Probability of Profit

Definition

Probability of Profit is the probability that an option strategy will profit at least $0.01 at expiration. In other words, the probability that the underlying security will close within the profit range on the expiration date.

For an option strategy with all option legs on the same expiration date, the break-even point at expiration can be calculated in advance, so the profit range of the option strategy can be determined in advance.

We assume that the distribution of stock prices in the future is subject to a log-normal distribution, that is, the return of stock prices obeys a normal distribution. As long as the distribution function and the profit interval are known, the probability of the stock price closing within the profit interval at expiration can be calculated. This probability is the Probability of Profit.

The distribution of stock prices over a period of time in the future depends on:

1. The current stock price.
2. The volatility of the stock price.
3. The length of time.

Variables

With the following variables:

1. u = price at which probability is to be calculated
2. t = days to expiration
3. price = last price of the underlying
4. v = Implied Volatility, the model has the characteristic that all options on the underlying must have the same BS implied volatility.

We use implied volatility as an expectation of future volatility of underlying security. We assure that the return of stock in the coming year follows a normal distribution: N (0, V²).

What needs to be confirmed now is the probability density of stock returns in the next t days, that is, the volatility of stock returns in the future t days needs to be confirmed.

According to the square root rule, the volatility of the stock return in the next t days = v*√(t ⁄ 365).

Therefore, the stock return in the future t days follows a normal distribution: N(0, V²*t ⁄ 365).

That is: ln⁡(u ⁄ price)~ N(0,V²*t ⁄ 365)

u = price at which probability is to be calculated.

So (ln⁡(u ⁄ price)) ⁄ ((v*√(t ⁄ 365))) follow the standard normal distribution: N(0,1).

That is: (ln⁡(u ⁄ price)) ⁄((v*√(t⁄365))) ~ N(0,1)

Calculations

Taking long call as an example, first calculate the breakeven point: breakeven = strike price + premium, the profit interval is [breakeven, +∞).

Therefore, Probability of Profit = 1-Φ(X) = 1-Φ((ln⁡(breakeven⁄price))⁄((v*√(t⁄365)))) Φ(X) is the standard normal distribution function.

Remember, the calculation of Profit probability is based on the probability distribution of stock prices. It only represents a theoretical value, not a probability that it will actually happen.

Disclaimer:

The Probability of Profit display is not guaranteed and only a calculation using a set of assumptions. This tool is not designed to produce definite returns for your investments and will not guarantee profits or returns. The information supplied by this tool is subject to error and only gives investors a general estimate of possible outcomes. Details rendered by this tool may not be exact or act as an accurate representation of the potential gains or losses made from your investments. Webull Financial is not liable for miscalculations or inaccuracies, including computational or human-made oversights and errors, contrived as a result of using the probability of ITM display. Additionally, Webull and its affiliates are not responsible for measures taken based upon these calculations. The ITM probability display does not constitute investment advice; it is heavily encouraged that all financial objectives and risks are carefully considered before investing in options.