Abstract
The pricing problem of geometric average Asian option under fractional Brownian motion is studied in this paper. The partial differential equation satisfied by the option’s value is presented on the basis of no-arbitrage principle and fractional formula. Then by solving the partial differential equation, the pricing formula and call-put parity of the geometric average Asian option with dividend payment and transaction costs are obtained. At last, the influences of Hurst index and maturity on option value are discussed by numerical examples.
Highlights
Option pricing theory has been an unprecedented development since the classic Black-Scholes option pricing model [1] was proposed
Since fractional Brownian motion has the properties of self-similarity, thick tail, and long-term correlation, that fractional Brownian motion has become a good tool to depict the process of underlying asset price
Because an Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter [8] was obtained by Christian Bender, it brought great convenience to option pricing
Summary
Option pricing theory has been an unprecedented development since the classic Black-Scholes option pricing model [1] was proposed. Vecer and Xu [4] extended the method of removing path correlation to the case in a semimartingale model and obtained the partial differential equation of the option value under the standard Brownian motion. Guasoni [10] studied the standard option with transaction costs under the fractional Brownian motion, but he did not obtain option pricing formula. Asian option pricing problems with transaction costs and dividends under fractional Brownian motion are studied. 2. Geometric Average Asian Options Pricing Model under Fractional Brownian Motion. Suppose that the underlying asset price St satisfied (3); the value of the geometric average Asian call at time t (0 ≤ t ≤ T), V(t, Jt, St), satisfies the following mathematical model:. For a single European Asian option, (13) and (15) can be represented as (δt)H−1)
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