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 {xt,t = t1, t2, …, tn} replaces the continuous variable. These prices constitute a set of all possible future outcomes. The set of probabilities {p(xti,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, Xt1) > 0}, where the payoff function is positive, and I- = {i : PF(B, S, Xt1) ≤ 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:
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?
Solving the puzzle
To find the solution, we compare the time dynamics of implied and historical volatilities. It is commonly believed that during crises, option premiums (and, therefore, implied volatility) increase sharply because of rising uncertainty of market participants. Although historical volatility increases as well, this happens slower and with an evident time lag. “Trends in volatilities” (below) illustrates this phenomenon using Boeing stock as an example.
The divergence in cycles of two volatilities is because during crises, option premiums increase sharply, while historical volatility rises more slowly. This is because it is calculated with historical data that include prices from less volatile periods. Increased premiums imply higher values of the payoff function; relatively low historical volatility implies lower variance used in the lognormal probability density function. As it follows from the previous formulas, both factors cause PP to increase at higher levels of implied volatility.
Unjustifiably inflated values of PP, obtained in periods of high volatility, are inappropriate for estimating the profitability of option portfolios containing short positions. It is essential to develop methods that enable us to adjust PP according to the current volatility level. There are several approaches to solve this problem:
- Calculation of historical volatility — used as variance to build the probability density function — is based on an historical price series of a given length. The longer the price series, the greater the influence of old data — data belonging to the calm period preceding an extreme market — and the greater the divergence in historical and implied volatility cycles. This leads to overvaluation of PP. The distortions in probability estimates may be reduced significantly by regulating the length of the price series parameter, according to the current level of implied volatility. The parameter should relate inversely to the volatility.
- The standard method used for calculating historical volatility is based on historical prices, each of which has equal weight relative to all other prices. Alternatively, we can consider differential application of weight coefficients similar to what you would do in calculating an exponential moving average as opposed to a simple moving average. Higher weights would be assigned to recent prices, while older prices receive lower weight coefficients. As a result, recent price fluctuations would exert greater influence on the variance as compared to older price changes. The function setting the weights can be of any form — linear, exponential, etc.
- One of the main factors determining the payoff function of an option combination is the premium obtained by the trader as proceeds from opening the short position. During crises, the premium grows faster than historical volatility, leading to the divergence of volatilities and distortion of probability. Thus, to obtain unbiased PP it is possible to reduce the divergence between two volatilities by artificially decreasing the payoff function profile. This can be achieved by introducing the adjusting coefficient that lowers premium values by some fixed amount or by a certain coefficient.
High volatility is a fact in today’s market. What’s more, it often manifests without warning. It is critical that traders recognize this and adjust their analysis techniques accordingly. The methods discussed here are viable solutions to the problem of high volatility distorting profit expectations in options positions.
Sergey Izraylevich, Ph.D., and Vadim Tsudikman are authors of “Systematic Options Trading” (Financial Times Press, 2010), where you can find an extended discussion of practical application of the option profit probability indicator and the methods used to determine the optimal values of its parameters. Contact the authors at izraylevich.hti@numericable.fr.
