A Novel Analytical Formula for the Discounted Moments of the ECIR Process and Interest Rate Swaps Pricing

Abstract

This paper presents an explicit formula of conditional expectation for a product of polynomial functions and the discounted characteristic function based on the Cox–Ingersoll–Ross (CIR) process. We also propose an analytical formula as well as a very efficient and accurate approach, based on the finite integration method with shifted Chebyshev polynomial, to evaluate this expectation under the Extended CIR (ECIR) process. The formulas are derived by solving the equivalent partial differential equations obtained by utilizing the Feynman–Kac representation. In addition, we extend our results to derive an analytical formula of conditional expectation of a product of mixed polynomial functions and the discounted characteristic function. The accuracy and efficiency of the proposed scheme are also numerically shown for various modeling parameters by comparing them with those obtained from Monte Carlo simulations. In addition, to illustrate applications of the obtained formulas in finance, analytical pricing formulas for arrears and vanilla interest rate swaps under the ECIR process are derived. The pricing formulas become explicit under the CIR process. Finally, the fractional ECIR process is also studied as an extended case of our main results.

Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimension

“…n ] in terms of generalized hypergeometric functions. We remark that computing E[Y γ n ] relies on the c k coefficients which can be obtained from the recursive Formulas ( 4)-( 7), and we demonstrate later in our numerical study in Section 5 that implementing our Formula (8) for computing E[Y γ n ] consumes significantly less time and effort than employing MC simulations.…”

“…In hypothesis testing, several test statistics converge in distribution toward a conic combination of independent noncentral chi-square random variables (see, e.g., [2][3][4][5]). Moreover, f Y n (y) and E[Y γ n ] play an interesting role in financial applications; see, e.g., [6][7][8][9][10][11]. Very recently, Rujivan and Rakwongwan [11], Chumpong et al [6], and Rujivan [10] showed that the log-return realized variance when the underlying asset follows the extended Black-Scholes model can be expressed in terms of a conic combination of independent noncentral chi-square random variables.…”

Simple Closed-Form Formulas for Conditional Moments of Inhomogeneous Nonlinear Drift Constant Elasticity of Variance Process

“…In the context of future works, our proposed closed-form formulas under the IND-CEV process have further beneficial aspects for pricing financial derivatives, such as moment swaps and the asset whose payoff can be generated by the conditional moments, see more details in [23,30]. In addition, since the transition PDF of process ( 2) is complicated and does not exist in closed form, our closed-form formulas can also be applied for parameter estimations of the behavior and dynamic of observed data; see more details in [9].…”

“…There are several empirical studies confirming that a mean-reverting drift process, such as the Vašíček, Ornstein-Uhlenbeck (OU) [5] and Cox-Ingersoll-Ross (CIR) [6] processes, should not necessarily be linear. Indeed, the behaviors and dynamics of interest rate and its derivatives prefer nonlinearity in the mean-reverting drift rather than linear drift processes; see for more details in [7][8][9][10]. In order to extend the OU process, a nonlinear diffusion process was introduced by Cox [11], namely, the constant elasticity of variance (CEV) process.…”

Finite integration method using chebyshev expansion for solving heat equation with non-local boundary conditions

“…This method, later known as the FIM using the Chebyshev polynomial expansion (FIM-CPE), published in 2020 [3], significantly outperformed all preceding versions of FIM, marking a major leap forward in the field. There are applied to solve many problems, see [3][4][5][6][7][11][12][13][14][15][16] for more details. Nevertheless, it is important to note that extensive applications of this FIM-CPE…”