This paper deals with the problem of pricing European currency options in the mixed fractional Brownian environment. Both the pricing formula and the mixed fractional partial differential equation for European call currency options are obtained. Some Greeks and the estimator of volatility are also provided. Empirical studies and simulation results confirm the theoretical findings and show that the mixed fractional Brownian pricing model is a reasonable one.

Pricing currency options in the mixed fractional Brownian motion

In this paper we propose a general foreign equity option pricing framework that unifies the vast foreign equity option pricing literature and incorporates the stochastic volatility into foreign equity option pricing. Under our framework, the time-changed Levy processes are used to model the underlying assets price of foreign equity option and the closed form pricing formula is obtained through the use of characteristic function methodology. Numerical tests indicate that stochastic volatility has a dramatic effect on the foreign equity option prices.

Pricing foreign equity option with stochastic volatility

The evaluation of geometric Asian power options under time changed mixed fractional Brownian motion

Parameter identification in mixed Brownian–fractional Brownian motions using Powell's optimization algorithm

Empirical evidence suggest that fractional stochastic based models are well suited for modelling systems and phenomenons exhibiting memory and hereditary properties. Assuming that the stock market exhibits some unexplained memory structures, described by a non-random fractional stochastic process governed under a standard Brownian motion, we derive a time-fractional Black-Scholes (tfBS) partial differential equation for pricing option contracts on such stocks. We further propose a corresponding robust numerical method which is based on the extension of a Crank Nicholson finite difference method for solving tfBS-PDEs. Through rigorous theoretical analysis, we established that the method is unconditionally stable and convergent up to order O ( k 2 + h 2 ) . Two numerical examples are presented using realistic market parameters. Our results confirm theoretical observations and general consensus in literature that, stock market dynamics are of a power law nature and follow heavy tailed distributions with memory.

A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics

In this paper, we construct equity warrants pricing model under Fractional Brownian motion, deduce the European options pricing formula with a simple method, then propose the warrants pricing formula, and extend it to cover equity warrants on a stock providing dividends. Finally, taking Changdian warrant in Chinese stock market as an example, we illustrate that the results based on the new warrants pricing formula is more accuracy than the classical results based on traditional pricing model.

Wei-Lin Xiao

Papers

Equity warrants pricing model under Fractional Brownian motion and an empirical study