Direct computation of distribution functions from characteristic functions using the fast Fourier transform

A.A.G. Requicha

doi:10.1109/proc.1970.7875

PRICING HOLDER-EXTENDABLE CALL OPTIONS WITH MEAN-REVERTING STOCHASTIC VOLATILITY

The ANZIAM Journal ◽

10.1017/s1446181119000142 ◽

2019 ◽

Vol 61 (4) ◽

pp. 382-397

Author(s):

S. N. I. IBRAHIM ◽

A. DÍAZ-HERNÁNDEZ ◽

J. G. O’HARA ◽

N. CONSTANTINOU

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Stochastic Volatility ◽

Fourier Method ◽

Heston Model ◽

Computational Time ◽

Characteristic Functions ◽

Underlying Asset ◽

Call Options ◽

Integral Forms

Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black–Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls.

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Pricing holder-extendable call options with mean-reverting stochastic volatility

ANZIAM Journal ◽

10.21914/anziamj.v61i0.12090 ◽

2020 ◽

Vol 61 ◽

pp. 382-397

Author(s):

Siti Nur Iqmal Ibrahim ◽

Adan Diaz-Hernandez ◽

John G. O'Hara ◽

Nick Constantinou

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Stochastic Volatility ◽

Fourier Method ◽

Heston Model ◽

Computational Time ◽

Characteristic Functions ◽

Underlying Asset ◽

Call Options ◽

Integral Forms

Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black–Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls. doi:10.1017/S1446181119000142

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Computing the Stationary Distribution of Queueing Systems with Random Resource Requirements via Fast Fourier Transform

Mathematics ◽

10.3390/math8050772 ◽

2020 ◽

Vol 8 (5) ◽

pp. 772 ◽

Cited By ~ 1

Author(s):

Valeriy A. Naumov ◽

Yuliya V. Gaidamaka ◽

Konstantin E. Samouylov

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Cumulative Distribution Function ◽

Queueing Systems ◽

Distribution Functions ◽

Cumulative Distribution ◽

Queuing Systems ◽

Cumulative Distribution Functions ◽

Resource Requirements ◽

Convolution Powers

Queueing systems with random resource requirements, in which an arriving customer, in addition to a server, demands a random amount of resources from a shared resource pool, have proved useful to analyze wireless communication networks. The stationary distributions of such queuing systems are expressed in terms of truncated convolution powers of the cumulative distribution function of the resource requirements. Discretization of the cumulative distribution function and the application of the fast Fourier transform are a traditional way of calculating convolutions. We suggest finding truncated convolution powers of the cumulative distribution functions by calculating the convolution powers of the truncated cumulative distribution functions via fast Fourier transform. This radically decreases computational complexity. We introduce the concept of resource load and investigate the accuracy of the proposed method at low and high resource loads. It is shown that the proposed method makes it possible to quickly and accurately calculate truncated convolution powers required for the analysis of queuing systems with random resource requirements.

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