Call warrants pricing formula under mixed-fractional Brownian motion with Merton jump-diffusion

Abstract

Mixed fractional Brownian motion (MFBM) is a linear combination of a Brownian motion and an independent fractional Brownian motion which may overcome the problem of arbitrage, while a jump process in time series is another problem to be address in modeling stock prices. This study models call warrants with MFBM and includes the jump process in its dynamics. The pricing formula for a warrant with mixed-fractional Brownian motion and jump, is obtained via quasi-conditional expectation and risk-neutral valuation.

Fractional Brownian motion in financial engineering models

“…However, as it will be shown, the transition probability density for stochastic value r(τ ) (16) that is built based on Gaussian measure does not match the solution to the Fokker-Planck equation (18). Let us illustrate this for the case of stochastic equation (16)…”

“…Based on measure (25), let us build a measure for stochastic process r(τ ) that is given by a stochastic equation (16). For this, we shall use the approach for Brownian motion given in [19,25].…”

“…Expression (31) gives transition probability density for the supplementary stochastic process dx(τ ) (30). In order to obtain transition probability density K(r, r 0 , t) of stochastic process dr(τ ) (16) let us use connections between them…”

“…In comparison to Brownian motion, for which increments on time intervals that do not intersect do not correlate, whereas, for fBm, a correlation takes place, which is also called a strong after action. Properties of fBm and its application to a number of applied problems were researched in many works [11][12][13][14][15][16]. In particular, in works [11,12], a generalized Ito formula for fBm is given.…”

Penalty method for pricing American-style Asian option with jumps diffusion process

“…[6] demonstrated that the accuracy of the Black-Scholes model should be improved by including discontinuous jump processes into the conventional stochastic process of geometric Brownian motion (GBM). Notably, [7] pioneered such models for the study of option valuation, and some studies that included jumps in the pricing model are [12][13][14]. Hence, we focused on developing a penalty method for solving a PIDE problem.…”