By Stefan Rostek
The clinical debate of contemporary years approximately alternative pricing with admire to fractional Brownian movement was once fascinated by the feasibility of the no arbitrage pricing process. because the unrestricted fractional industry surroundings allows arbitrage, the traditional reasoning is that fractional Brownian movement doesn't qualify for modeling cost process.
In this publication, the writer issues out that arbitrage can merely be excluded in case that marketplace costs circulation a minimum of a little bit quicker than any industry player can react. He clarifies that non-stop tradability consistently gets rid of the danger of the fractional cost technique, regardless of the translation of the stochastic fundamental as an quintessential of Stratonovich or Itô type.
Being left with an incomplete marketplace atmosphere, the writer indicates that alternative valuation with admire to fractional Brownian movement might be solved by means of utilizing a probability choice dependent strategy. The latter offers us with an intuitive closed-form resolution for ecu innovations in the fractional context.
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Additional resources for Option Pricing in Fractional Brownian Markets
Sample text
By name, we state fractional versions of the Girsanov theorem and the Itˆo formula. We only provide the results and skip all proofs referring to the relevant literature. For a summarizing discussion of the topic, see Bender (2003a). 26 2 Fractional Integration Calculus The classical Girsanov theorem discusses the properties of classical Brownian motion—or more generally classical Brownian integrals—under change of measure. It gives the possibility of changing a Brownian motion with drift into one without any drift.
Duncan et al. (2000) show that for any p ≥ 0, the random variables of Lp can be approximated with arbitrary exactness by linear combinations of so-called Wick exponentials ε(f ) that are defined via fractional integrals with deterministic integrands f : ∞ ε(f ) := exp 0 1 f (t)dBtH − |f |2ϕ , f ∈ L2ϕ . 2 Note that such an exponential is of course a random variable. As the stochas∞ tic integral 0 f (t) dBtH is normally distributed with zero mean and vari∞ ance |f |2ϕ (see Gripenberg and Norros (1996), p.
Corresponding to Chap. 2, we have the following representation of the fractional Brownian motion: B0H = 0, B1H = k(1, 1)ξ1 , B2H = k(2, 1)ξ1 + k(2, 2)ξ2 . 46 3 Fractional Binomial Trees Therefore, the Brownian increments dBj = Bj − Bj−1 are dB1H = k(1, 1)ξ1 , dB2H = (k(2, 1) − k(1, 1))ξ1 + k(2, 2)ξ2 . With S0 = 1, the price process of the geometric fractional Brownian motion (P ) in the pathwise sense, denoted by St , develops as follows: (P ) S0 =0 (P ) S1 (P ) S2 = S0 (1 + dB1H ) = 1 + k(1, 1)ξ1 , (P ) (P ) = S1 (1 + dB2H ) = (1 + k(1, 1)ξ1 ) (1 + (k(2, 1) − k(1, 1))ξ1 + k(2, 2)ξ2 ) = 1 + k(2, 1)ξ1 + k(2, 2)ξ2 + k(1, 1)k(2, 2)ξ1 ξ2 +k(1, 1)(k(2, 1) − k(1, 1))ξ12 .