By Stefan Rostek
The medical debate of modern years approximately choice pricing with appreciate to fractional Brownian movement was once fascinated by the feasibility of the no arbitrage pricing strategy. because the unrestricted fractional industry environment permits arbitrage, the normal reasoning is that fractional Brownian movement doesn't qualify for modeling fee process.
In this publication, the writer issues out that arbitrage can basically be excluded in case that marketplace costs flow at the least a little speedier than any marketplace player can react. He clarifies that non-stop tradability continually gets rid of the chance of the fractional expense strategy, regardless of the translation of the stochastic indispensable as an crucial of Stratonovich or Itô type.
Being left with an incomplete marketplace surroundings, the writer indicates that alternative valuation with recognize to fractional Brownian movement should be solved through making use of a possibility choice dependent strategy. The latter presents us with an intuitive closed-form answer for eu strategies in the fractional context.
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Extra info for Option Pricing in Fractional Brownian Markets
Duncan et al. (2000) now deﬁne the Wick product implicitly on these Wick exponentials by postulating that for two functions f, g ∈ L2ϕ the following 20 2 Fractional Integration Calculus equation holds: ε(f ) ε(g) = ε(f + g). From this deﬁnition on, the Wick product can be extended to random variables in Lp . For an explicit deﬁnition of the Wick product based on a representation using Hermite polynomials, see for example Hu and Øksendal (2003). We stress that the Wick product of two random variables is only deﬁned as a multiplication of two complete random variables and cannot be interpreted in a pathwise sense.
Consequently, even a negative evolution in time step j (that is ξj = −1) cannot invert the upward trend. Hence, a one-step buy-and-hold strategy will always yield a positive gain. The described situation occurs with positive probability, as we are given a ﬁnite number of states (paths). So, we have an arbitrage possibility also in the binomial Wick-based setting. Albeit the existence of arbitrage possibilities within the framework described above seems disillusioning at ﬁrst glance, the situation is not as bad as it seems.
5) where t Mt = and c1 = 2HΓ 0 1 1 c1 s 2 −H (t − s) 2 −H dBsH , 3 −H Γ 2 H+ 1 2 −1 . The process Mt is a martingale with independent increments, zero mean and variance function where EMt2 = c22 t2−2H , cH √ . c2 = 2H 2 − 2H It is called the fundamental martingale. 5) of the Radon–Nikodym derivative by dP a 1 = exp −aMt − a2 c22 t2−2H dP 2 . From this representation it is easy to see that for the case H = 12 we obtain the well-known change of measure formula. The generalization of this drift removal theorem from the simple fractional Brownian motion to fractional integrals can also be done (see Bender (2003b), p.