Long range dependence, no arbitrage and the Black-Scholes formula

A bond and stock model is considered where the driving process is the sum of a Wiener process W and a continuous process Z with zero generalized quadratic variation. By means of forward integrals a hedge against Markov type claims is contructed. If Z is independent of W under some natural assumptions on Z and the admissible portfolio processes the model is shown to be arbitrage free. The fair price of the above claims appears to be the same as in the classical case Z ≡ 0. In particular, the Black Scholes formula remains valid for non semimartingale models with long range dependence. 0 Introduction A bond and stock model with price processes B(t) = B(0) exp    t ∫ 0 r(s) ds    and S(t) = S(0) exp    t ∫ 0 ( b(s)− σ2/2)ds + σ(W (t) + Z(t))    on the interval [0, T ] is considered. W is a Wiener process and Z is a continuous process with vanishing generalized quadratic variation. The interest rates r(t) and the mean rates of return b(t) are bounded measurable processes adapted to the ltration given by Y := W + Z. Related stochastic di erential equations are understood in the sense of stochastic forward integrals as studied in [12] [15]. 2 M. Zähle The case Z ≡ 0 corresponds to the classical model which has been investigated within martingale theory. In particular, the well-known Black Scholes formula for prices of options on the stock relies on martingale properties. In the present paper a process Z is added which needs not be a semimartingale. This also permits long range dependence in the stock price development. The corresponding portfolio valuing process X is introduced in Section 2 by means of self nancing strategies. If r is càglàd, a similar representation as known from semimartingale theory holds true (Proposition 2). For constant r in Theorem 1 a hedge against a terminal claim of the form C = h(S(T )) for a continuous function h with at most polynomial growth is constructed. It is determined by means of the same partial di erential equation as used for the case Z ≡ 0. The essential tool is the Itô formula for the forward integral which holds also for non semimartingale processes with generalized quadratic variations. In order to treat the problem of arbitrage and pricing a claim the conditions on Z ind Section 4 are further restricted. It is supposed that Z is independent of W and has with probability 1 (w.p.1) fractional derivatives of order α in L2 for some α > 1/2. Balls of radius ε are now determined by the sum of the L∞ norms of the trajectories of Z and of the L2 norms of their fractional derivatives. We suppose that for every ε > 0 the ε ball has positive probability. An example for Z is fractional Brownian motion with Hurst exponent H > 1/2. For such integrators Z the forward integral has nice continuity properties. (The relationship between the type of stochastic integration and presence or absence of arbitrage is discussed in Section 4.) Under some natural conditions on the portfolio processes (fractional di erentiability of order 1− α and a certain weak continuity w.r.t. the stock price development) we prove in Theorem 2 for deterministic rates r(t) and b(t) that there is no arbitrage opportunity. Moreover, Theorem 3 shows that the fair price of a claim C = h(S(T )) basing on strongly admissible portfolios of the above type is given by the corresponding value for Z ≡ 0 if r is constant. In particular, the classical Black Scholes formula for European call options remains valid for our extended arbitrage free model with long range dependence. (The situation is similar to the case when working with equivalent risk neutral probability measures.) In Theorem 3 it is also shown that the hedge process constructed in Theorem 1 under the additional assumptions on Z is strongly admissible in the above sense. 1 The bond and stock model On a basic probability space [Ω,F ,P] a Wiener process W and a (not necessarily independent) continuous process Z with vanishing generalized quadratic process [Z] and Z(0) = 0 are given. We consider a nite time interval [0, T ]. Long range dependence, no arbitrage and the Black-Scholes formula 3 The above bracket is de ned for continuous Z by [Z](t) := lim ε→0 (ucp) 1 ∫