that depend on the running maximum of an underlying price process and are triggered when the underlying price drops to a given level or alternatively, when the drawdown or relative drawdown of the underlying price attains a given height.

math.PR↗
arXiv2006

Small time path behavior of double stochastic integrals and applications to stochastic control

We study the small time path behavior of double stochastic integrals of the form $\int_0^t(\int_0^rb(u) dW(u))^T dW(r)$, where $W$ is a $d$-dimensional Brownian motion and $b$ is an integrable progressively measurable stochastic process taking values in the set of $d\times d$-matrices. We prove a law of the iterated logarithm that holds for all bounded progressively measurable $b$ and give additional results under continuity assumptions on $b$. As an application, we discuss a stochastic control problem that arises in the study of the super-replication of a contingent claim under gamma constraints.

math.PR↗
arXiv2004

Dynamic monetary risk measures for bounded discrete-time processes

We study time-consistency questions for processes of monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a process of monetary risk measures time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time, and we show how this property manifests itself in the corresponding process of acceptance sets. For processes of coherent and convex monetary risk measures admitting a robust representation with sigma-additive linear functionals, we give necessary and sufficient conditions for time-consistency in terms of the representing functionals.

math.PR↗