   # Computing Probability of Profit

The term "probability of profit" used in this site means the probability of an outcome which will result in the investor making a profit of at least one cent on the investment. Geometric Brownian Motion is used to model the price movement of underlying securities. To compute the "probability of profit" the lognormal cumulative distribution function for the underlying at the selected time period is shifted so that its 50% point aligns with the security price that would result given the specified risk-neutral growth rate. Then the value of the distribution is read for a break even outcome. If the investment is bullish, this value is subtracted from 100%.

Example:
An equity with a 28% annualized volatility has a price of \$100 and has a risk-neutral rate of growth of 10%/yr. We sell 60-day Put at \$90 for a premium of \$2, hoping the option will expire out-of-the-money.

The Break-even price for the investment is \$88 (\$90-\$2).

The price of the \$100 equity in 60 days under the 10% annualized growth condition is \$101.70 (using continuous compounding). The 50% point in the lognormal cumulative distribution function is shifted to match \$101.70. The lognormal cumulative distribution is shown below. The value of the function at \$88 is read to be 0.1. This means that the probability of the pricing being equal to or less than \$88 is 10%. Therefore the probability of profit for the investment is 90% (the investment is bullish so 100%-10% is used).

Decreased volatility or shorter expiration intervals result in compressing above curve in the horizontal dimension, making it more vertical and reducing its tails. Increased volatility or longer expiration intervals result in expanding above curve in the horizontal dimension, making it less vertical and increasing its tails. Decreasing the growth rate shifts it left. Increasing the growth rate shifts it right.

"Probability of Profit", by itself, is not a good indicator of the profitability of an option investment. An option investment with a "Probability of Profit" of 95% would be a very poor investment if losses occuring in the remaining 5% probability outweigh the profits. We prefer to judge an investment by its "Expected Profit" (mathematically known as the investment's "Expected Value"). This is discussed in the next section.

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