4A.9 Scaling in, scaling out and pyramiding

Motivation

Traders rarely trade one fixed size. They scale out (sell part at a first target), pyramid (add to winners), and sometimes average down (add to losers). Each changes the payoff distribution. None changes the sign of the expectancy.

4A.9.1Exposure-time and expectancy#

Setup

Let $\theta_t$ be the position (units held) at time $t$ , chosen using only past prices (a predictable process), bounded, and zero after a stopping time $\tau$ with $\mathbb E\tau<\infty$ . The P&L is the stochastic integral $G=\int_0^\tau\theta_t\,dX_t$ .

Proposition 4A.12expectancy and variance of any sizing rule

With $X_t=\mu t+\sigma W_t$ ,

\mathbb E[G]=\mu\,\mathbb E\!\int_0^\tau\theta_t\,dt,\qquad \mathbb E\Big[\Big(G-\mu\!\int_0^\tau\theta_tdt\Big)^2\Big]=\sigma^2\,\mathbb E\!\int_0^\tau\theta_t^2\,dt. \tag{4A.16}

For $\mu=0$ , $\mathbb EG=0$ and $\operatorname{Var}G=\sigma^2\mathbb E\int_0^\tau\theta_t^2dt$ .

Proof

$G=\mu\int\theta\,dt+\sigma\int\theta\,dW$ . The Itô integral of a bounded predictable integrand up to a stopping time with $\mathbb E\tau<\infty$ has mean zero and, by the Itô isometry, second moment $\mathbb E\int_0^\tau\theta^2dt$ .∎

So the expectancy is drift times average exposure-time (unit-days held), and the noise is volatility times root-mean-square exposure-time. Pyramiding raises exposure after gains; scaling out lowers it after gains; averaging down raises it after losses. With $\mu>0$ constant, adding exposure anywhere adds expectancy in proportion; the question is only where the extra variance lands.

4A.9.2Scaling out versus pyramiding#

Worked example 4A.8three ways to trade two units, no drift

Units of $\sigma$ , stop distance 1, everything from Proposition 4A.6.

Fixed bracket, 2 units, stop −1, target +2: outcomes −2 with probability 2/3, +4 with probability 1/3. Mean 0, variance 8.
Scale out: 2 units, stop −1 on both; at +1 sell one unit and move the stop on the other to breakeven (0); final target +3. From 0, $P(+1\text{ before }-1)=1/2$ . From +1, the remaining unit hits +3 before 0 with probability $1/3$ . Outcomes: −2 (prob 1/2), $+1+0=+1$ (prob 1/3), $+1+3=+4$ (prob 1/6). Mean 0, variance $4\cdot\frac12+1\cdot\frac13+16\cdot\frac16=5$ ; $P(\text{loss})=1/2$ .
Pyramid: 1 unit at 0 with stop −1; at +1 add a second unit and move the common stop to 0; target +3. Outcomes: −1 (prob 1/2), stopped at 0 after adding: $0+(-1)=-1$ (prob 1/3), target: $3+2=+5$ (prob 1/6). Mean 0, variance $1\cdot\frac56+25\cdot\frac16=5$ ; $P(\text{loss})=5/6$ .

All three have zero mean, as they must. Scaling out turns the bracket into a higher-win-rate, lower-variance trade. Pyramiding keeps the same variance as scaling out but concentrates it in a rare large win: it loses five times in six. These are different distributions, and a trader's tolerance for long losing streaks (M03 · 3.8) should choose between them. With positive drift, (4A.16) says the pyramid earns $\mu$ times its expected unit-days, which are concentrated in the paths that are already trending; if drift is stronger after a breakout has confirmed (a momentum hypothesis, M04 · 4.3), the pyramid allocates exposure where the drift is.

Averaging down is the mirror image: exposure rises after losses. With $\mu=0$ it has zero mean and negative skew (frequent small wins, rare large losses), the payoff profile of the martingale betting system. If the adverse move carries information (conditional drift negative after a decline, as under momentum), (4A.16) says its expectancy is negative in proportion to the exposure added in the losing state.