# If price is a random walk, is ok to use the binomial distribution to estimate a trading strategy?

Is it OK to assume a trading strategy should have a binomial distribution if the price is just a random walk?
using p of the event as:
$$\frac{AverageStopLossPercent}{AverageStopLossPercent + AverageTakeProfitPercent}$$

More details in case this is an XY problem:
I want to test some strategy (I think is not important which one, suppose just a normal RSI indicator), if I set the take profit to 1% and stop loss to 1%, and I execute 30 trades, what is the probability of getting 20 wining trades in a random walk market?

What I'm trying to calculate is the chance of a false positive. Because in that example 20/30 = 66%. And if the market is just random I guess all strategies will have 50% (if TP/SL ratio is 1:1). So at first it looks like is not a random walk but the strategy found a pattern. But how can we calculate if this is a false positive? I mean, if we flip a coin 30 times we could have 20 heads too, even in a fair coin right?

I was thinking in using the binomial as reference, in that example n=30, p=0.5, and k=20, it gives the probability of k to n (k>=20) of 2%
So we could say there are 2% of chance the strategy is giving a false positive?

It is a good idea to make an assumption of "no informational content" on prices to have a reference level for this $$H_0$$ hypothesis. The best is probably to make Monte-Carlo simulations, i.e. to simulate as many random walks as possible and to record the full distribution of the payoff, hence you will obtain a probability that your strategy is better than this $$H_0$$ one by simply counting the percentage of Monte-Carlo simulations that make worst than your idea.

Let me be more accurate:

1. you take the historical time series of prices your strategy uses
2. you estimate its covariance matrix and trend
3. using these two statistics you simulate 10,000 compatible price trajectories
4. you apply your strategy on them
5. you obtain 10,000 PnL
6. you compute as many scores on it as you want (like the Sharpe Ratio)
7. you score the score of your strategy vs this distribution of 10,000 scores, i.e. the percentage of these strategies that are worse than yours
8. the closer to 100%, the better.

Last remark: you can replace steps 2 and 3 by any other generative model, like bootstrap or GAN/autoencoder (since machine learning has to be cited in any answer nowadays!).

• thanks! I will search more about the sharpe ratio and bootstrap etc. But can you clarify why is not a good idea just to use the binomial distribution? I mean if is random walk should be just like flipping a coin right? also can you point me in the right direction for steps 2 and 3? maybe some good resource/tutorial to create those simulated prices – Enrique Sep 15 '19 at 15:13
• binomial is good, but will never provide you the flexibility of Monte-Carlo (especially if you want to preserve volatility or covariances). – lehalle Sep 15 '19 at 15:15
• but I think something is wrong, because there are some strategies giving 70% of good trades, using TP/SL ratio = 1, combining long and short, and with 100 trades. If I calculate the binomial with p=0.5, n=100 and k=70, then it says is almost impossible to get that for example flipping a coin. That means the strategy is by sure not flipping a coin and doing something good? and hence the price is not random walk? – Enrique Sep 15 '19 at 15:27
• A random walk is different from brownian motion. Prices do not go up and down by the same increment each timestep. The increments are Normally distributed so they will each have a different size. – roz Sep 15 '19 at 16:56
• it is true @roz , nevertheless it is possible to design random walks with different increments. In any case a Monte-Carlo simulation (or a bootstrapping of time series) seems to be more adequate to simulate prices. – lehalle Sep 15 '19 at 18:01

What you are describing is not a binomial. It is incidentally expressed in percentages but percentages are not what makes something a binomial process. While I have no idea what the likelihood function is here, what you are creating is some form of ratio distribution. It wouldn't be very helpful because your model doesn't show how it is generated. What you are seeing are results.