# Probability of Profit

The Skinny On Options Math

Options involve risk and are not suitable for all investors. Please read Characteristics and Risks of Standardized Options before deciding to invest in options.

This segment features a discussion about the probability of profit, how it applies to trading and the Black-Scholes model and how to use simple math to calculate it. While the TOS platform and Dough can supply these numbers for you, a greater understanding of probability of profit should help your trading.

One of the most important features to look at when placing a trade is the probability of profit. To calculate the probability of profit you need to adjust the strike by the credit received and then determine the probability that the underlying is on the profitable side of that value at expiration. We then see that computing the probability of profit depends on your model for the motion of the underlying. Generally, any model with mean reverting volatility will only disagree with Black-Scholes for very far out of the money options.

Assuming that you are in a fair game, if you could risk \$2 to possibly profit \$1, you would know that the probability of winning was ⅔. This exact argument should apply to binary options according to the efficient market hypothesis. Additionally, if we are willing to approximate defined risk trades in general by assuming they either will be max losers or max winners, then we can do it for spreads also..The probability of profit would be the width of the strikes – credit received divided by the width of the strikes. So assuming that the market is fair then higher probability of profit comes either by reducing the maximum profit or by increasing the maximum loss.

It is also worth noting that the delta of a naked option is very close to it’s probability of profit, and many traders use it as a proxy. Jacob displayed with the two very similar calculations. If you do decide to use delta as a proxy for probability of profit, it is worth noting that it will always slightly overstate the probability of profit since cumulative distribution functions are always increasing.

Managing winners is affected by this too. The symmetry of Brownian motion, something like which is entailed by an efficient market hypothesis, means that once an underlying reaches a price point, it has probability of being above or below there later. Using algebra, we get what is known as the reflection principle which is that the probability of a touch is twice the probability of profit. This means that by managing winners we can almost double our probability of profit. That would require taking winners off the moment they were up a penny, which is not a good plan, but it shows the scale on which managing winners can supplement your probability of profit. Conversely, attempts to manage losers will hurt your probability of profit, since some of those losers you closed out were going to turn around.

Watch this segment of “The Skinny on Options Math” with Tom Sosnoff, Tony Battista and tastytrade's Math Genius, Jacob Perlman for an in depth discussion of probability of profit.