Cai, Song, and Kou (2015) [Cai, N., Y. Song, S. Kou (2015) A general framework for pricing Asian options under Markov processes. Oper. Res. 63(3): 540-554] made a breakthrough by proposing a general framework for pricing both discretely and continuously monitored Asian options under one-dimensional Markov processes. In this note, under the setting of continuous-time Markov chain (CTMC), we explicitly carry out the inverse Z-transform and the inverse Laplace transform respectively for the discretely and the continuously monitored cases. The resulting explicit single Laplace transforms improve their Theorem 2, p.543, and numerical studies demonstrate the gain in efficiency.

Single-Transform Formulas for Pricing Asian Options in a General Approximation Framework under Markov Processes

A general continuous time Markov chain approximation for multi-asset option pricing with systems of correlated diffusions

This paper is concerned with the solution of the optimal stopping problem associated to the value of American options driven by continuous-time Markov chains. The value-function of an American option in this setting is characterised as the unique solution (in a distributional sense) of a system of variational inequalities. Furthermore, with continuous and smooth fit principles not applicable in this discrete state-space setting, a novel explicit characterisation is provided of the optimal stopping boundary in terms of the generator of the underlying Markov chain. Subsequently, an algorithm is presented for the valuation of American options under Markov chain models. By application to a suitably chosen sequence of Markov chains, the algorithm provides an approximate valuation of an American option under a class of Markov models that includes diffusion models, exponential Lévy models, and stochastic differential equations driven by Lévy processes. Numerical experiments for a range of different models suggest that the approximation algorithm is flexible and accurate. A proof of convergence is also provided.

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American option valuation under continuous time Markov Chains

In this paper a class of statistics based on high frequency observations of oscillating Brownian motions and skew Brownian motions is considered. Their convergence rate towards the local time of the underling process is obtained in form of a Central Limit Theorem.

Rates of convergence to the local time of Oscillating and Skew Brownian Motions

Markov chain approximation and measure change for time-inhomogeneous stochastic processes

In this paper, we propose a general valuation framework for option pricing problems related to skew diffusions based on a continuous-time Markov chain approximation to the underlying stochastic pro...

A Markov chain approximation scheme for option pricing under skew diffusions

In this paper, we propose a novel simultaneous two-dimensional continuous-time Markov chain (CTMC) approximation method, in contrast to the existing double-layer approach, to approximate the general fully coupled Markov diffusion processes which cover all the classical models. Extensive simulation studies on different kinds of financial option pricing problems in the European, American, and barrier settings, confirm that the proposed methodology has superior accuracy and outperforms the widely applicable Monte Carlo (MC) simulation approach consistently.

Simultaneous Two-Dimensional Continuous-Time Markov Chain Approximation of Two-Dimensional Fully Coupled Markov Diffusion Processes

In this paper, we propose a new explicit series expansion formula for the price of an arithmetic Asian option under the Black–Scholes model and Merton's jump-diffusion model. The method is based on an equivalence in law relation together with the diffusion operator integral method proposed by Heath and Platen. The method yields explicit series expansion formula for the Asian options' prices. The theoretical convergence of the expansion to the true value is established. We also consider the American Asian option (i.e., Amerasian option) and derive the corresponding expansion formula through the early exercise premium representation. Numerical results illustrate the accuracy and efficiency of the method as compared with benchmarks in the literature. 

Kailin Ding

Papers

A Markov chain approximation scheme for option pricing under skew diffusions

Markov chain approximation and measure change for time-inhomogeneous stochastic processes

Simultaneous Two-Dimensional Continuous-Time Markov Chain Approximation of Two-Dimensional Fully Coupled Markov Diffusion Processes

Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach

Multilayer-perceptron-based prediction of sand-over-clay bearing capacity during spudcan penetration