ARBITRAGE WITH FRACTIONAL BROWNIAN MOTION?1 - Helsinki.fi

4 CHRISTIAN BENDER, TOMMI SOTTINEN, AND ESKO VALKEILA
The set of nds-admissible portfolios is denoted A nds . Here nds stands for
‘no doubling strategy’.
As a direct consequence of the fundamental theorem of asset pricing (in
the version of Delbaen and Schachermeyer [9]), we obtain:
Theorem 1. If S is not a semimartingale, then there exists an approximative
arbitrage in the class A si .
Here, an approximative arbitrage is meant in the sense of a ‘free lunch
with vanishing risk’. We refer the reader to [9] for the exact definition
of this notion. The previous theorem implies that one needs to restrict the
class of self-financing strategies, if one wants to consider non-semimartingale
models, for instance the models (fully or partially) driven by a fractional
Brownian motion that we consider in the following sections.
3. Arbitrage in fractional Black-Scholes model
Recall that fractional Brownian motion B H t with Hurst parameter 0 <
H < 1 is a continuous centered Gaussian process with covariance structure
E[B H t B H s ] = 1 2
(
t 2H + s 2H − |t − s| 2H) .
We state some well known properties of the fractional Brownian motion:
1. For H = 1/2 fractional Brownian motion is a classical Brownian motion.
2. If H ≠ 1/2 fractional Brownian motion is not a semimartingale.
3. If H > 1/2, fractional Brownian motion has zero pathwise quadratic
variation along appropriate sequences of partitions, i.e. for all 0 ≤
t ≤ T ,
lim
∑
n→∞
t i ∈π n ; t i ≤t
|B H t i
− B H t i−1
| 2 =: 〈B H 〉 t = 0; P -almost surely.
4. For H > 1/2 fractional Brownian motion has a long memory, in the
sense that the covariance function
ρ H (n) := Cov(B H k
− B H k−1, B H k+n − B H k+n−1)
satisfies ∑ ∞
n=1 |ρ H (n)| = ∞.
From now on B H will always denote a fractional Brownian motion with
H > 1/2. Since Itô’s formula carries over to the forward integral, with the
quadratic variation interpreted in the above pathwise sense, (see [13]), we
obtain
∫ t
f(Bt H ) = f(0) + f ′ (Bu H )dBu H ,
0