Exact results for the roughness of a finite size random walk

We consider the role of finite size effects on the value of the effective Hurst exponent H. This problem is motivated by the properties of the high-frequency daily stock-prices. For a finite size random walk we derive some exact results based on Spitzer's identity. The conclusion is that finite size effects strongly enhance the value of H and the convergency to the asymptotic value (H=12) is rather slow. This result has a series of conceptual and practical implication which we discuss.