Optimal Trend Following Trading Rules
The first rigorous justification of trend following: in a bull/bear regime-switching market, the provably optimal rule buys when the filtered probability of a bull market crosses an upper threshold and sells below a lower one.
The stock's drift switches between a bull state and a bear state via an unobservable two-state Markov chain. The Wonham filter turns the hidden regime into an observable quantity: the conditional probability p of being in a bull market, which rises as the price rises — so the optimal rule is mathematically "buy high, sell higher," genuine trend following rather than contrarian trading.
Solving the HJB variational inequalities with transaction costs splits the state space into a buy region, a sell region and a no-trade band, separated by smooth threshold curves on p. Calibrated to historical index statistics, the thresholds are roughly p* ≈ 0.95 to buy and p* ≈ 0.80 to sell.
In Monte-Carlo simulation the rule earns about 11.4% annualized versus 4.4% for buy-and-hold, and it dominates buy-and-hold on nearly every simulated path. Ex-ante tests confirm the edge: S&P 500 (1972–2011) 11.0% versus 9.8% buy-and-hold, and Shanghai Composite (2001–2011) 14.0% versus 2.6%.
References
- Dai, M., Zhang, Q. & Zhu, Q. J. (2010). "Trend Following Trading under a Regime Switching Model." SIAM Journal on Financial Mathematics, 1(1), 780–810. https://epubs.siam.org/doi/10.1137/090770552
- Working paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1630903