Superreplication of the Best Pairs Trade in Hindsight

Abstract

For a market with m assets and T discrete trading sessions, Cover and Ordentlich (1998) found that the “Cost of Achieving the Best Rebalancing Rule in Hindsight” is p(T, m) = n1 ···Σ nm=T (n1,T...,nm)(n1/T)(n1 · · · (nm/T)nm. Their super-replicating strategy is impossible to compute in practice. This paper gives a workable generalization: the cost (read: super-replicating price) of achieving the best s−stock rebalancing rule in hindsight is (m/s) p(T, s). In particular, the cost of achieving the best pairs rebalancing rule in hindsight is (m/2) TΣn=0 (T/n) (n/T)n ((T − n)/T)T−n = O( √ T). To put this in perspective, for the Dow Jones (30) stocks, the Cover and Ordentlich (1998) strategy needs a 10,000-year horizon in order to guarantee to get within 1% of the compoundannual growth rate of the best (30-stock) rebalancing rule in hindsight. By contrast, it takes 1,000 years (in the worst case) to enforce a growth rate that is within 1% of the best pairs rebalancing rule in hindsight. For any preselected pair (i, j) of stocks it takes 320 years. Thus, the more modest goal of growth at the same asymptotic rate as the best pairs rebalancing rule in hindsight leads to a practical trading strategy that still beats the market asymptotically, albeit with a lower asymptotic growth rate than the full-support universal portfolio.