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.
Next Section: Applying
Expected Value Concepts to Option Investing
Previous Section: The
Role of Probability in Modeling Investments
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