Adaptive Wave Models for Sophisticated Option Pricing

Abstract

Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black-Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equation. The nonlinear approach comes in two flavors: for the case of constant volatility, it is defined by a single adaptive nonlinear Schrödinger (NLS) equation, while for the case of stochastic volatility, it is defined by an adaptive Manakov system of two coupled NLS equations. The linear quantum approach is defined in terms of de Broglie's plane waves and free-particle Schrödinger equation. In this approach, financial variables have quantum-mechanical interpretation and satisfy the Heisenberg-type uncertainty relations. Both models are capable of successful fitting of the Black-Scholes data, as well as defining Greeks.

Nonlinear Schrödinger approach to European option pricing

“…However, the general model can predict option prices more precisely. Recently, based on an adaptive market hypothesis as well as Elliott wave market theory, a more general option pricing model based on a nonlinear Schrödinger equation has been proposed in [30,31]. In [32] this model has been solved numerically using a domain decomposition method.…”

“…Therefore, in this paper the nonlinear Schrödinger equation [30] is used to price European call options on index WIG20 listed on the Warsaw Stock Exchange (WSE). The model used in this paper contains an additional quantum potential as well as an internal potential rather than only an internal potential as in [30,31]. The nonlinear boundary value problem is solved numerically using a Runge-Kutta method.…”

Analytical Solutions of the Ivancevic Option Pricing Model With a Nonzero Adaptive Market Potential

“…The Black-Scholes model (1.2) can be applied to a reasonable number of one dimensional option models ascribed to u and S, say for puts/calls and stocks/dividends respectively [2]. As noted in [14,15], one could consider the associated probability density function (PDF) resulting from the backward Fokker-Planck equation using the classical Kolmogorov probability method instead of the market value of an option obtained via the Black-Scholes equation.…”

“…In mathematical finance, the classical Black-Scholes model serves as hallmark financial model; it describes the time-evolution of the market value of financial equity such as stock option [1,2,3]. The basic assumptions under which this classical arbitrage pricing theory is formulated include the following: the asset price S (or the underlying asset) following a geometric Brownian motion (GBM), the drift parameter, µ and the volatility rate, σ are assumed constants, lack of arbitrage opportunities (no risk-free profit), frictionless and competitive markets [4,5,6].…”

Pricing Options on Foreign Currency with a Preset Exchange Rate

“…Much of these innovations are discussed in Gong, Thavanewswaran and Liang [23] where they use partial differential equations for various stochastic diffusion models to study option pricing with the pure jump process, jump diffusion process, stochastic volatility and jump diffusion with stochastic volatility. Ivancevic [24] shifts from the Black and Scholes option pricing equation to a Kolmogorov probability approach to develop an adaptive waveform nonlinear stochastic option pricing model.…”