Modern option pricing theories — from the basic Black-Scholes model to advanced mathematical algorithms — are based on probability concepts. One of the most popular indicators used to evaluate the investment attractiveness of both single options and their combinations is the probability to earn profit. This gauge expresses the probability that at the time of options expiration (or any other time point prior to expiration) a given option will make money. Whether the payoff function is positive or negative is determined by the future price of the underlying asset.

Applying classic probability theory, the profit probability of an option, or combination of different options, can be calculated by integrating the product of the combination payoff function by the probability density function over the price range (or ranges) for which the payoff function is positive:

where *x* is the underlying asset price; *PF*(*B*,*S*,*x*) is the payoff function of combination *S* with the underlying asset *B*; *LogN*(*Mean*,*σ*,*x*) is the probability density function of lognormal distribution with parameters *Mean* (mathematical expectation of the price); and variance *σ*; *θ*(*y*) is the theta-function with argument *y* = *PF*(*B*,*S*,*x*), which has the following values: *θ*(*y*) = 1, if *y* > 0, and *θ*(*y*) = 0 in other cases.

The above formula applies if the price is considered as a continuous variable.

In the discrete case, a finite price series {*x _{t}*,

*t*=

*t*,

_{1}*t*, …,

_{2}*t*} replaces the continuous variable. These prices constitute a set of all possible future outcomes. The set of probabilities {

_{n}*p*(

*x*,

_{ti}*t*),

*i*= 1, 2, …,

*n*}, assigned to the corresponding prices, replaces the probability density function. Price index

*i*forms two subsets:

*I*= {

^{+}*i*:

*PF*(

*B*,

*S*,

*X*) > 0}, where the payoff function is positive, and

_{t1}*I-*= {

*i*:

*PF*(

*B*,

*S*,

*X*) ≤ 0}, where the payoff is negative or equal to zero. Profit probability can be estimated as a sum of probabilities of all elements forming the first subset:

_{t1}

Estimating profit

Profit probability (PP) is widely used by traders and is included in almost all software products developed for analyzing options. It represents one of the main criteria used in different market scanners and rankers designed for identifying potential trading opportunities appearing on options exchanges. The popularity of this indicator is accounted for by the relative simplicity of the PP calculation and by sufficiently high effectiveness of its practical application. Combined with another important indicator, expected profit, PP enables accurate estimation of option profitability.

At the same time, according to longstanding observations, PP of short option combinations may be overvalued significantly during highly volatile periods. It is common knowledge that shorting options is one of the riskiest option strategies. In periods of high market volatility, and especially during financial crises, risks inherent in shorting options increase manifold. Because overstated estimates of PP inevitably lead to risk underestimation, it is absolutely essential to determine the extent to which the volatility of the underlying asset affects the probability values.

To accomplish this task, statistical studies were conducted using a five-year database containing prices of options and their underlying assets (from 2005 to 2010). This period includes data pertaining to both calm and extreme market conditions (the last financial crisis). Using these data, horizontal and vertical analyses were performed involving relationships between profit probability and underlying asset volatility.

In the vertical analysis, 1,000 of the most liquid U.S. stocks were used as underlying assets. For each of them, we created three short straddle combinations (using strike prices that are closest to the current underlying price) for the first, second and third weeks before the expiration date. All combinations were constructed for September 2010 expiration. In total, we estimated 3,000 short straddles.

For the horizontal analysis, five stocks were selected with actively traded options: Apple, Boeing, Ford Motor, General Electric and IBM. For each of these stocks, one short straddle was created on each trading day during a five-year period. By analogy with the vertical analysis, all combinations were created using the most liquid option contracts (the nearest expiration date and strike prices that are closest to the current underlying asset price). For all combinations, profit probability was calculated and recorded for implied and historical volatility values. Altogether, about 6,000 combinations were evaluated.

“Profit probability vs. volatility” (below) illustrates the relationships between profit probability and implied volatility. In both vertical and horizontal analyses the probability to earn profit increases as volatility increases. Although these relationships are not strong (correlation coefficients range from 0.36 to 0.66), they are statistically significant in all cases. At extremely high volatility levels, corresponding to crisis periods, PP rises to 80%-90%, and in some cases to nearly 100%. Obviously, such estimates are biased. Intuition suggests — and the experience of the latest financial crises proves this — that in periods of extreme market fluctuations, the probability to gain profit from short option positions decreases rather than increases. If this is the case, how can the existence of the direct relationship between probability and volatility be explained?