Trend-Following PnL Under a Random-Walk Price Model
Summary
The document asks how to characterize the profit and loss of a strategy that buys after an uptick or shorts after a downtick, then holds the position. It assumes a stock begins at a fixed price and moves up or down by equal increments with equal probability. The discussion contrasts a single-share entry after the first move with a repeated trading scheme that changes the position as the stock moves.
One answer treats the terminal PnL as a mixture of conditional price changes and concludes it has a discretized normal-like distribution with zero mean. Another derives PnL from the changing position and argues it is related to the square of the terminal price, yielding an adjusted chi-squared-like distribution under a normal approximation. The answers disagree because they interpret the trading rule differently. The document provides no empirical test, and its approximations depend on the assumed independent price moves; it notes that trend-following rationale generally relies on positive autocorrelation.
Key ideas
- The strategy’s PnL distribution depends on whether the position is opened once or adjusted after every price move.
- Under independent, symmetric price changes, the first answer expects no directional edge from following the initial move.
- A repeated position adjustment can make PnL depend on the square of the terminal price.
- The proposed Gaussian and chi-squared descriptions are approximations, not empirical findings.
- Trend-following would require price persistence that the independent-move assumption excludes.
Tags
Full text
# What is the distribution of the trend-following strategy PnL?
# What is the distribution of the trend-following strategy PnL?
Suppose you start with zero dollars; and a stock is at \$100 and goes up and down \$1 equally likely, i.e., both with probability 50%.
A trend-following strategy, during a period of 31 days, works as follows:
- If the stock goes up you buy one share and sell it at the end of the 30-day period.
- If the stock goes down you sell (short if necessary) one share and buy it back at the end of the 30-day period.
What is the distribution of your PnL at the end of the 31-day period?
## Answer by lehalle (score 1, accepted)
https://quant.stackexchange.com/a/69277
I do not really understand the purpose of the question but the answer (if I understood the question) is straitforward:
- event +1: given the price goes up (at \$101) you buy and wait 30 days, the distribution of the PnL is a discretized Gaussian ${\cal N}(0, 30)$ because the price at $T=31$ is centered at \$101 and with a std of $\sqrt{30}$.
- event -1: given the price goes up (at \$99) you sell and wait 30 days, the distribution of the PnL is a discretized Gaussian ${\cal N}(0, 30)$ because the price at $T=31$ is centered at \$99 and with a std of $\sqrt{30}$.
Since these events occur with the same probs of $1/2$, the PnL is a mixture of these two independent Gaussians, it is itself a Gaussian ${\cal N}(0, 30)$.
I do not understand the purpose of the question because if you play a trend, it should be because you believe the price dynamics have a positive auto-correlation, but here you assume they are independent.
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/69678
The scheme you describe implies that the number of shares in position at any time t is equal to the current stock price minus 100, since you buy another share every time it upticks and sell one every time it downticks. Therefore the cumulative p/l is of the form $\Sigma{(S_t-100)dS_t}$ and if you sum this you should get that the final p/l is of the form $S_T^2 - E[S_T^2]$. If $S_T$ is normally distributed, which your scheme would approximately satisfy , then the p/l is a chi squared distribution with an adjusted mean such that the expected p/l is zero.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.