A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results

Baaquie, Belal E.

doi:10.1051/jp1:1997167

Cited by 75 publications

(85 citation statements)

References 13 publications

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“…The fundamental principles governing the financial and economical systems are not completely uncovered. In recent years the physical community has started applying concepts and methods of statistical and quantum physics of complex systems to analyze economical problems [3,4,5,6] (and references therein). The improvement of the BS model itself did not stand still too.…”

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confidence: 99%

Black–Scholes model under subordination

Stanislavsky¹

2003

Physica A: Statistical Mechanics and its Applications

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In this paper we consider a new mathematical extension of the Black-Scholes model in which the stochastic time and stock share price evolution is described by two independent random processes. The parent process is Brownian, and the directing process is inverse to the totally skewed, strictly α-stable process. The subordinated process represents the Brownian motion indexed by an independent, continuous and increasing process. This allows us to introduce the long-term memory effects in the classical Black-Scholes model. Key words:Continuous-time random walk, Brownian motion, Lévy process, Subordination, Fractional calculus, Econophysics PACS: 02.50.-r, 02.50.Ey, 02.50. Wp, 89.90.+n The option trading has the long history. The mission of the options as financial instruments is to protect investors from the stock market randomness. Since the early seventies the option market rapidly became very successive in development. The theoretical study of options was directed in finding a fair and presumably riskless price of these instruments. Without questions, the works of Black and Scholes [1] and Merton [2] are a turning-point in the study. Their method has been proven to be very useful for investors trading in option markets. On the other hand, the approach is fruitful for extending the option pricing theory in many ways. Therefore, nowadays the Black-Scholes (BS) model is very popular in finance.The BS equation is nothing else as a diffusion equation. In fact, their option price formula is a solution of the diffusion equation with the initial and boundary conditions given by the option contract terms. The fundamental principles governing the financial and economical systems are not completely uncovered. In recent years the physical community has started applying concepts and methods of statistical and quantum physics of complex systems to *

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confidence: 99%

Black–Scholes model under subordination

Stanislavsky¹

2003

Physica A: Statistical Mechanics and its Applications

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“…The existence results for di erent types of linear Schrödinger equations can be found in book [22]. Stock options pricing models based on linear Schrödinger equations and their relation to Black-Scholes models are reported in many papers [23][24][25][26][27][28][29]. Among others in the author's previous paper [29], the European call option price based on the linear Schrödinger equation has been calculated.…”

Section: U(t S(t)) = Max{ S(t) − K}mentioning

confidence: 99%

Nonlinear Schrödinger approach to European option pricing

Wróblewski

2017

Open Physics

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This paper deals with numerical option pricing methods based on a Schrödinger model rather than the Black-Scholes model. Nonlinear Schrödinger boundary value problems seem to be alternatives to linear models which better re ect the complexity and behavior of real markets. Therefore, based on the nonlinear Schrödinger option pricing model proposed in the literature, in this paper a model augmented by external atomic potentials is proposed and numerically tested. In terms of statistical physics the developed model describes the option in analogy to a pair of two identical quantum particles occupying the same state. The proposed model is used to price European call options on a stock index. the model is calibrated using the Levenberg-Marquardt algorithm based on market data. A Runge-Kutta method is used to solve the discretized boundary value problem numerically. Numerical results are provided and discussed. It seems that our proposal more accurately models phenomena observed in the real market than do linear models.

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“…For processes involving a stochastic volatility (y = log(V )) the expression of the path integral is more complicated and can be found in [2]. From now on we will just consider the case of a constant volatility.…”

Section: Monte Carlo Simulationsmentioning

confidence: 99%

Hamiltonian and potentials in derivative pricing models: exact results and lattice simulations

Baaquie

Corianò

Srikant

2004

Physica A: Statistical Mechanics and its Applications

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The pricing of options, warrants and other derivative securities is one of the great success of financial economics. These financial products can be modeled and simulated using quantum mechanical instruments based on a Hamiltonian formulation. We show here some applications of these methods for various potentials, which we have simulated via lattice Langevin and Monte Carlo algorithms, to the pricing of options. We focus on barrier or path dependent options, showing in some detail the computational strategies involved.

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A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results

Cited by 75 publications

References 13 publications

Black–Scholes model under subordination

Black–Scholes model under subordination

Nonlinear Schrödinger approach to European option pricing

Hamiltonian and potentials in derivative pricing models: exact results and lattice simulations

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