Optimal trading algorithms and selfsimilar processes: a p-variation approach

Almgren and Chriss ("Optimal execution of portfolio transactions", Journal of Risk, Vol. 3, No. 2, 2010, pp. 5-39) and Lehalle ("Rigorous strategic trading: balanced portfolio and mean reversion", Journal of Trading, Summer 2009, pp. 40-46.) developed optimal trading algorithms for assets and portfolios driven by a brownian motion. More recently, Gatheral and Schied ("Optimal trade execution under geometric brownian motion in the Almgren and Chriss framework", Working paper SSRN, August 2010) addressed the same problem for the geometric brownian motion. In this article we extend these ideas for assets and portfolios driven by a discrete version of a selfsimilar process of exponent H in (0,1), which can be either a fractional brownian motion of Hurst exponent H or a truncated Lévy distribution of index 1/H. The cost functional we use is not the classical expectation-variance one: instead of the variance, we use the p-variation, i.e. the Lp equivalent of the variance. We find explicitly the trading algorithm for any p>1 and compare the resulting trading curve (that we call p-curve) with the classical expectation-variance curve (the 2-curve). If p2 then the p-curve is above the 2-curve at the beginning of the execution and below at the end. Therefore, this pattern minimizes the market impact. We also show that the value of p in the p-variation is related to the exponent H of selfsimilarity via p=1/H. In consequence, one can find the right value of p to put into the trading algorithm by calibrating the exponent H via real time series. We believe this result is interesting applications for high-frecuency trading.