The Effect of Moneyness and Stock Price on Theta

Theta is not a constant. As variables influencing option values change, theta can change, too. One such variable is the option’s moneyness. Exhibit 2.10 shows theoretical values (theos), time values, and thetas for 3-month options on Adobe (ADBE). In this example, Adobe is trading at $31.30 a share with threemonths until expiration. The more ITM a call or a put gets, the higher its theoretical value.Butwhen studying an option’s time decay, one needs to be concerned only with the option’s time value, because intrinsic value is not subject to time decay.

The ATM options shown here have higher time value than ITM or OTM options. Hence, they have more time premium to lose in the same three-month period. ATM options have the highest rate of decay, which is reflected in higher thetas. As the stock price changes, the theta value will change to reflect its change in moneyness.If this were a higher-priced stock, say, 10 times the stock price used in this example, with all other inputs held constant, the option values, and therefore the thetas, would be higher.

If this were a stock trading at $313, the 325-strike call would have a theoretical value of 16.39 and a one-day theta of 0.189, given inputs used otherwise identical to those in the Adobe example.The Effects of Volatility and Time on Theta:Stock price is not the only factor that affects theta values. Volatility and time to expiration come into play here as well. The volatility input to the pricing model has a direct relationship to option values.

Higher-valued options decay at a faster rate than lower-valued options—they have to; their time values will both be zero at expiration. All else held constant, the higher the volatility assumption,the higher the theta.The days to expiration have a direct relationship to option values as well.As the number of days to expiration decreases, the rate at which an option decays may change, depending on the relationship of the stock price to the strike price.

ATM options tend to decay at a nonlinear rate—that is, they lose value faster as expiration approaches—whereas the time values of ITM and OTM options decay at a steadier rate.Consider a hypothetical stock trading at $70 a share. Exhibit 2.11 shows how the theoretical values of the 75-strike call and the 70-strike call decline with the passage of time, holding all other parameters constant.

The OTM 75-strike call has a fairly steady rate of time decay over this 26-week period.The ATM70-strike call, however, begins to lose its value at an increasing rate as expiration draws nearer. The acceleration of premium erosion continues until the option expires.

Exhibit 2.12 shows the thetas for this ATM call during the last 10 days before expiration.Incidentally, in this example, when there is one day to expiration, the theoretical value of this call is about 0.44. The final day before expiration ultimately sees the entire time premium erode.Vega:Over the past decade or so, computers have revolutionized option trading.

Options traded through an online broker are filled faster than you can say,“Oops! I meant to click on puts.” Now trading is facilitated almost entirely online by professional and retail traders alike. Market and trading information is disseminated worldwide in subseconds, making markets all the more efficient. And the tools now available to the common retail trader are very powerful as well.

Many online brokers and other web sites offer highpowered tools like screeners, which allow traders to sift through thousands of options to find those that fit certain parameters.Using a screener to find ATM calls on same-priced stocks—say, stocks trading at $40 a share—can yield a result worth talking about here. One $40 stock can have a 40-strike call trading at around 0.50, while a different $40 stock can have a 40 call with the same time to expiration trading at more like 2.00.

Why? The model doesn’t know the name of the company, what industry it’s in, or what its price-to-earnings ratio is. It is a mathematical equation with six inputs. If five of the inputs—the stock price, strike price,time to expiration, interest rate, and dividends—are identical for two different options but they’re trading at different prices, the difference must be the sixth variable, which is volatility.