This section on binary call option theta, as with the binary put option theta section, is in two parts:. The equation for the binary call options theta can be found at the bottom of the page. Figure 1 shows binary call option price profiles at different times to expiry. Figure 2 shows how with seven static underlying prices, the binary call options change in value as the days to expiry fall from 25 to 0, so in effect a profile from Figure 2 is a vertical cross section at that underlying price in Figure 1.
When the underlying price is When the underlying price is above The theta as represented by the above formula measures the gradient of the slopes in Figure 2. When there is over 20 days to expiry price decay whether negative or positive is very low; as time passes the theta increases in absolute value with that increase dependent on how close to the strike the underlying is. Chords have been added centred around five days to expiry so that, for example, the five-day chord stretches from 7.
Since the price profile is decreasing exponentially, the gradient of the chords decrease the longer the length of the chord. The theta is therefore the first differential of the binary call fair value with respect to time to expiry and can be stated mathematically as:. Figure 1 illustrates 5. Irrespective of the days to expiry the theta when at-the-money is always zero.
When out-of-the-money the binary call theta is always negative as with out-of-the-money conventional call options but when in-the-money the binary call options theta is positive unlike in-the-money conventional call options.
With sufficient days to expiry 25 days in Figure 4 the binary call option theta is almost flat at close to zero. As time passes the absolute maximum value of the theta increases with the peak and trough progressively closing on the strike.
This can be explained by the case where there is just 0. As is usual the implied volatility has a similar effect on the price profiles but there are some subtle differences between the binary call theta profiles of Figs. The maximum absolute theta in Figure 6 is fairly steady at around 2. Irrespective of implied volatility the binary call theta travels through zero for the now familiar reason that at-the-money binaries are priced at 50, or very close to it.
From Figure 3 above it is hopefully visually apparent that an equal measure of time backwards provides an increase in call option value which is less than the decrease in option value for an equivalent jump forwards in time, e. So from the 6th day to the 5th day the option loses:. Figure 7a shows that as the binary call option price decay either positive or negative is negligible then the theoretical theta almost overlaps the practical theta, especially when implied volatility is low.
With 10 and 4 days to expiry the theoretical theta gradually becomes more inaccurate as a measure of actual option price change with the actual time decay being absolutely greater at the peaks and troughs of the theta binary call options theta profiles but becoming lesser as the underlying moves away from the strike. The left hand scales of Figures 7a-c are gradually increasing in value as the theta increases over time.
Should a theta be required for binary call option prices that range between 0 and then the theta should be multiplied by If theta is solely represented by the results of Eq 1 then it is a useful tool for establishing daily time decay if divided by plus there is sufficient time to expiry. The delta can be hedged away by trading the underlying; until time itself becomes a tradable entity a future?
As with deltas, as expiry approaches the theta can reach ludicrously high numbers so one should always observe the tenet: You must be logged in to post a comment. Binary Call Option Theta The Binary Call Option Theta measures the change in the price of a binary call option over time and is the gradient of the slope of the binary options price profile versus time decay. This section on binary call option theta, as with the binary put option theta section, is in two parts: More posts to check out: Binary Put Option Delta.
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