Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Lévy market

We study an integro-differential parabolic problem modeling a process with jumps arising in financial mathematics. Under suitable conditions, we prove the existence of weak solutions to a more general integro-differential equation by using the Schaefer’s fixed point theorem and generalize the result to unbounded domains.     

Viscosity solutions and the pricing of European-style options in a Markov-modulated exponential Lévy model

ABSTRACT

In this paper, we investigate the pricing problem of a European-style contingent claim under a Markov-modulated exponential Lévy model. One of the main feature of this model is the modulator factor which takes into account the empirical facts observed in asset prices dynamics such as the long-term (stochastic) variability and time inhomogeneities. Using the viscosity solutions framework, we show that the value of a European-style option is the unique viscosity solution of a system of coupled linear Partial Integro-Differential Equations when the payoff function satisfies a Lipschitz condition. Moreover, we propose a numerical scheme for approximating solution of this system and discuss its stability, consistency and convergence.     

Representations for conditional expectations and applications to pricing and hedging of financial products in Lévy and jump-diffusion setting

In this article, we derive expressions for conditional expectations in terms of regular expectations without conditioning but involving some weights. For this purpose, we apply two approaches: the conditional density method and the Malliavin method. We use these expressions for the numerical estimation of the price of American options and their deltas in a Lévy and jump-diffusion setting. Several examples of applications to financial and energy markets are given including numerical examples.     

Approximate indifference pricing in exponential Lévy models

Financial markets based on Lévy processes are typically incomplete and option prices depend on risk attitudes of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the non-linear partial integro-differential equation associated to the indifference price. In this work, we develop closed-form approximations to exponential utility indifference prices in exponential Lévy models. To this end, we first establish a new non-asymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity. Next, we use this formula to derive a closed-form approximation of the indifference price by treating the Lévy model as a perturbation of the Black–Scholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of Lévy processes (?erný et al. 2013) to non-linear and non-smooth functionals. Our formula represents the indifference price as the linear combination of the Black–Scholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying Lévy process, and the derivatives of the Black–Scholes price. As a by-product, we obtain a simple approximation for the spread between the buyer’s and the seller’s indifference price. This formula allows to quantify, in a model-independent fashion, how sensitive a given product is to jump risk when jump size is small.     

Solutions to Integro-differential Problems Arising on Pricing Options in a Lévy Market

We study an integro-differential parabolic problem arising in Financial Mathematics. Under suitable conditions, we prove the existence of solutions for a multi-asset case in a general domain using the method of upper and lower solutions and a diagonal argument. We also model the jump in the related integro differential equation and give a solution procedure for that model assuming that the brownian motions are not correlated. For a bounded domain, this model for the jump gives an elegant expression of the solution in terms of hyper-spherical harmonics.     

Variational Solutions of the Pricing PIDEs for European Options in Lévy Models

Abstract

One of the fundamental problems in financial mathematics is to develop efficient algorithms for pricing options in advanced models such as those driven by Lévy processes. Essentially there are three approaches in use. These are Monte Carlo, Fourier transform and partial integro-differential equation (PIDE)-based methods. We focus our attention here on the latter. There is a large arsenal of numerical methods for efficiently solving parabolic equations that arise in this context. Especially Galerkin and Galerkin-inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

The contribution of this paper is therefore to analyse weak solutions of the Kolmogorov backward equations which are related to prices of European options in (time-inhomogeneous) Lévy models and to establish a precise link between the prices and the weak solutions of these equations. The resulting relation is a Feynman–Kac representation of the solution as a conditional expectation. Our special concern is to provide a framework that is able to cover both, the common types of European options and a wide range of advanced models in which these derivatives are priced.

An application to financial models requires in particular to admit pure jump processes such as generalized hyperbolic processes as well as unbounded domains of the equation. In order to deal at the same time with the typical pay-offs that can arise, the weak formulation of the equation is based on exponentially weighted Sobolev–Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.     

Swing options in commodity markets: a multidimensional Lévy diffusion model

We study valuation of swing options on commodity markets when the commodity prices are driven by multiple factors. The factors are modeled as diffusion processes driven by a multidimensional Lévy process. We set up a valuation model in terms of a dynamic programming problem where the option can be exercised continuously in time. Here, the number of swing rights is given by a total volume constraint. We analyze some general properties of the model and study the solution by analyzing the associated HJB-equation. Furthermore, we discuss the issues caused by the multi-dimensionality of the commodity price model. The results are illustrated numerically with three explicit examples.     

An optimal investment strategy for a stream of liabilities generated by a step process in a financial market driven by a Lévy process

In this paper we investigate an asset–liability management problem for a stream of liabilities written on liquid traded assets and non-traded sources of risk. We assume that the financial market consists of a risk-free asset and a risky asset which follows a geometric Lévy process. The non-tradeable factor (insurance risk or default risk) is driven by a step process with a stochastic intensity. Our framework allows us to consider financial risk, systematic and unsystematic insurance loss risk (including longevity risk), together with possible dependencies between them. An optimal investment strategy is derived by solving a quadratic optimization problem with a terminal objective and a running cost penalizing deviations of the insurer’s wealth from a specified profit-solvency target. Techniques of backward stochastic differential equations and the weak property of predictable representation are applied to obtain the optimal asset allocation.     

Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let $\{Y^{(a)}_{n}:n\ge1\}In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let {Y^(a)_n:n 3 1}\{Y^{(a)}_{n}:n\ge1\} be a sequence of independent and identically distributed random variables and {X^(a)_t:t 3 0}\{X^{(a)}_{t}:t\ge0\} be a Lévy process such that X₁^(a)=^dY₁^(a)X_{1}^{(a)}\stackrel{d}{=}Y_{1}^{(a)}, \mathbbEX₁^(a) < 0\mathbb{E}X_{1}^{(a)}<0 and \mathbbEX₁^(a)-0\mathbb{E}X_{1}^{(a)}\uparrow0 as a↓0. Let S^(a)_n=?_k=1ⁿ Y^(a)_kS^{(a)}_{n}=\sum _{k=1}^{n} Y^{(a)}_{k}. Then, under some mild assumptions, , for some random variable and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.     

Boundary-value problems for a nonlinear parabolic equation with Lévy Laplacian resolved with respect to the derivative

We present the solutions of boundary-value and initial boundary-value problems for a nonlinear parabolic equation with Lévy Laplacian ∆ _L resolved with respect to the derivative

Pricing American currency options in an exponential Lévy model

In this article the problem of the American option valuation in a Lévy process setting is analysed. The perpetual case is first considered. Without possible discontinuities (i.e. with negative jumps in the call case), known results concerning the currency option value as well as the exercise boundary are obtained with a martingale approach. With possible discontinuities of the underlying process at the exercise boundary (i.e. with positive jumps in the call case), original results are derived by relying on first passage time and overshoot associated with a Lévy process. For finite life American currency calls, the formula derived by Bates or Zhang, in the context of a negative jump size, is tested. It is basically an extension of the one developed by Mac Millan and extended by Barone‐Adesi and Whaley. It is shown that Bates' model generates pretty good results only when the process is continuous at the exercise boundary.     

Properties of positive solutions to a nonlinear parabolic problem

This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vazquez and the comparison principle, we deduce that the blow-up occurs only on the boundary (?)Ω. In addition, for a bounded Lipschitz domainΩ, we establish the blow-up rate estimates for the positive solution to this problem with a= 0.     

The riquier problem for a nonlinear equation unresolved with respect to the Lévy iterated laplacian

We present a method of solving for the nonlinear equationf(U(x),Δ _L ² U(x)) = Δ _L U(x) (Δ _L is an infinite-dimensional Laplacian) unresolved with respect to an iterated infinite-dimensional Laplacian and for the Riquier problem for this equation. Ukrainian NII MOD, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 423–427, March, 1999.     

Lévy processes with values in a space of continuous functions arising as limits of independent triangular arrays

In this paper we investigate the limit distribution of the functions of independent triangular arrays X_nj, 1≤j≤k(n), n≥1. According to LeCam's theorem, if f belongs to the class of functionsPD[0,2] (which is slightly weaker than the assumptions that f(0)=0, and f has the second derivative at zero), then the distribution of is shift convergent. He also gives the explicit form of the characteristic function of the limit infinitely divisible distribution. We consider the class of functionsPD[0,1] and prove a similar statement. Since in the definition of the sequence of centering constants the truncation points depend only on the value of X_nj and not on the function f, this makes the analysis of the joint distribution of random variables in the above form considerably easier. Also we analyze the process of partial sums , 0≤u≤1. where f(x,t) is a parametric family of functions depending continuously on the parameter t. In the case of power functions we give an explicit representation of the limit process in term of Poissonian integrals. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II. Eger. Hungary. 1994.     

Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs

We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose that transactions between the assets incur a transaction cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps intertemporal preferences. This portfolio optimisation problem is formulated as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity solutions. The associated dynamic programming equation is a second order degenerate elliptic integro-differential variational inequality subject to a state constraint boundary condition. The main result is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation. Emphasis is put on providing a framework that allows for a general class of Lévy processes. Owing to the complexity of our investment-consumption model, it is not possible to derive closed form solutions for the value function. Hence, the optimal policies cannot be obtained in closed form from the first order conditions for the dynamic programming equation. Therefore, we have to resort to numerical methods for computing the value function as well as the associated optimal policies. In view of the viscosity solution theory, the analysis found in this paper will ensure the convergence of a large class of numerical methods for the investment-consumption model in question.     

Stochastic control of SDEs associated with Lévy generators and application to financial optimization

This paper is concerned with the optimal control of jump type stochastic differential equations associated with (general) Lévy generators. The maximum principle is formulated for the solutions of the equations, which is inspired by N. C. Framstad, B. Øksendal and A. Sulem [J. Optim. Theory Appl., 2004, 121: 77–98] (and a continuation, J. Bennett and J. -L. Wu [Front. Math. China, 2007, 2(4): 539–558]). The result is then applied to optimization problems in financial models driven by Lévy-type processes.     

The optimal stopping problem concerned with ultimate maximum of a Lévy process

For a Lévy process X = (X _t )₀≤t<∞ we consider the time θ = inf{t ≥ 0: sup _s≤t X _s = sup _s≥0 X _s }. We study an optimal approximation of the time θ using the information available at the current instant. A Lévy process being a combination of a Brownian motion with a drift and a Poisson process is considered as an example.     

A note on optimal investment–consumption–insurance in a Lévy market

In Shen and Wei (2014) an optimal investment, consumption and life insurance purchase problem for a wage earner with Brownian information has been investigated. This paper discusses the same problem but extend their results to a geometric Itô–Lévy jump process. Our modelling framework is very general as it allows random parameters which are unbounded and involves some jumps. It also covers parameters which are both Markovian and non-Markovian functionals. Unlike in Shen and Wei (2014) who considered a diffusion framework, ours solves the problem using a novel approach, which combines the Hamilton–Jacobi–Bellman (HJB) and a backward stochastic differential equation (BSDE) in a Lévy market setup. We illustrate our results by two examples.