In this paper we show that optimal trading results can be achieved if we can forecast a key summary statistic of future prices

largest change before reversal is recommended to be used as an output in a neural network for the generation of trading signals. When neural network forecasting is applied to a dataset of Hang Seng Index Futures Contract traded in Hong Kong, it is shown that forecasting the largest change before reversal outperforms the k-step-ahead forecast in achieving higher trading pro®ts. Copyright # 2000 John Wiley & Sons, Ltd.

KEYWORDS           ®nance; forecasting; trading rules; largest change before reversal; neural network

`Buy low, sell high' is an investment dictum which is easier said than done. Trading decisions have to be made without the knowledge of future prices and therefore it is impossible to determine whether the current price is a low or a high. For a trading rule to be practicable, it has to satisfy the Markov property as de®ned in Neftci (1991), which is equivalent to saying that trading decisions can only utilize past but not future price information. For the pro®tability of some technical trading rules using past information only, see Taylor (1994) and Corrado and Lee (1992). However, even if we are allowed to make use of future information, it may still be a nontrivial problem to devise an optimal strategy, as we will explain below.

Consider an asset whose prices ¯uctuate from day to day and the closing price on the tth day compounded return on day(t ˆ 0, 1, 2,..., n) is q_t. Let tp. An investment decision is a vector_t ˆ ln q_t be the log-price and r_t ˆ p_t ^ÿd ˆp_tÿ(d11be the continuously_{, d2,..., dn), where}

* Correspondence to: Kin Lam, Department of Finance and Decision Sciences, School of Business, Hong Kong Baptist University, 224 Waterloo Road, Kowloon, Hong Kong.

CCC 0277±6693/2000/010039±14$17.50                   Received July 1997 Accepted December 1998

d_i ˆ 0 or 1, meaning that the investor maintains a `neutral' or a `long' position respectively on day i. Here, we assume that the strategy of short-selling the asset is not available to the investor. This restriction can be lifted by allowing d_i ˆ ÿ1, but we will carry out the discussion here using the simpli®ed version of restricting d_i to be 0 or 1. Assuming that the investor starts with a certain sum of money and is fully invested when a `long' position is taken, the investment return corresponding to the decision d is given by r ˆ Pniˆ1 ridi. Assuming that r1,..., rn are known, r can be maximized by choosing di ˆ 1, if ri 40 and di ˆ 0, if ri 40.

The optimization problem becomes non-trivial when transaction cost is taken into account. Whenever the investment decision d^i‡1 is such that dⁱ 6ˆ d^i‡1, a transaction cost of 100c% is incurred. Assuming that the investor has a neutral position in the beginning and in the end, i.e. d₀ ˆ d_n‡1 ˆ 0, the investment return is given by

r ˆ Xiˆ1 r_id_i ÿ c Xiˆ1 jd_i ÿ d_iÿ1j n n‡1

The mathematical programming problem (*) of maximizing r with given r₁,..., r_n and d₀ ˆ dIn the next section, we give a justi®cation of why (*) is of practical interest and why it has_n‡1 ˆ 0 becomes non-trivial. The solution of (*) will be discussed in later sections of this paper. important implications of forecasting for the generation of trading signals in a ®nancial market.

FORECASTING FOR THE GENERATION OF TRADING SIGNALS

At ®rst sight, the solution of (*) is of no practical interest because the optimal trading decision at

(1991)time t. However, the solution of (*) can give great insights as to which quantity is to be forecasteddepends on the values of ^r_t‡1, ^r_t‡2,... and is hence not Markov in the sense of Neftci for the generation of trading signals. When the market satis®es the weak-form market eciency in the sense of Fama (1971), the e€orts of generating trading signals would be futile. However, according to Brock, Lakonishok and LeBaron (1992), the ecient market hypothesis has come under serious siege in recent years. Serial correlation in returns over various investment horizons are reported for individual stocks as well as for various portfolio of stocks; see, for example, Fama and French (1986), Poterba and Summers (1988), Jegadeesh (1990), and Cutler, Poterba and Summers (1990). The correlations are statistically signi®cant and sometimes