# When Volatility Distorts Probability: Profit Probability Can Become Inflated during Periods of High Volatility. Here's How to Keep Your Expectations in Check

Izraylevich, Sergey, Tsudikman, Vadim, Modern Trader

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:

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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,[sigma],x) is the probability density function of lognormal distribution with parameters Mean (mathematical expectation of the price); and variance [sigma]; [theta](y) is the theta-function with argument y = PF(B,S,x), which has the following values: [theta](y) = 1, if y > 0, and [theta](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.sub.t], t = [t.sub.1], [t.sub.2], [t.sub.n]} replaces the continuous variable. These prices constitute a set of all possible future outcomes. The set of probabilities [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], assigned to the corresponding prices, replaces the probability density function. Price index i forms two subsets: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the payoff function is positive, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], 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:

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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. …

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