Pricing American Drawdown Options under Markov Models

Pricing American drawdown options under Markov models

The Linear Dynamic Complementarity Problem is a special case of the Extended Linear Complementarity Problem

We present a new difference of convex functions algorithm (DCA) for solving linear and nonlinear mixed complementarity problems (MCPs). The approach is based on the reformulation of bilinear complementarity constraints as difference of convex (DC) functions, more specifically, the difference of scalar, convex quadratic terms. This reformulation gives rise to a DC program, which is solved via sequential linear approximations of the concave term using standard DCA techniques. The reformulation is based on a generalization of earlier results on recasting bilinearities and leads to a novel algorithmic framework for MCPs. Through extensive numerical experimentation, the proposed approach, referred to as DCA-BL, proves to be an efficient heuristic for complementarity problems. For linear complementarity problems (LCPs), we test the approach on a number of randomly generated instances by varying the size, the density, and the eigenvalue distribution of the LCP matrix, providing insights into the numerical properties of DCA-BL. In addition, we apply the framework to a market equilibrium problem and find that DCA-BL scales well on realistic LCP instances. Also, through experimentation, we find that that DCA-BL performs particularly well compared with other DC-based complementarity approaches in the literature if the LCP is highly dense, asymmetric, or indefinite. Lastly, the method is successfully applied to a set of equilibrium problems with second-order cone constraints, which give rise to nonlinear complementarity problems, with applications to stochastic equilibrium problems in water infrastructure and finance. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2113891], the Division of Electrical, Communications and Cyber Systems [Grant 2114100], the Office of Naval Research Global [Grant 00014-22-1-2649], Petrobras [Grant 4324713], the Division of Mathematical Sciences [Grant 2318519], and the Danmarks Frie Forskningsfond [Grant 0217-00009B]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0822 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0822 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The contribution first describes the implementation of Continuous-Time Markov Chains (CTMC) in the language Modelica; this enables the simulation by powerful solvers making typical Functional Safety applications numerically more efficient and reliable than with classically used Discrete-Time Markov Chains (DTMC). Secondly, the CTMC are extended to Multi-Phase Markov Chains (CTMC MP ) enabling the inclusion of maintenance (restoration) actions as well as online diagnostics, which is finally elaborated to the online prediction of an individual system's safety quantities.

A general framework is proposed for pricing both continuously and discretely monitored Asian options under one-dimensional Markov processes. For each type (continuously monitored or discretely monitored), we derive the double transform of the Asian option price in terms of the unique bounded solution to a related functional equation. In the special case of continuous-time Markov chain (CTMC), the functional equation reduces to a linear system that can be solved analytically via matrix inversion. Thus the Asian option prices under a one-dimensional Markov process can be obtained by first constructing a CTMC to approximate the targeted Markov process model, and then computing the Asian option prices under the approximate CTMC by numerically inverting the double transforms. Numerical experiments indicate that our pricing method is accurate and fast under popular Markov process models, including the CIR model, the CEV model, Merton’s jump diffusion model, the double-exponential jump diffusion model, the variance gamma model, and the CGMY model.

A number of equivalent characterizations for the existence and boundedness of solutions of the linear complementarity problem: Mx+q≥0, x≥0, x T(Mx+q)=0 where M is an n×n real matrix and q is an n-vector, are given for the case when M is copositive plus. The special case when M is skew-symmetric covers the linear programming case. One useful characterization of existence and boundedness of solutions is given by solving a simple linear program. Other important characterizations are the Slater constraint qualification and the stability condition that for all arbitrary but sufficiently small perturbations of the data M and q which maintain copositivity plus, the perturbed linear complementarity problem is solvable and its solutions are uniformly bounded. An interesting sufficient condition for boundedness of solutions is that the linear complementarity problem have a nondegenerate vertex solution. Another result is that the subclass ™ of copositive plus matrices for which the linear complementarity problem has a solution for each q in R n, that is ™⊂Q, coincides with the subclass of copositive plus matrices for which the linear complementarity problem has a nonempty bounded solution set for each q in R n.Key wordsLinear Complementarity ProblemSolution SetBoundednessCopositive-plus Matrices

We present in this paper a mathematical method based on stochastic comparisons of Markov chains in order to compute in a mobile network performance measures that are important Q parameters for users: the dropping handover of voice and the blocking probability of a new voice call. The key idea of this methodology is that given a complex system represented by a multi-dimensional Markov chain which is too large to be solved, we propose to reduce the state space, and so to define a new Markov chain which is a simplified version of the original one. This reduced Markov chain is defined as an aggregated one, which represents a stochastic bound for performance measures written as increasing reward function on the stationary distribution. The main steps of the construction of the aggregated Markov chain applied into mobile networks are presented in this paper. As the number of mobile users with different kinds of applications increases, the associated model is more complex and it can be represented by multi-dimensional continuous time Markov chain with a very large size. Thus, we define an aggregated Markov chain represented by a multi-dimensional birth and death process which is very easy to solve. We have proved that there is a weak ordering between the original Markov chain and the aggregated one using increasing set formalism. We have computed upper bounds of dropping handover and blocking probability for different values of input parameters. Numerical results prove that upper bounds give good results and so the stochastic methodology is an interesting mathematical tool for the performance evaluation of complex systems. Keywords: Mobile Networks, Quality of Service, Stochastic ordering, stochastic comparisons, Continuous Time Markov Chains.

We consider the susceptible-infective (SI) epidemiological model, a variant of the Kermack-McKendrick models, and let the contact rate be a function of the number of infectives, an indicator of disease spread during the course of the epidemic. We represent the resultant model as a continuous-time Markov chain. The result is a pure death (or birth) process with state-dependent rates, for which we find the probability distribution of the associated Markov chain by solving the Kolmogorov forward equations. This model is used to find the analytic solution of the SI model as well as the distribution of the epidemic duration. We use the maximum likelihood method to estimate contact rates based on observations of inter-infection time intervals. We compare the stochastic model to the corresponding deterministic models through a numerical experiment within a typical household. We also incorporate different intervention policies for vaccination, antiviral prophylaxis, isolation, and treatment considering both full and partial adherence to interventions among individuals.

This work develops asymptotic expansions for solutions of systems of backward equations of time-inhomogeneous Markov chains in continuous time. Owing to the rapid progress in technology and the increasing complexity in modeling, the underlying Markov chains often have large state spaces, which make the computational tasks infeasible. To reduce the complexity, two-time-scale formulations are used. By introducing a small parameter ɛ > 0 and using suitable decomposition and aggregation procedures, it is formulated as a singular perturbation problem. Both Markov chains having recurrent states only and Markov chains including also transient states are treated. Under certain weak irreducibility and smoothness conditions of the generators, the desired asymptotic expansions are constructed. Then error bounds are obtained.

We study a quantum entanglement switch that serves $k$ users in a star topology. We model variants of the system using Markov chains and standard queueing theory and obtain expressions for switch capacity and the expected number of qubits stored in memory at the switch. While it is more accurate to use a discrete-time Markov chain (DTMC) to model such systems, we quickly encounter practical constraints of using this technique and switch to using continuous-time Markov chains (CTMCs). Using CTMCs allows us to obtain a number of analytic results for systems in which the links are homogeneous or heterogeneous and for switches that have infinite or finite buffer sizes. In addition, we can model the effects of decoherence of quantum states fairly easily using CTMCs. We also compare the results we obtain from the DTMC against the CTMC in the case of homogeneous links and infinite buffer, and learn that the CTMC is a reasonable approximation of the DTMC. From numerical observations, we discover that decoherence has little effect on capacity and expected number of stored qubits for homogeneous systems. For heterogeneous systems, especially those operating close to stability constraints, buffer size and decoherence can have significant effects on performance metrics. We also learn that in general, increasing the buffer size from one to two qubits per link is advantageous to most systems, while increasing the buffer size further yields diminishing returns.

In this paper, we present a new interior-point method of convergence order six to solve the linear complementarity problem. Computational efficiency in its general form is discussed and a comparison between the efficiency of the proposed method and existing ones is made. The performance is tested through numerical experiments on some test problems and a practical example of bio-economic equilibrium model.

A stationary point for a convex polyhedral set and a continuously differentiable function with positive semi-definite derivatives is computed by iteratively solving the linearized problem which is a linear complementarity problem (LCP). The procedure is shown to be a mixing of a finite number of Newton methods all converging to the same points, and consequently, to have convergence properties like Newton’s methods.

Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations

A kth partition reduction method is used to obtain new upper bounds for the inverses of H-matrices with their subclasses. The estimates are expressed via the determinants of kth order matrices, which is sharper than the existed ones. Numerical experiments with various random matrices show that they are stable and better than the estimates presented in literature. In addition, by reduce the order of the variables, the error bound of linear complementarity problems relies on less variables and reaches at the vertex on the reduced closed region. We use the upper bounds in order to improve known error estimates for linear complementarity problems with H-matrices.

Solving piecewise linear systems in abs-normal form