ANALYTICAL PATH-INTEGRAL PRICING OF DETERMINISTIC MOVING-BARRIER OPTIONS UNDER NON-GAUSSIAN DISTRIBUTIONS

2020 ◽  
Vol 23 (01) ◽  
pp. 2050005
Author(s):  
ANDRÉ CATALÃO ◽  
ROGÉRIO ROSENFELD
Keyword(s):  
Path Integral ◽  
First Passage Time ◽  
Passage Time ◽  
Barrier Options ◽  
Barrier Option ◽  
Entropy Model ◽  
Gaussian Distributions ◽  
Pricing Options ◽  
Barrier Option Pricing ◽  
Non Gaussian

In this work, we present an analytical model, based on the path-integral formalism of statistical mechanics, for pricing options using first-passage time problems involving both fixed and deterministically moving absorbing barriers under possibly non-Gaussian distributions of the underlying object. We adapt to our problem a model originally proposed by De Simone et al. (2011) to describe the formation of galaxies in the universe, which uses cumulant expansions in terms of the Gaussian distribution, and we generalize it to take into account drift and cumulants of orders higher than three. From the probability density function, we obtain an analytical pricing model, not only for vanilla options (thus removing the need of volatility smile inherent to the Black & Scholes (1973) model), but also for fixed or deterministically moving barrier options. Market prices of vanilla options are used to calibrate the model, and barrier option pricing arising from the model is compared to the price resulted from the relative entropy model.

scholarly journals First passage dynamics of stochastic motion in heterogeneous media driven by correlated white Gaussian and coloured non-Gaussian noises

2021 ◽  
Author(s):  
Nicholas Mwilu Mutothya ◽  
Yong Xu ◽  
Yongge Li ◽  
Ralf Metzler ◽  
Nicholas Muthama Mutua
Keyword(s):  
Diffusion Coefficient ◽  
Gaussian Noise ◽  
First Passage Time ◽  
Passage Time ◽  
Mean First Passage Time ◽  
Stochastic Motion ◽  
First Passage ◽  
The Mean ◽  
Non Gaussian ◽  
Markovian Systems

Abstract We study the first passage dynamics for a diffusing particle experiencing a spatially varying diffusion coefficient while driven by correlated additive Gaussian white noise and multiplicative coloured non-Gaussian noise. We consider three functional forms for position dependence of the diffusion coefficient: power-law, exponential, and logarithmic. The coloured non-Gaussian noise is distributed according to Tsallis' $q$-distribution. Tracks of the non-Markovian systems are numerically simulated by using the fourth-order Runge-Kutta algorithm and the first passage times are recorded. The first passage time density is determined along with the mean first passage time. Effects of the noise intensity and self-correlation of the multiplicative noise, the intensity of the additive noise, the cross-correlation strength, and the non-extensivity parameter on the mean first passage time are discussed.

scholarly journals Universal First-Passage-Time Distribution of Non-Gaussian Currents

2019 ◽  
Vol 122 (23) ◽  
Author(s):  
Shilpi Singh ◽  
Paul Menczel ◽  
Dmitry S. Golubev ◽  
Ivan M. Khaymovich ◽  
Joonas T. Peltonen ◽  
...  
Keyword(s):  
Time Distribution ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Non Gaussian

scholarly journals A Semigroup Expansion for Pricing Barrier Options

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Takashi Kato ◽  
Akihiko Takahashi ◽  
Toshihiro Yamada
Keyword(s):  
Expansion Method ◽  
Stochastic Volatility Model ◽  
Approximation Formula ◽  
Parabolic Partial Differential Equations ◽  
Barrier Options ◽  
Barrier Option ◽  
Volatility Model ◽  
Barrier Option Pricing ◽  
Asymptotic Expansion Method ◽  
Expansion Scheme

This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develop a semigroup expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments.

ESCAPE THROUGH A FLUCTUATING ENERGY BARRIER IN THE PRESENCE OF NON-GAUSSIAN NOISE

2007 ◽  
Vol 07 (02) ◽  
pp. L151-L161 ◽  
Author(s):  
GURUPADA GOSWAMI ◽  
PRADIP MAJEE ◽  
BIDHAN CHANDRA BAG
Keyword(s):  
Gaussian Noise ◽  
First Passage Time ◽  
Passage Time ◽  
Mean First Passage Time ◽  
First Passage ◽  
Resonant Condition ◽  
Non Gaussian ◽  
Turnover Behavior ◽  
Resonant Activation ◽  
Limiting Value

In this paper we have studied how barrier crossing dynamics is affected by colored additive non-Gaussian noise if the barrier fluctuates deterministically. Our investigation indicates that resonant activation(RA) is either enhanced or it becomes robust if noise characteristic is deviated from the Gaussian behavior. We find that additive colored non Gaussian noise can induce the RA-like phenomenon. Another interesting observation is that the turnover behavior persists even in presence of barrier fluctuations at finite rate. Finally, it is observed that mean first passage time(MFPT) decreases with increase of non-Gaussian characteristic of the additive colored noise for a given noise strength and noise correlation time and ultimately reaches to a limiting value. The limiting value remains almost the same if the barrier fluctuating frequency is zero or far from the resonant condition. But near the resonant condition the mean first passage time initially decreases and then increases passing through a minimum as the non Gaussian parameter grows.

Population system with coupling between non-Gaussian and Gaussian colored noise under Allee effect

2018 ◽  
Vol 32 (24) ◽  
pp. 1850279 ◽  
Author(s):  
Yachao Yang ◽  
Dongxi Li
Keyword(s):  
Allee Effect ◽  
Colored Noise ◽  
First Passage Time ◽  
Planck Equation ◽  
Single Species ◽  
Passage Time ◽  
Population System ◽  
Stationary Probability Distribution ◽  
Gaussian Colored Noise ◽  
Non Gaussian

We investigate a stochastic model for single species population growth with strong and weak Allee effects subjected to coupling between non-Gaussian and Gaussian colored noise as well as nonzero cross-correlation in between. Stationary probability distribution of population model is obtained depending on the Fokker–Planck equation. The mean first-passage time is also calculated in order to quantify the time of transition between survival state and extinction state with Allee effect in population. The intensity of non-Gaussian colored noise can induce phase transition, and population may be vulnerable to extinction due to the increase in the intensity of non-Gaussian colored noise. Whether Allee effect is strong or weak, the increase in Allee threshold will not contribute to the survival and stability of the population. Further, the phenomenon of resonant activation is firstly discovered in the study of population dynamics with Allee effect. These behaviors can be interpreted on the basis of a biological model of population evolution.

scholarly journals Semi-Analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab® codes)

2018 ◽  
Vol 9 (1) ◽  
pp. 42-67 ◽  
Author(s):  
C. Guardasoni
Keyword(s):  
Analytical Method ◽  
Boundary Integral ◽  
Time Dependent ◽  
Barrier Options ◽  
Barrier Option ◽  
Boundary Integral Methods ◽  
Integral Methods ◽  
Interest Rate Volatility ◽  
Black Scholes ◽  
Barrier Option Pricing

Abstract A Semi-Analytical method for pricing of Barrier Options (SABO) is presented. The method is based on the foundations of Boundary Integral Methods which is recast here for the application to barrier option pricing in the Black-Scholes model with time-dependent interest rate, volatility and dividend yield. The validity of the numerical method is illustrated by several numerical examples and comparisons.

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