# Binary Put Option Theta

## Theta measures the change in the price of an option over time and is the gradient of the slope of the binary put option price profile due to time decay.

This section on binary put option theta, as with the binary call option theta section, is in two parts:

i.    the first section covers the formula, derivation of the formula from first principles, plus the binary put option theta with respect to time to expiry and implied volatility,

ii.    while the second section analyses the theta as reflected by the formula as a useful analytical tool, discusses its drawbacks and provides an alternative ‘practical’ theta.

### Binary Put Option Theta and Finite Theta

The theta ϴ of any option is defined by:

ϴ = δP / δt

where:

P     =     price of the option

t     =     time in years to expiry

δP    =    a change in the value of P

δt     =    a change in the value of t

Figure 1 shows binary put option price profiles at different times to expiry. Figure 2 shows how with seven static underlying prices, the binary put 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 100.00 the option is at-the-money and the passing of time has no effect on the price of the binary option as it is always 50. When the underlying price is below 100.00 the price profiles all slope upwards reflecting a positive theta, whereas the out-of-the-money profiles, i.e. where S > 100.00, the price profiles all slope down meaning a negative theta.

Fig.1 – Binary Put Option Price profiles w.r.t. Time to Expiry

Fig.2 – Binary Put Option Price profiles with Fixed Underlying Prices

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.

Figure 3 is the S=99.75 price profile over the last 11 days of its life. Chords have been added centred around five days to expiry so that, for example, the five-day chord stretches from 7.5 days to 2.5 days to expiry. Since the price profile is increasing exponentially, the gradient of the chords increase the longer the length of the chord.

The gradient of the chord is defined by:

Gradient     =     ‒ ( P2 – P1 ) / ( S2 – S1 )

where:

t2    =    t + δt

t1    =    t – δt

P2    =     Binary Put value at t2

P1    =     Binary Put value at t1

i.e.         8-Day Gradient    =     ― (62.6554 ― 83.0906) / (9 ‒ 1)      =    2.5544

Fig.3 – Slope of the Theta at \$99.75 plus approximating Theta ‘chords’

as indicated in the bottom row of the central column of Table 1.

The gradients of the ‘5 day chord’ and ‘2 day  chord’ are calculated in the same manner and are also presented in the central column of Table 1.

Table 1 - From Gradient of Chord to Put Theta
Days to Expiry9.07.56.05.04.02.51.0
δt=01.5446
δt=265.30881.579868.4685
δt=563.78911.803572.8067
δt=862.65542.554483.0906

As the time difference narrows (as reflected by δt = 5 and δt = 2) the gradient tends to the theta of 1.5446 at 5 days to expiry, i.e. where δt = 0. The theta is therefore the first differential of the binary put fair value with respect to time to expiry and can be stated mathematically as:

as δt → 0,     ϴ = dP / dt

which means that as δt falls to zero the gradient approaches the tangent (theta) of the price profile of Figure 2 at 5 days.

### Binary Put Option Theta w.r.t. Time to Expiry

Figure 1 illustrates 5.0% implied volatility binary put profiles with Figure 4 providing the associated thetas for the same days to expiry.

Irrespective of the days to expiry the theta when at-the-money is always zero. When out-of-the-money the binary put option theta is always negative (as with out-of-the-money conventional put options) but when in-the-money the binary put option theta is positive (unlike in-the-money conventional put options).

With sufficient days to expiry (25 days in Figure 4) the binary put 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 0.5 days to expiry where at an underlying price of 99.90 the binary put option is worth 70.5941, so that 100―70.5941=29.4059 is the amount that the option will increase by over the next half-day if the underlying remains at 99.90.

Fig.4 – Binary Put Option ‘Theoretical’ Theta w.r.t. Time to Expiry

Although at 99.90 and 1-day to expiry the binary put option is worth 64.9362 and will therefore appreciate by 35.0638 to expiry, (5.6579 more than at the half-day to expiry) the binary put option theta is lower as the theta is an annual measurement, not necessarily a practical one.

### Binary Put Option Theta w.r.t. Implied Volatility

Figures 5 & 6 provide the binary put options price profiles over a range of implied volatilities with the associated binary put option theta. As is usual the implied volatility has a similar effect on the price profiles but there are some subtle differences between the binary put option theta profiles of Figs. 4 & 6.
The maximum absolute theta in Figure 6 is fairly steady at around 2.43 irrespective of the implied volatility, although the implied volatility does determine how close to the strike the peak and trough in theta is.

Fig.5 – Binary Put Option Price profiles w.r.t. Implied Volatility

Fig.6 – Binary Put Option ‘Theoretical’ Theta w.r.t. Implied Volatility

Irrespective of implied volatility the binary put option theta travels through zero for the now familiar reason that at-the-money binaries are priced at 50, or very close to it.

### ‘Theoretical’ Theta and ‘Practical’ Theta

From Figure 3 above it is (hopefully) visually apparent that an equal measure of time backwards provides a decrease in put option value which is less than the increase in option value for an equivalent jump forwards in time, e.g. at time 5 days to expiry the binary put option fair value is 66.6643, so using the example with δt=2, the 6-day and 4-day options are worth respectively 65.3088 and 64.4685. So from the 6th day to the 5th day the option gains:

Price appreciation from Day 6 to Day 5  =  (66.6643―65.3088) = 1.3555

while from the 5th day to the 4th day the option loses:

Price appreciation from Day 5 to Day 4  =  (68.4685―66.6643) = 1.8042

Table 2 presents the option value at days to expiry from 7 to 0 with the daily difference plus the ‘theoretical’ theta; it is apparent that the actual appreciation from one day to the next is greater than the theoretical theta. The ‘theoretical’ binary put option theta in this instance is derived from the formula of Eq(1) above divided by 365 (Eq(1) provides an annual rate) and multiplied by 100 (Eq(1) assumes a binary option price range between 0 and 1, not 0 and 100).

Table 2 - Binary Call Option Fair Value with associated day's decay and theta
Days to Expiry76543210
Binary Put Option Price64.243165.308866.664368.468571.037175.116583.09060
Day's Appreciation1.06581.35551.80422.56874.07937.974216.9094N/A
Theoretical Theta0.95451.19151.54462.11283.13555.344912.0436N/A

This again begs the question mooted in Binary Call Option Theta  as to the efficacy of using the formula of Eq(1) when might it not be simpler to compute the theta as calculated from the ‘Day’s Appreciation’ row of Table 2. Not particularly mathematically elegant, but there are a number of equally inelegant adjustments made by market practitioners to ‘elegant’ mathematical models in order to make them work, with volatility ‘skew’ being one of the more obvious. To be even deeper, the CAPM financial model is dependent on a ‘risk-free’ rate of interest…………is there such a thing as a ‘risk-free’ rate of interest?: what if the IMF was downgraded by Moody’s over the PIGS?!

Figures 7a-f offer graphical illustrations of the difference between ‘theoretical’ theta and ‘practical’ theta, a term I’ve coined to simply describe the actual change in price from one day to the next. Figure 7a shows that as the binary put option price decay (either positive or negative) is negligible then the theoretical theta almost overlaps the practical theta, especially when implied volatility is low.

Fig.7a – Binary Put Option Theta, ‘Theoretical’ & ‘Practical’, 5-Days to Expiry w.r.t. Implied Volatility

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 put option theta profiles but becoming lesser as the underlying moves away from the strike. This ‘smoothing’ is what might be expected when comparing the actual price changes of the ‘practical’ theta and the notional price changes portrayed by the ‘theoretical’ theta which itself is an annualised rate and in effect has a built in averaging mechanism.
The left hand scales of Figures 7a-c are gradually increasing in value as the theta increases over time.

Fig.7b – Binary Put Option Theta, ‘Theoretical’ & ‘Practical’, 2.5-Days to Expiry w.r.t. Implied Volatility

Fig.7c – Binary Put Option Theta, ‘Theoretical’ & ‘Practical’, 1-Day to Expiry w.r.t. Implied Volatility

When there is one day to expiry (Figure 7d) the undervaluation of time decay as generated by the ‘theoretical’ theta is at its most pronounced because at this point the ‘practical’ theta is in fact the binary put option premium when out-of-the-money and 100 less the binary put option premium when in-the-money.

Fig.7d – Binary Put Option Theta, ‘Theoretical’ & ‘Practical’, 0.5-Day to Expiry w.r.t. Implied Volatility

Finally Figures 7e & 7f illustrate the absolute ‘theoretical’ theta rising aggressively while the absolute ‘practical’ theta is now falling, the latter due to the lower premium of the option.

Fig.7e – Binary Put Option Theta, ‘Theoretical’ & ‘Practical’, 0.2-Days to Expiry w.r.t. Implied Volatility

Fig.7f – Binary Put Option Theta, ‘Theoretical’ & ‘Practical’, 0.1-Days to Expiry w.r.t. Implied Volatility

The scales of Figures 7e & 7f are worth noting, in particular Fig 7f where the ‘theoretical’ theta now rises above 100, which is an interesting concept since the maximum range of the binary put option is limited to 100!

Points of note are:

1)    Whereas conventional put option thetas are always negative as time value is always positive, time value with binary put options can be positive or negative dependent on whether they are in- or out-of-the-money.

2)    Whereas with conventional put options theta is always at its absolute highest when at-the-money, the binary put option theta when at-the-money is always zero.

3)    Out-of-the-money binary put options have negative or zero theta, in-the-money binary put options have zero or positive theta.

4)    Using Eq(1) to calculate theta can generate theta in excess of 100.

### Formula

$\textup{Binary&space;Put&space;Option&space;Theta}=-re^{-rt}N\left&space;(&space;d_{2}&space;\right&space;)+e^{-rt}{N}'\left&space;(&space;d_{2}&space;\right&space;)\left&space;(&space;\frac{d_{1}}{2t}-\frac{r-D}{\sigma&space;\sqrt{t}}&space;\right&space;)$

where:

$d_{1}=\frac{log\left&space;(&space;\frac{S}{E}&space;\right&space;)+\left&space;(&space;r-D+\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

$d_{2}=\frac{log\left&space;(&space;\frac{S}{E}&space;\right&space;)+\left&space;(&space;r-D-\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

${N}'\left&space;(&space;x&space;\right&space;)=\frac{1}{\sqrt{2\pi&space;}}.e^{-0.5x^{2}}$

and:

$\textup{S}=\textup{price&space;of&space;the&space;underlying}$

$\textup{E}=\textup{strike/exercise&space;price}$

$\textup{r}=\textup{risk&space;free&space;rate&space;of&space;interest}$

$\textup{D}=\textup{continuous&space;dividend&space;yield&space;of&space;underlying}$

$\textup{t}=\textup{time&space;in&space;years&space;to&space;expiry}$

$\sigma&space;=\textup{annualised&space;standard&space;deviation&space;of&space;asset&space;returns}$

N.B.

(i)    The theta generated by the above equation is an annualised number, so should a daily theta be required as an approximation then the theta needs to be divided by 365.

(ii)    This formula is based on binary put option prices that range between 0 and 1. Should a theta be required for binary put option prices that range between 0 and 100 then the theta should be multiplied by 100.

### Summary

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 365 plus there is sufficient time to expiry. But as time to expiry falls this ‘theoretical’ theta becomes increasingly inaccurate as a tool for forecasting the binary option price change over time.

The delta can be hedged away by trading the underlying; until time itself becomes a tradable entity (a future?) hedging theta can only be achieved by trading other options.

As with deltas, as expiry approaches the theta can reach ludicrously high numbers so one should always observe the tenet: “Beware Greeks bearing silly analysis numbers…” (as ever).

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