Joint Characteristic Function Research Articles

Forward starting options pricing under a regime-switching jump-diffusion model with Wishart stochastic volatility and stochastic interest rate

Vanilla options become effective immediately after they are entered, while some exotic options will only come to effective some time after they are bought or sold. Forward starting options are one kind of such exotic options started actually at some pre-specified future date. This paper presents an extension of regime-switching jump diffusion model, in which the parameters are driven by a continuous time and stationary Markov chain on a finite state space, by introducing the Wishart process into the instantaneous variance-covariance matrix of the risky asset price and stochastic interest rate. We derive the discounted conditional joint characteristic function and the forward characteristic function of the log-asset price and its the instantaneous variance-covariance process, and thereby the price of forward starting options are well evaluated by the probabilistic approach combined with the Fourier-cosine (COS) method. We also provide efficient Monte Carlo simulation of this proposed model, and simulated solutions to forward starting options pricing within a two-state regime switching framework. Numerical results show that the COS method is accurate and efficient for pricing forward starting options. Finally, we analyse impacts of some main parameters (especially, parameters in the Wishart stochastic volatility) in this proposed model on option prices and Δ values. Also, we consider the forward implied volatility. Furthermore, the forward starting options under the regime-switching jump diffusion model with Wishart stochastic volatility and stochastic interest rate which we derived are more generalized than those recently appeared in the derivatives pricing literature, and thus have wider application.

Extremum Options Pricing of Two Assets under a Double Nonaffine Stochastic Volatility Model

In this paper, we consider the pricing problem for the extremum options by constructing a double nonaffine stochastic volatility model. The joint characteristic function of the logarithm of two asset prices is derived by using the Feynman–Kac theorem and one-order Taylor approximation expansion. The semiclosed analytical pricing formulas of the European extremum options including option on maximum and option on minimum of two underlying assets are derived by using measure change technique and Fourier transform approach. Some numerical examples are provided to analyze the pricing results of extremum options under affine model, nonaffine model, Black–Scholes model, and the influences of some model parameters on the option. Numerical results show that the analytical pricing formulas have higher computational efficiency and accuracy than those of Monte Carlo simulation method. Also, results of sensitivity analysis report that the nonaffine models are more effective than other existing models in capturing the effect of volatility on option pricing.

Hyperspectral Image Few-Shot Classification Network With Brownian Distance Covariance

At present, how to achieve high precision hyperspectral image classification (HSIC) under the condition of few samples is a hot research issue. Metric-based meta-learning methods have proved to be very successful in this field. However, in terms of quantifying the dependencies between embedded features of hyperspectral samples, previous methods either only model marginal distribution and ignore joint distribution, limiting expressive capability of feature representation, or bring large computational cost though considering joint distribution. In this paper we propose a novel few shot learning (FSL) method based on Brownian distance covariance (BDC) for HSIC, which learns hyperspectral images’ representations by measuring the discrepancy between joint characteristic functions of embedded features and product of the marginals. In addition, a lightweight feature extraction network based on tied block convolution is proposed to better model cross channel correlation and aggregate global spectral-spatial features across channels. Extensive evaluations on several datasets show the effectiveness of the proposed method.

An efficient cross-domain few-shot website fingerprinting attack with Brownian distance covariance

Website fingerprinting (WF) attacks can infer the specific websites from Tor encrypted traffic. However, most existing WF attacks require a large number of training samples (e.g., hundreds) per website to achieve the desirable accuracy. To lessen the training data scale, the low-data WF (LDWF) attacks are proposed. Whereas existing LDWF attacks generally require gathering an auxiliary dataset, which increases the bootstrap time to launch a WF attack. To address this limitation, we present the cross-domain LDWF attack problem and put forward the first cross-domain LDWF attack, namely WFBDC (WF with Brownian Distance Covariance), which can use a historical gathered dataset to be the auxiliary dataset. The primary advantage of WFBDC is the introduction of the BDC metric to measure the similarity between two samples. The key to BDC lies in that it defines the similarity by measuring the discrepancy between joint characteristic functions of embedded features and the product of the marginals. To mitigate the domain deviation, the transfer learning and multi-similarity loss techniques are also adopted. We conduct 10 experiments based on 15 datasets to evaluate the performance and efficiency of WFBDC. Evaluation results show that WFBDC can improve the performance of the state-of-the art LDWF attacks by up to 9% and 19% in the closed-world and open-world scenarios, respectively. Meanwhile, WFBDC can significantly reduce the pre-training time of existing LDWF attacks.

The characteristic function of Gaussian stochastic volatility models: an analytic expression

Stochastic volatility models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylised facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample paths. This is made possible by virtue of the flexibility introduced in the choice of the covariance function of the Gaussian process. The price to pay is that in general, such models are no longer Markovian nor semimartingales, which limits their practical use. We derive, in two different ways, an explicit analytic expression for the joint characteristic function of the log-price and its integrated variance in general Gaussian stochastic volatility models. That analytic expression can be approximated by closed-form matrix expressions. This opens the door to fast approximation of the joint density and pricing of derivatives on both the stock and its realised variance by using Fourier inversion techniques. In the context of rough volatility modelling, our results apply to the (rough) fractional Stein–Stein model and provide the first analytic formulas for option pricing known to date, generalising that of Stein–Stein, Schöbel–Zhu and a special case of Heston.

Pricing of spread and exchange options in a rough jump–diffusion market

Asset dynamics with rough volatility recently received a great deal of attention in finance because they are consistent with empirical observations. This article provides a detailed analysis of the impact of roughness on prices of spread and exchange options. We consider a bivariate extension of the rough Heston model with jumps and derive the joint characteristic functions of asset log-returns under the risk neutral measure and under a measure using a risky asset as numeraire. These characteristic functions are expressed in terms of solutions of fractional differential equations (FDE’s). To infer these FDE’s, we rewrite the rough model as an infinite dimensional Markov process and propose a finite dimensional approximation. Next, we show that characteristic functions of log-returns admit a representation in terms of forward differential equations. FDE’s are obtained by passing to the limit. Spread and exchange options are valued with a two or a one dimensional discrete Fourier Transform. The numerical illustration reveals that considering a rough instead of Brownian volatility does not systematically increase exchange option prices.

Skew Brownian motion with dry friction: Joint density approach

This note concerns the distribution of Skew Brownian motion with dry friction and its occupation time. These distributions were obtained in Berezin and Zayats (2019) by using the Laplace transform and joint characteristic functions. We provide an alternative approach, which is based on the use of the joint density for Skew Brownian motion, its last visit to the origin, its local and occupation times derived in Gairat and Shcherbakov (2017).

Valuation of European-style vulnerable options under the non-affine stochastic volatility and double exponential jump

The pricing of European-style vulnerable option when the price process of the underlying asset follows non-affine stochastic volatility and double exponential jump is investigated. An approximate expression for the joint characteristic function of the log-price of underlying asset and the log-value of counterparty asset is derived. An analytical approximate price of European-style vulnerable option is also obtained by means of Fourier-cosine method. Numerical experiments are given to confirm the accuracy and efficiency of the proposed result for pricing the European-style vulnerable option compared with Monte Carlo simulation. Finally, sensitivity analysis is presented to further explain the theoretical results.

Vulnerable European option pricing in a Markov regime-switching Heston model with stochastic interest rate

This paper considers pricing of European-style vulnerable options under the Heston stochastic volatility and stochastic interest rate model in which the mean-reversion levels of both variance and interest rate processes are modulated by a continuous-time Markov process with a finite state space. An analytical pricing formula is derived by using the Esscher transform, joint characteristic function and multivariate Fourier transform technique, where the closed-form solution of the characteristic function is obtained by the law of iterated expectation. Then we provide the efficient approximation to calculate the analytical pricing formula of option using the FFT approach and examine the accuracy of the approximation by Monte Carlo simulation. Finally, the sensitivity analysis of different parameters in the proposed model on the vulnerable call option price and its Delta value are provided, and the difference between the proposed model and the Heston and stochastic interest rate model with non-Markov regime-switching are presented by some numerical experiments, which shows the influence of introducing regime-switching into Heston model with stochastic interest rate.

On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations

Abstract We investigate joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient α ∈ (0, 1) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ(x) (1 − x) β , x ∈ (0, 1), with β ∈ (−1, ∞) and lim x ↑ 1 ψ ( x ) = ψ 1 ∈ ( 0 , ∞ ) $\lim\limits_{x\uparrow 1} \psi(x) = \psi_1 \in (0, \infty)$ , different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β ∈ (−1, 0] in the so-called simultaneous case, i.e., when both N and the time scale n increase to infinity at a given rate. The case β ∈ (0, ∞) remains open. We also give a new explicit formula for the joint characteristic functions of finite dimensional distributions of the appropriately centered aggregated process in question.

Pricing energy quanto options in the framework of Markov-modulated additive processes

Abstract Energy quanto options are risk management tools that have a payoff similar to the product of the payoffs of two options, each written on an energy-related underlying. These options, as opposed to standardized contracts that only account for price risk, are designed to manage both volumetric and price risk in energy markets. Since the use of such options enables actors in the energy market also to hedge against production volume risk, they are becoming very popular. This paper considers the valuation of such an option on futures when the underlying futures prices are governed by Markov-modulated additive processes, which have independent but non-stationary increments within each regime. We derive a valuation formula by using the Fast Fourier Transform (FFT) technique under the assumption that the joint characteristic function of the log-futures prices is known analytically. We study this approximation under different regime-switching models. Several numerical case studies illustrate that our FFT-based valuation has a high precision and is much faster than Monte Carlo estimates.

A closed-form pricing formula for catastrophe equity options

In this article, we derive a closed-form pricing formula for catastrophe equity put options under a stochastic interest rate framework. A distinguishing feature of the proposed solution is its simplified form in contrast to several recently published formulae that require evaluating several layers of infinite sums of $n$-fold convoluted distribution functions. As an application of the proposed formula, we consider two different frameworks and obtain the closed-form formula for the joint characteristic function of the asset price and the losses, which is the only required ingredient in our pricing formula. The prices obtained by the newly derived formula are compared with those obtained using Monte-Carlo simulations to show the accuracy of our formula.

Efficiency large deviation function of quantum heat engines

The efficiency of small thermal machines is typically a fluctuating quantity. We here study the efficiency large deviation properties of two exemplary quantum heat engines, the harmonic oscillator and the two-level Otto motors. To this end, we analytically compute their joint characteristic functions for heat and work based on the two-projective-measurement approach. We investigate work–heat correlations within the respective engine cycles and find, for generic scale-invariant quantum heat engines, that work and heat are perfectly anticorrelated for adiabatic driving. In this limit, the effects of thermal as well as quantum fluctuations are suppressed, the large deviation functions are singular and the stochastic efficiency is equal to the macroscopic efficiency.

Weighted naïve Bayes text classification algorithm based on improved distance correlation coefficient

This paper proposes an innovative method to improve the attribute weighting approaches for naive Bayes text classifiers using the improved distance correlation coefficient. The resulted model is called improved distance correlation coefficient attribute weighted multinomial naive Bayes, denoted by IDCWMNB. Unlike the traditional correlation statistical measurements that consider the cumulative distribution function of random vectors, the improved distance correlation coefficient tests the joint correlation of random vectors by describing the distance between the joint characteristic function and the product of the marginal characteristic functions. Specifically, a measurement of inverse document frequency that considers the distribution information of document concentrating and scattering has been proposed. Then, the measurement and the distance correlation coefficient between attributes and categories have been combined to measure the importance of attributes to categories, to allocate different weights to different terms. Meanwhile, the learned attribute weights are incorporated into the posterior probability estimates of the multinomial naive Bayes model, which is known as deep attribute weighting. This measurement is more effective than the traditional statistical measurements in the presence of nonlinear relationship between two random vectors. Experimental results taking benchmark and real-world data indicate that the new attribute weighting method can achieve an effective balance between classification accuracy and execution time.

Maximal‐ratio combining detection in massive multiple‐input multiple‐output systems with accurate probability distribution function

This letter derives the probability distribution function (PDF) of received symbols for orthogonal frequency-division multiplexing (OFDM) based massive multiple-input, multiple-output (MMIMO) systems, which uses maximal-ratio combining (MRC) detection. The effects of noise and interferences are evaluated through random variables, and the PDF is then derived from their joint probability and characteristic functions. The bit error rate (BER) using Binary and Quadrature Phase Shift Keying (BPSK, QPSK), and M-ary Quadrature Amplitude Modulation (QAM) waveforms is then analyzed by using this PDF. Simulation and analytical results confirm that the derived equations provided accurate PDF and BER, and therefore, can be used efficiently to evaluate performance of OFDM-MMIMO systems.

FFT-network for bivariate Lévy option pricing

We propose a two-dimensional fast Fourier transform (FFT) network to retrieve the prices of options that depend on two Levy processes. Applications include, but are not limited to, the valuation of options on two stocks under the Levy processes, and options on a single stock under a random time-change Levy process. The proposed numerical scheme can be applied to different multivariate Levy constructions such as subordination and linear combination provided that the joint characteristic function is available. The proposed FFT-network can be thought of as a lattice approach implemented through the characteristic function. With the prevalent implementation of FFT, the network approach results in significant computational time reduction while maintaining satisfactory accuracy. Furthermore, we investigate option pricing on a single asset where the asset return and its volatility are driven by a pair of dependent Levy processes. Such a model is also called the random time-changed Levy process. Numerical examples are given to demonstrate the efficiency and accuracy of FFT-network applied to exotic and American-style options.

Rainbows and Transforms: Semi-Analytic Formulae

In this work we show how the techniques introduced by Hurd and Zhou can be used to derive a pricing framework for rainbow options by using the joint characteristic function of the logarithm of the underlying assets. Semi-analytic formulae will be achieved by splitting the symbol, applying different dampings to its constituent components, and finally calculating the Fourier transforms of said components allowing the pricing of call and put options on the maximum and minimum of multiple assets; we shall also revisit Margrabe’s classical formula. Our work extends a number of classical results that focus on Geometric Brownian motion dynamics, and provides an alternative pricing methodology to that proposed in some of the most recent literature.

Tests for validity of the semiparametric heteroskedastic transformation model

There exist a number of tests for assessing the nonparametric heteroskedastic location-scale assumption. The goodness-of-fit tests considered are for the more general hypothesis of the validity of this model under a parametric functional transformation on the response variable, specifically testing for independence between the regressors and the errors in a model where the transformed response is just a location/scale shift of the error is considered. The proposed criteria use the familiar factorization property of the joint characteristic function under independence. The difficulty is that the errors are unobserved and hence one needs to employ properly estimated residuals in their place. The limit distribution of the test statistics is studied under the null hypothesis as well as under alternatives, and also a resampling procedure is suggested in order to approximate the critical values of the tests. This resampling is subsequently employed in a series of Monte Carlo experiments that illustrate the finite-sample properties of the new test. The performance of related test statistics for normality and symmetry of errors is also investigated, and application of our methods on real data sets is provided.

Characterizations of random walks on random lattices and their ramifications

We analyze the behavior of a particle moving along a d-dimensional lattice and representing a “doubly stochastic” random walk. Unlike the classical random walk, the lattice is randomly generated upon particle’s landing at a node. At any time, the particle is enclosed in a rectangular cylinder (i.e., with a bounded a-dimensional rectangle Ra at its base) it attempts to escape. Thus, the particle moves from one node to another at random epochs of time and, with some probability, leaves The particle may jump arbitrarily far beyond the boundary at its passage time. Furthermore, the particle’s location is not assumed to be known in real time, but only upon certain random epochs Of a key interest is the location of the particle at any time, where is the linear interpolation of particle’s positions at times τn’s. If denotes the first observed escape time (virtual first passage time), where we target the joint characteristic function of and itself, where or in tractable forms, thereby attempting to enhance as much as possible the probabilistic data lost due to the crudeness of the observations and to couple and with deterministic time intervals Among various applications, we discuss and treat antagonistic games of two active and several passive players as well as situations that occur in complex queueing systems.

On the Ratio of Two Correlated Complex Gaussian Random Variables

In this letter, we derive new results for the statistics of the ratio of two complex Gaussian random variables (RVs), where the numerator and denominator may have arbitrary means and are possibly correlated. Exact expressions are derived for the joint probability density function (pdf) of the real and imaginary parts, for the joint pdf of the amplitude and phase, and also for the joint characteristic function (cf) of the real and imaginary parts, which generalize the existing results. Then, we show an example application of the derived pdf to the symbol error probability (SEP) analysis for a single antenna communication system with imperfect channel state information (ICSI).