Fourier Cosine Series Expansion Research Articles

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.

Robust Recursive Least-Squares Fixed-Point Smoother and Filter using Covariance Information in Linear Continuous-Time Stochastic Systems with Uncertainties

This study develops robust recursive least-squares (RLS) fixed-point smoothing and filtering algorithms for signals in linear continuous-time stochastic systems with uncertainties. The algorithms use covariance information, such as the cross-covariance function of the signal with the observed value and the autocovariance function of the degraded signal. A finite Fourier cosine series expansion approximates these functions. Additive white Gaussian noise is present in the observation of the degraded signal. A numerical simulation compares the estimation accuracy of the proposed robust RLS filter with the robust RLS Wiener filter, showing similar mean square values (MSVs) of the filtering errors. The MSVs of the proposed robust RLS fixed-point smoother are also compared to those of the proposed robust RLS filter.

Stochastic stability and the moment Lyapunov exponent for a gyro-pendulum system driven by a bounded noise

Abstract. The stochastic stability of a gyro-pendulum system parametrically excited by a real noise is investigated by the moment Lyapunov exponent in the paper. Using the spherical polar and non-singular linear stochastic transformations and combining these with Khasminskii's method, the diffusion process and the eigenvalue problem of the moment Lyapunov exponent are obtained. Then, applying the perturbation method and Fourier cosine series expansion, we derive an infinite-order matrix whose leading eigenvalue is the second-order expansion g2(p) of the moment Lyapunov exponent. Thus, an infinite sequence for g2(p) is constructed, and its convergence is numerically verified. Finally, the influences of the system and noise parameters on stochastic stability are given such that the stochastic stability is strengthened with the increased drift coefficient and the diffusion coefficient has the opposite effect; among the system parameters, only the increase in k and A0 strengthens moment stability.

The valuation of arithmetic Asian options with mean reversion and jump clustering

It is known that the prices of many commodities exhibit mean reversion and are subject to jumps. Particularly, there is mounting empirical evidence that suggests the existence of clustered jumps. In this paper, we consider the valuation problem of arithmetic Asian options under a Hawkes jump diffusion model which captures both the mean reversion and jump clustering phenomena. We compute option prices by means of characteristic functions and Fourier-cosine series expansion. Numerical studies justify the validity and efficiency of our pricing method, and demonstrate the nonnegligible effects of mean reversion and jump clustering on Asian option prices.

Valuation of guaranteed minimum maturity benefits under mean reversion and jump models with surrender risk

We propose an efficient valuation approach for guaranteed minimum benefits embedded in variable annuity (VA) contracts, where the price of underlying asset is modeled by a mean reversion process with jumps. We consider the case in which early surrender is permitted, and an intensity-based framework is used to capture surrender behavior and other asset price volatilities. To analyze the contract within this complex stochastic setting, we rely on the Fourier cosine series expansion method to approximate the expectation of the value function, which serves as an estimation of the value of VA benefits. We conduct numerical examples to demonstrate the efficiency and accuracy of the proposed method. Sensitivity analysis of involved parameters is given.

Option Pricing with Fractional Stochastic Volatilities and Jumps

Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models.

Valuation of variable annuities with guaranteed minimum maturity benefits and periodic fees

This paper focuses on the valuation of variable annuities with a guaranteed minimum maturity benefit under a regime-switching Lévy model. The model allows policyholders to surrender their annuities and receive a surrender benefit at predetermined tenor times before maturity. Additionally, we consider a state-dependent periodic fee structure where fees are deducted from the policyholder's account if it exceeds a certain level at discrete time points. Incorporating this fee structure, the Fourier cosine series expansion method based on characteristic functions is employed to determine the values and optimal surrender strategies for variable annuity contracts. Finally, we provide a comprehensive set of numerical examples to demonstrate and assess the effectiveness of our approach thoroughly.

Finite-time expected present value of operating costs until ruin in a Cox risk model with periodic observation

In this paper, we use a Cox risk model to describe the surplus flow of an insurance company, where the intensity process in the Cox process is assumed to follow a general stochastic differential equation. Suppose that the insurer observes the surplus process periodically with constant observation frequency. Whenever the observed surplus level is larger than a critical level b2>0, the excess amount is paid as a lump sum of dividends; whenever the observed surplus level is between zero and another critical level b1 (0<b1<b2), capital is injected to the surplus process so that it return to the level b1; whenever the observed surplus level is less than zero, ruin is declared and the process is stopped. Under these assumptions, we study the finite-time expected present value of operating costs until ruin. The continuous time Markov chain (CTMC) approximation technique is used to approximate the intensity process in the Cox model, and the Fourier cosine series expansion (COS) method is applied to approximate the function of interest. A lot of numerical results are given to show accuracy and efficiency of our method, and impacts of some parameters are also analyzed.

Fast Barrier Option Pricing by the COS BEM Method in Heston Model (with Matlab Code)

Abstract In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process. A Matlab code implementing this technique is attached at the end of the paper.

Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model

&lt;p style='text-indent:20px;'&gt;In this paper, we investigate the valuation problem of equity-linked death benefits with a threshold expense structure. Specifically, a regime-switching Lévy process is used to describe the underlying asset price process, which is monitored periodically. The fees are assumed to be continuously deducted at some constant rate from the policyholder's account between the current and next monitoring times, if the account value is smaller than a pre-specified level at the current observation time point. Under the modified threshold expense structure, some explicit valuation expressions for life-contingent call options are derived by the Fourier cosine series expansion method. Numerical results demonstrate the accuracy and efficiency of our method.&lt;/p&gt;

Efficient valuation of guaranteed minimum maturity benefits in regime switching jump diffusion models with surrender risk

We present an efficient valuation approach for guaranteed minimum maturity benefits (GMMBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We allow early surrender of the VA contract and impose surrender charges, which are important in practice to discourage early termination/lapse of the contract. We consider both continuously-monitored and discretely-monitored surrender behaviors before maturity, and utilize an intensity-based framework. Based on the continuous-time Markov chain (CTMC) approximation combined with the Fourier cosine series expansion method, we find that the valuation problem can be solved under a regime-switching jump diffusion framework. Both error analysis and numerical experiments demonstrate the accuracy and efficiency of the proposed method.

Valuation of a DB underpin hybrid pension under a regime-switching Lévy model

This paper studies the valuation problem of the defined benefit (DB) underpin guarantee. We consider that the salary process follows a geometric Brownian motion, and the stochastic price index process of the funds in the defined contribution (DC) account is modeled by a regime-switching Lévy process. Under this framework, the explicit valuation formula of the DB underpin option is derived by the Fourier cosine series expansion (COS) method, and the corresponding error analysis is provided. Numerous simulation experiments are performed to illustrate the accuracy and efficiency of the proposed method. In addition, the convergence of this method and its sensitivity with respect to various model paraments are analyzed.

Valuing variable annuities with path-dependent surrender guarantees under regime-switching Lévy models

Variable annuities with complex surrender features are nowadays increasingly popular for managing longevity risks. The study of their accurate pricing and sensitivity analysis is one of the main actuarial research topics. This paper studies the valuation problem of variable annuity contracts with guaranteed minimum maturity benefits on a set of predetermined discrete tenor dates under regime-switching Lévy models. Extending from existing vanilla payoffs, we consider the guaranteed minimum maturity benefits with lookback and geometric average features. We customise the dynamic programming principle to solve the corresponding optimal stopping problem, relying on some semi-analytical valuation formulae resulting from an acute Fourier cosine series expansion. Finally, numerical illustrations are provided to show the accuracy and efficiency of the proposed method. We also demonstrate the use of our proposed method in a range of sensitivity analysis exercises, which shed light on the pricing and risk management of complex variable annuity products.

Pricing some life-contingent lookback options under regime-switching Lévy models

In this paper, we study the valuation problem of life-contingent lookback options embedded in variable annuity with guaranteed minimum death benefit (GMDB). Specifically, the underlying asset price process is assumed to be an exponential regime-switching Lévy process, which is observed periodically. The Fourier cosine series expansion method is applied to compute exponential moments of the discretely monitored maximum and minimum of the regime-switching Lévy process. Furthermore, some explicit pricing formulas for the life-contingent lookback options embedded in GMDB products are derived. Finally, numerical experiments confirm the accuracy and efficiency of our method.

The SINC way: a fast and accurate approach to Fourier pricing

The goal of this paper is to investigate the method outlined by one of us (P. R.) in Cherubini, U., Della Lunga, G., Mulinacci, S. and Rossi, P. [Fourier Transform Methods in Finance, 2009 (John Wiley & Sons Inc.).] to compute option prices. We name it the SINC approach. While the COS method by Fang, F. and Oosterlee, C.W. [A novel pricing method for european options based on Fourier-cosine series expansions. SIAM. J. Sci. Comput., 2009, 31(2), 826–848.] leverages the Fourier-cosine expansion of truncated densities, the SINC approach builds on the Shannon Sampling Theorem revisited for functions with bounded support. We provide several results which were missing in the early derivation: (i) a rigorous proof of the convergence of the SINC formula to the correct option price when the support grows and the number of Fourier frequencies increases; (ii) ready to implement formulas for put, Cash-or-Nothing, and Asset-or-Nothing options; (iii) a systematic comparison with the COS formula for several log-price models; iv) a numerical challenge against alternative Fast Fourier specifications, such as Carr, P. and Madan, D. [Option valuation using the fast Fourier transform. J. Comput. Finance, 1999, 2(4), 61–73.] and Lewis, A.L. [Option Valuation Under Stochastic Volatility with Mathematica Code, 2000 (Newport Beach: Finance Press).]; (v) an extensive pricing exercise under the rough Heston model of Jaisson, T. and Rosenbaum, M. [Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab., 2015, 25(2), 600–631.]; (vi) formulas to evaluate numerically the moments of a truncated density. The advantages of the SINC approach are numerous. When compared to benchmark methodologies, SINC provides the most accurate and fast pricing computation. The method naturally lends itself to price all options in a smile concurrently by means of Fast Fourier techniques, boosting fast calibration. Pricing requires resorting only to odd moments in the Fourier space.

Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite.

Finite-time dividend problems in a Lévy risk model under periodic observation

In this paper, we use a Lévy process to model the surplus flow of an insurance company. It is assumed that the surplus level is observed at a sequence of fixed times and dividend decisions are made at each observation time. If the observed surplus level is larger than a given barrier, then the excess amount would be paid off as a lump sum of dividends. Further, we assume that ruin is declared as soon as the observed surplus level is negative. Using the Fourier cosine series expansion method, we propose some numerical methods for computing the finite-time expected discounted dividend payments before ruin and the finite-time expected discounted penalty function. Both error analysis and numerical examples are given to show accuracy and efficiency of our method.

A regime switching fractional Black–Scholes model and European option pricing

In this paper, we investigate the European option pricing problem under a regime switching FMLS (finite moment log-stable) model. This model is not only able to capture the main characteristics of asset returns, it also incorporates the effect of regime switching being consistent with market observations. However, option prices under this model are governed by a coupled FPDE (fractional partial differential equation) system, and the difficulty in seeking for analytical solution arises from the combination of the coupled system and the spatial-fractional derivative. To deal with this difficulty, we develop a two-step solution procedure; we firstly assume that the future information of the Markov chain is known, and we derive the conditional option price by analytically solving a time dependent FPDE, based on which an exact and explicit pricing formula for the unconditioned price is successfully worked out by using the Fourier cosine series expansion. It is also shown through the numerical experiments that it converges very rapidly and has potential to be applied in practice.

Effect of general coupling conditions on the vibration and power flow characteristics of a two-plate built-up plate structure

This paper presents the modeling approach, dynamic response analysis and the effect of coupling junction properties on the vibration characteristics of two plates elastically coupled in the form a T shaped structure with elastic boundary conditions. These built-up plate structures are the most commonly used configurations in the design of engineering platforms both floating and stationary. The coupling junction has been represented as a combination of translational and rotational spring stiffness’s, thus making the junction state general in nature. The analysis has been carried out using a Fourier cosine series representation for the bending and in-plane plate displacements modified with inclusion of derived auxiliary function. The derived auxiliary functions are also presented in form of the product of a single Fourier cosine series expansion and one polynomial function. The Rayleigh-Ritz method is employed to the Lagrangian function to derive the natural frequency and dynamic response matrix. Both the free vibration and power transmission characteristics of the plate structure has been numerically presented. Validation of the modified Fourier function has been presented for an L shaped plate structure in comparison with existing literature results for free vibration analysis. Finally, the effect of variation of the coupling junction condition on the vibration and power flow characteristics of the T-plate structure has been presented. The modified Fourier function has proved to generate simple theoretical formulations and accurate results in numerical analysis of a T plate coupled configuration. The modified Fourier series function can be universally and efficiently extended to undertake vibration analysis of any type of coupled plate system with any configuration of boundary or coupling configurations. Communicated by J. Logo

Stochastic Stability of a Planner Gyropendulum System with Synchronous Motor Driven by Gaussian White Noises

In this paper, the stochastic moment stability and almost-sure stability of a planner gyropendulum system with synchronous motor under the white noises are investigated. By applying the theory of diffusion process, an eigenvalue problem for the moment Lyapunov exponent is formulated. Then, through a perturbation method and a Fourier cosine series expansion, the second-order expansion of the moment Lyapunov exponent is solved, which is just the leading eigenvalue of an infinite matrix. Finally, the convergence and validity of the procedure are numerically verified, and the effects of system and noise parameters on the moment Lyapunov exponent are discussed. It was found that the increase in both the noise intensity and coefficient of the synchronous motor torque will weaken the stability of the gyropendulum system, and when they reach certain values, the system becomes unstable. In addition, according to the relationship between the moment Lyapunov exponent and maximal Lyapunov exponent, the stable thresholds are also given.