Empirical investigation of the pattern of time value decay of options 34

Hypothesis 2 examines the shape of the expected time value curve. Tannous and Lee-Sing (2008) consider the effects of stochastic stock price movements on the decay of time value of options. Using simulation, they find that the time value is expected to decay at a decreasing rate. Using transaction data, I depict the expected time value curve of one- and three-month options and explore whether the shape is consistent with that predicted by Tannous and Lee-Sing (2008).

First, I compare the expected time value curve of one-month and three-month op- tions during the same month. Figure 4.1 shows the time value curves for calls while Fig- ure 4.2 shows the time value curve of put options. The horizontal axis denotes the days to maturity of one-month options and the days to maturity of three-month options are the corresponding value plus 60 days. The horizontal axis is contrary to ordinary measure- ment because the number decreases from left to right. The vertical axis denotes the ex- pected time value. The one- and three-month expected time value curves are highly corre- lated and the correlation coefficient is almost equal to one.

= = = = = Insert Figure 4.1 Here = = = = = = = = = = Insert Figure 4.2 Here = = = = =

Figures 5.1 and 5.2 depict the full expected time value curve for three-month op- tions. The objective is to investigate whether the time value decays at a decreasing rate as time passes. The shapes of the curve for call and put options are very similar and they are both similar to the curve obtained from simulation and shown in Figure 2.

35

= = = = = Insert Figure 5.2 Here = = = = =

I run OLS regressions to see whether there is a nonlinear relation between an op- tion’s days to maturity and the time value. I set the time value as the dependent variable and days to maturity, days to maturity squared, and days to maturity raised to power 3 as the independent variables. The higher powers of the days to maturity variable are used to examine the existence of a nonlinear relation. The regression results show that the op- tion’s time value is well explained by days to maturity and its quadratic and cubic terms. The adjusted R2 of each of the two regressions is almost equal to one. The positive coef- ficient of squared days to maturity of the first regression, which is significant at 1% level for both call and put options, implies that the expected time value curve is predominantly convex. Therefore Hypothesis 2 is supported. However, when I consider the impact of days to maturity, days to maturity squared and cubic terms together, the coefficient of the squared term is negative and only significant for the put option. The cubic term of days to maturity is positive and significant. The results suggest that the time value curve slightly changes convexity over the option’s life.

According to the theory, the time value of a three-month option decays most dur- ing the first month and least during the last month. I examine this proposition. I compare option time value cumulative decay during the first month with the cumulative decay dur- ing the second month, and the cumulative time value decay during the second month with the cumulative time value decay during the last month. I find that the mean time value decay is highest in the first 30 days and lowest in the last 30 days for both call and put options. The mean difference between the decay during the first 30 days of a three-month option and the decay during the last 30 days is significant at the 1% level. Similarly, the mean difference between the decay during the second 30 days of a three-month option and the decay during the last 30 days is significant at the 1% level. These results suggest that the option time value do decay at a decreasing rate.

Since the option market closes on weekends, I also test whether option time value decays more on weekends than on weekdays. Based on end of day data, three days pass from Friday to Monday (Saturday, Sunday and Monday) and time value is expected to decay more on weekends than weekdays. Our findings are consistent with the expectation and the results are significant for both call and put options.

36

The empirically observed expected time value curves seem to be flatter than and slightly different from the expected time value curves obtained by simulation. I propose several possible explanations. First, the simulation method runs thousands of trails based on the same input parameters, so the initial stock price and stock return volatility are the same in each trail. In the empirical tests, I choose a portfolio of stock options to test the time value rather than a single stock option. The stock price volatility changes continu- ously unlike the constant volatility of the simulation model. Second, the simulation uses at-the-money options to test time value. As shown in Tannous and Lee-Sing (2008), the expected time value curve of an at-the-money option has greater convexity than that of an out-of-the-money option or an in-the-money option.16 I use the nearest out-of-the-money options to calculate the observed expected time value curve to approximate the simula- tion assumption of at-the-money options. However, the remaining difference may have an impact on the degree of convexity. Third, the simulation is based on Merton’s (1976) option pricing formula. Model misspecifications may also lead to differences between the theoretical time value curve and the curve which is observed from transaction data. Fourth, it is possible that arbitrage opportunities are forcing a near linear relationship rather than a significantly convex one. As shown earlier, in the presence of transaction costs, the performance of a strategy of writing a three-month call and closing the position after one month is better than writing a one-month option and holding it until expiry.