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 email@example.com.