Continuous Time Markov Chain Approximation Research Articles

A Weak MLMC Scheme for Lévy-Copula-Driven SDEs with Applications to the Pricing of Credit, Equity and Interest Rate Derivatives

This paper develops a novel weak multilevel Monte-Carlo (MLMC) approximation scheme for Lévy-driven Stochastic Differential Equations (SDEs). The scheme is based on the state space discretization (via a continuous-time Markov chain approximation [Mijatović, Aleksandar, Matija Vidmar, and Saul Jacka. 2012. “Markov Chain Approximations for Transition Densities of Lévy Processes.” Electronic Journal of Probability 19]) of the pure-jump component of the driving Lévy process and is particularly suited if the multidimensional driver is given by a Lévy copula. The multilevel version of the algorithm requires a new coupling of the approximate Lévy drivers in the consecutive levels of the scheme, which is defined via a coupling of the corresponding Poisson point processes. The multilevel scheme is weak in the sense that the bound on the level variances is based on the coupling alone without requiring strong convergence. Moreover, the coupling is natural for the proposed discretization of jumps and is easy to simulate. The approximation scheme and its multilevel analogous are applied to examples taken from mathematical finance, including the pricing of credit, equity and interest rate derivatives.

VIX Options Valuation via Continuous-Time Markov Chain Approximation and Ito-Taylor Expansion

VIX Options Valuation via Continuous-Time Markov Chain Approximation and Ito-Taylor Expansion

Efficient valuation of guaranteed minimum accumulation benefits in regime switching jump diffusion models with lapse risk

We present a streamlined valuation method for the guaranteed minimum accumulation benefits incorporated within variable annuity contracts. At each contract renewal date, the insurer updates the policyholder's account value to the higher of the guaranteed value and the equity-linked investment value. In addition, we introduce lapse risks into the variable annuity contract, modeling the lapse decision under the assumption of stochastic intensity. Utilizing a combination of continuous-time Markov chain approximation and the Fourier cosine series expansion method, we derive closed-form valuation formulas under regime switching jump diffusion models. Numerical simulations showcase the precision and effectiveness of the proposed approach.

Continuous-time Markov chain approximation for pricing Asian options under rough stochastic local volatility models

We propose a general framework for pricing both discretely and continuously monitored arithmetic average Asian options whose underlying asset price satisfies the rough stochastic local volatility model. We use a semimartingale approximation approach for the rough stochastic local volatility model to obtain its Markovian representation. For each type of Asian options, we use the double-layer continuous-time Markov chain to approximate the transformed underlying asset price, and derive the double transform of the Asian option price in terms of the unique bounded solution to a related functional equation. We invert the analytical double transforms of Asian option prices under the approximate continuous-time Markov chain numerically to obtain the approximations for Asian option prices. Monte Carlo simulation demonstrates the accuracy and efficiency of the proposed method for Asian option pricing.

Power utility maximization problem under rough stochastic local volatility models with continuous-time Markov chain approximation

This paper studies the portfolio optimization problem based on rough stochastic local volatility models. Due to the challenge that this model is neither a semimartingale nor a Markov process, we first introduce the semimartingale approximate optimization problem and present the convergence analysis. We use the classic stochastic control method and continuous-time Markov chain approximation approach to solve the approximate optimization problem respectively. Then the optimal investment strategy and the HJB equation satisfied by the value function are obtained respectively. Furthermore, for the classic stochastic control method, we demonstrate the existence of solutions for the associated partial differential equation via the nonlinear Feynman–Kac formula. For continuous-time Markov chain approximation method, we prove the convergence of the value function. Finally, we provide a numerical analysis to illustrate the above results by using the power utility function. The numerical analysis shows that continuous-time Markov chain approximation method is effective in solving the portfolio optimization problem under rough stochastic local volatility models.

A general approach for lookback option pricing under Markov models

We propose a computationally efficient method for pricing various types of lookback options under Markov models. We utilize the model-free representations of lookback option prices as integrals of first passage probabilities. We combine efficient numerical quadrature with continuous-time Markov chain approximation for the first passage problem to price lookbacks. Our method is applicable to a variety of models, including one-dimensional time-homogeneous and time-inhomogeneous Markov processes, regime-switching models and stochastic local volatility models. We demonstrate the efficiency of our method through various numerical examples.

A General Framework to Simulate Diffusions with Discontinuous Coefficients and Local Times

In this article, we propose an efficient general simulation method for diffusions that are solutions to stochastic differential equations with discontinuous coefficients and local time terms. The proposed method is based on sampling from the corresponding continuous-time Markov chain approximation. In contrast to existing time discretization schemes, the Markov chain approximation method corresponds to a spatial discretization scheme and is demonstrated to be particularly suited for simulating diffusion processes with discontinuities in their state space. We establish the theoretical convergence order and also demonstrate the accuracy and robustness of the method in numerical examples by comparing it to the known benchmarks in terms of root mean squared error, runtime, and the parameter sensitivity.

Analysis of Markov Chain Approximation for Diffusion Models with Nonsmooth Coefficients

In this study, we analyze the convergence of continuous-time Markov chain approximation for one-dimensional diffusions with nonsmooth coefficients. We obtain a sharp estimate of the convergence rate for the value function and its first and second derivatives, which is generally first order. To improve it to second order, we propose two methods: applying the midpoint rule that places all nonsmooth points midway between two neighboring grid points or applying harmonic averaging to smooth the model coefficients. We conduct numerical experiments for various financial applications to confirm the theoretical estimates. We also show that the midpoint rule can be applied to achieve second-order convergence for some jump-diffusion and two-factor short rate models.

Convergence analysis for continuous-time Markov chain approximation of stochastic local volatility models: Option pricing and Greeks

This paper establishes the precise second order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing options under the general framework of stochastic local volatility (SLV) models. The stochastic local volatility models studied in this paper include Heston model, 4/2 model, α-Hypergeometric model, stochastic alpha beta rho (SABR) model, Heston-SABR model and quadratic SLV model. Using the stochastic flow theorem, the closed-form CTMC approximation formula for the Greeks are obtained and the second order convergence rates are proved. Numerical examples confirm the theoretical findings.

Analysis of VIX-linked fee incentives in variable annuities via continuous-time Markov chain approximation

We consider the pricing of variable annuities (VAs) with general fee structures under a class of stochastic volatility models which includes the Heston, Hull-White, Scott, α-Hypergeometric, 3/2, and 4/2 models. In particular, we analyze the impact of different VIX-linked fee structures on the optimal surrender strategy of a VA contract with guaranteed minimum maturity benefit (GMMB). Under the assumption that the VA contract can be surrendered before maturity, the pricing of a VA contract corresponds to an optimal stopping problem with an unbounded, time-dependent, and discontinuous payoff function. We develop efficient algorithms for the pricing of VA contracts using a two-layer continuous-time Markov chain approximation for the fund value process. When the contract is kept until maturity and under a general fee structure, we show that the value of the contract can be approximated by a closed-form matrix expression. We also provide a quick and simple way to determine the value of early surrenders via a recursive algorithm and give an easy procedure to approximate the optimal surrender surface. We show numerically that the optimal surrender strategy is more robust to changes in the volatility of the account value when the fee is linked to the VIX index.

Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models

Rough volatility models have recently been empirically shown to provide a good fit to historical volatility time series and implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with Hurst parameter less than half. Due to the challenge that it is neither a semimartingale nor a Markov process, there is no unified method that not only applies to all rough volatility models, but also is computationally efficient. This paper proposes a semimartingale and continuous-time Markov chain (CTMC) approximation approach for the general class of rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semimartingale approximation for the RSLV model and establish its existence , uniqueness and Markovian representation. We propose a fast CTMC algorithm and prove its weak convergence. Numerical experiments demonstrate the accuracy and high efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.

A Markov chain approximation scheme for option pricing under skew diffusions

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 process. We obtain an explicit closed-form approximation of the transition density of a general skew diffusion process, which facilitates the unified valuation of various financial contracts written on assets with natural boundary behavior, e.g. in the foreign exchange market with target zones, and equity markets with psychological barriers. Applications include valuation of European call and put options, barrier and Bermudan options, and zero-coupon bonds. Motivated by the presence of psychological barriers in the market volatility, we also propose a novel ‘skew stochastic volatility’ model, in which the latent stochastic variance follows a skew diffusion process. Numerical results demonstrate that our approach is accurate and efficient, and recovers various benchmark results in the literature in a unified fashion.

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

Continuous time Markov Chain (CTMC) approximation techniques have received increasing attention in the option pricing literature, due to their ability to solve complex pricing problems, although existing approaches are mostly limited to one or two dimensions. This paper develops a general methodology for modeling and pricing financial derivatives which depend on systems of stochastic diffusion processes. This is accomplished with a general decorrelation procedure, which reduces the system of correlated diffusions to an uncorrelated system. This enables simple and efficient approximation of the driving processes by univariate CTMC approximations. Weak convergence of the approximation is demonstrated, with second order convergence in space. Numerical experiments demonstrate the accuracy and efficiency of the method for various European and early-exercise options in two and three dimensions.

A General Continuous Time Markov Chain Approximation for Multi-Asset Option Pricing With Systems of Correlated Diffusions

Continuous time Markov Chain (CTMC) approximation techniques have received increasing attention in the option pricing literature, due to their ability to solve complex pricing problems, although existing approaches are mostly limited to one or two dimensions. This paper develops a general methodology for modeling and pricing financial derivatives which depend on systems of stochastic diffusion processes. This is accomplished with a general de-correlation procedure, which reduces the system of correlated diffusions to an uncorrelated system. This enables simple and efficient approximation of the driving processes by uni-variate CTMC approximations. Weak convergence of the approximation is demonstrated, with second order convergence in space. Numerical experiments demonstrate the accuracy and efficiency of the method for various European and early-exercise options in two and three dimensions.

Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models

Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime switching Levy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines continuous-time Markov chain approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi, $$\alpha $$ -Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a ‘unified’ approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.

Convergence Analysis for Continuous-Time Markov Chain Approximation of Stochastic Local Volatility Models: Option Pricing and Greeks

This paper establishes the precise second order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing options and calculating its Greeks under the general framework of stochastic local volatility models, which include the Heston and SABR models as special cases. Numerical examples confirm the theoretical findings.

Numerical Validation of the Continuous Time Markov Chain Approximation as a Service Level Estimation Method for Heterogeneous Customer Demand Classes

In this article, we study a threshold level based inventory allocation and replenishment problem for two priority-demand classes under the lot-for-lot replenishment policy. Demands of low-priority class are backordered when on-hand stock is at or below a certain threshold level, while high-priority class demands are lost in case of stock-out situations. We assume a continuous review lot-for-lot ordering policy with both deterministic and stochastic replenishment lead times. Both priority classes exhibit mutually independent, stationary, Poisson demand processes. There is no exact solution yet in the literature except for the special case of exponentially distributed lead times. In this complementary study, we aim to numerically validate the Continuous-Time Markov Chain (CTMC) approximation as a service level approximation method by thoroughly analyzing the effect of the type of lead time distribution, lead time variability, and value of system parameters on the quality of the class-specific service level estimations. As long as expected lead times are identical, we also show empirically the near-insensitivity of the effect of the type of lead time distribution and the lead time variability on the achieved service levels. The CTMC approximation works well under variety of system parameters and lead time distributions. For the cases considered in the constant lead time scenarios with service levels which we expect to see in practice (with fill rate requirements at least 60 % and 90 % for the low-priority and high priority customer classes, respectively), the average absolute errors for the CTMC approximations are calculated as 0.09 % (for the low-priority class fill rate) and 0.05 % (for the high-priority class fill rate); while the maximum absolute errors for the CTMC approximations are calculated as 0.37 % (for the low-priority fill rate) and 0.18 % (for the high-priority fill rate). The CTMC approach also provides quality approximations for the Erlang, lognormal and gamma distributed lead times.

A Markov Chain Approximation Scheme for Option Pricing Under Skew Diffusions

In this paper, we propose a general valuation framework for option pricing problems related to diffusions based on a continuous-time Markov chain approximation to the underlying stochastic process. We obtain an explicit closed-form approximation of the transition density of a general diffusion process, which facilitates the unified valuation of various financial contracts written on assets with natural boundary behaviors, e.g. in foreign exchange market with target zones, and equity markets with psychological barriers. Applications include valuation of European call and put options, barrier and Bermudan options, zero-coupon bonds, and arithmetic Asian options. Motivated by the presence of psychological barriers in the market volatility, we also propose a novel skew stochastic volatility model, in which the latent stochastic variance follows a diffusion process. The framework is shown to be able to also handle the case of jump diffusions. Numerical results demonstrate that our approach is accurate and efficient, and recover various benchmark results in the literature in a unified fashion.

Pricing Discretely Monitored Barrier Options Under Markov Processes Using a Markov Chain Approximation

We propose an explicit closed-form approximation formula for the price of discretely monitored single or double barrier options whose underlying asset evolves according to a generic one-dimensional Markov process. This set of stochastic processes includes, but is not limited to, diffusion and jump diffusion processes commonly used in derivative pricing applications. The formula’s derivation combines the integral equation method, the Z−transform technique, and a continuous-time Markov chain approximation of the underlying Markov process. It does not require one to perform an intermediate numerical quadrature or related potentially runtime-intensive, error-prone, or otherwise complicated numerical procedure that may require a high degree of tuning to ensure appropriate accuracy (e.g. an inverse Laplace transform or inverse Z−transform). Rather, the price and Greeks of a discretely-monitored double barrier option may be explicitly expressed in terms of rudimentary matrix operations. In addition, this framework may be extended to include additional features of barrier options often encountered in practice. Examples including time-dependent barriers and non-uniform monitoring time intervals, may be seamlessly incorporated into this framework. In addition, by limiting the monitoring frequency to a large value, we also obtain an accurate closed-form formula for the price and Greeks of continuously-monitored double barrier options with time-dependent barriers under general Markov processes. Finally, we provide many numerical examples to demonstrate the accuracy and efficiency of the proposed formula as well as its ability to reproduce existing benchmark results in the relevant literature.

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

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.