Probability of Profit (POP)

Probability of profit (POP) is an estimate of the chance that a trade, usually an options position, will be worth more than it cost when it is closed or when it expires. It is expressed as a percentage. A POP of 70 percent means that, under the model's assumptions, the position finishes with at least a small gain in 70 cases out of 100.

The figure is shown on many options trading platforms and is widely used by retail options traders to compare strategies. It is derived from the prices of options themselves, through implied volatility, and from an assumed distribution of the future price of the underlying. It measures how often a trade wins, not how much it wins or loses, and that distinction is the key to using it correctly.

Every options position has one or more break-even prices at expiry. POP is the probability that the underlying finishes on the profitable side of them. Under the standard assumption that the price at expiry follows a lognormal distribution, the probability that the underlying ends above a break-even price $B$ is:

• $S$ is the current price of the underlying and $S_T$ is its price at expiry.
• $B$ is the break-even point of the position.
• $r$ is the risk-free interest rate and $T$ is the time to expiry in years.
• $\sigma$ is the implied volatility.
• $N$ is the cumulative standard normal distribution function.

The expression is the same as the term usually written $N(d_2)$ in the Black-Scholes-Merton model, with the break-even price in place of the strike. For a position that profits below a price, the probability is one minus this value. For a position that profits between two prices, it is the difference between the two probabilities.

A share trades at 100. A trader buys a three-month call with a strike of 100 for 5, so the position breaks even at 105. Implied volatility is 25 percent and the interest rate is 4 percent. The numerator is $\ln(100 / 105) + (0.04 - 0.03125) \times 0.25 = -0.0488 + 0.0022 = -0.0466$, and the denominator is $0.25 \times \sqrt{0.25} = 0.125$. The ratio is -0.373, and $N(-0.373) = 0.355$. The long call has a POP of about 35 percent.

The trader on the other side, who sold the call for 5, profits whenever the share finishes below 105, so that position has a POP of about 65 percent. The two figures sum to 100 percent because the option is a transfer between buyer and seller. This symmetry holds generally: selling options gives a POP above one half and buying them a POP below it.

Traders often use an option's delta as a quick estimate of the probability that it expires in the money. A call with a delta of 0.30 is treated as having roughly a 30 percent chance of finishing above its strike. The approximation is close for short-dated options, though delta is not exactly this probability. POP differs from the probability of expiring in the money because it uses the break-even price, which accounts for the premium. A seller of that call profits not only when it expires worthless but also when it finishes slightly in the money by less than the premium received, so the seller's POP is a little above 70 percent.

• Bought options. Long calls, long puts and a long straddle have a POP below 50 percent, since the underlying must move enough to recover the premium.
• Sold options and credit spreads. Positions that collect premium, such as a covered call, a short straddle or a short strangle, have a POP above 50 percent.
• Range strategies. An iron condor profits when the underlying stays between its two break-even prices. Its POP is the probability of finishing inside that range, and it rises as the short strikes are placed further from the current price.

POP says nothing about the size of gains and losses. The measure that combines frequency and size is expected value:

Consider a credit spread with a POP of 80 percent that makes 100 when it wins and loses 450 when it loses. Its expected value is $0.80 \times 100 - 0.20 \times 450 = -10$. The trade wins four times in five and still loses money on average. In an efficiently priced market, a higher POP is paid for with a worse ratio of loss to gain, and the expected value of any position before costs is close to zero. Choosing trades by POP alone leads systematically towards positions with frequent small gains and rare large losses.

The probability is a model output. It assumes lognormal prices, whereas actual returns have fat tails, so the chance of an extreme move is higher than the calculation implies. It uses implied volatility, which is set by supply and demand for options and has on average exceeded the volatility later realized. That gap, the volatility risk premium, means the probabilities embedded in option prices are not the same as real-world probabilities. POP also refers to expiry. It ignores what happens along the way, including early assignment, margin calls and the decision to close a position early, and many platforms report a separate probability of touching a price before expiry for that reason. Commissions and bid-ask spreads, which matter for multi-leg trades, are left out as well.