Eachway Call Theta

European Binary OptionsEachway Call OptionsEachway Call DeltaEachway Call GammaEachway Call Vega

Eachway call theta describes the change in the fair value of an eachway call due to a change in time to expiry. The eachway call theta is the first derivative of the eachway call with respect to a change in time to expiry. It is depicted as:

Θ=\frac{dP}{dt}

Evaluating Eachway Call Theta

Eachway Call Theta = 0.4 x Binary Call Option Theta(K1) + 0.6 x  Binary Call Option Theta(K2)

where the terms to the right are the binary call theta with strikes K1 and K2 respectively.

Eachway Call Theta Over Time

The eachway call theta is quite difficult to visualise so the settlement price and eachway call price over time have been reintroduced to the reader. They are the same as Figures 1a and 2a of Eachway Calls.

Eachway Call Settlement Prices
Figure 1 – Eachway Call Settlement Prices

 

Eachway Call w.r.t. Time to Expiry
Figure 2 – Eachway Call w.r.t. Time to Expiry

 

Eachway Call Theta. w.r.t. Time to Expiry
Figure 3 – Eachway Call Theta. w.r.t. Time to Expiry

Eachway call theta is displayed against time to expiry in Figure 3. This displays the price movements of Figure 2 over time. The easiest way to follow Figure 3 is by having Figs. 1 & 2 at hand and just look at the interaction between 0.1 and 0.5 days to expiry.

If Figure 1 is compared with Figure 2 each time to expiry profile of Figure 2 has to move  towards Figure 1. This entails Figure 2 profiles moving up or down to reach the settlement profile of Figure 1.

In Figure 2 from 97.80 to 99.00 the red 0.5 days is above the black 0.1 days. This means that the eachway call price will fall from the 0.5-day value to the 0.1 day value. This fall provides the negative theta in Figure 3 below.

In Figure 2, at 99.00, the red and black intersect and until 100.00 black is higher than red. This means that between 99.00 and 100.00 and during the time between 0.5 days and 0.1 days the eachway call rises in value. This is portrayed in Figure 3 with the positive red 0.5 day profile.

At 100.00 the 0.5 and 0.1 intersect again and as Figure 2 shows the 0.5 profile (red) above the 0.1 profile (black) we know 0.5 has a negative theta again.

At 101.00 the two price profiles intersect yet again, black 0.1 price profile rises above the red 0.5 price profile meaning that the 0.5 theta profile is positive again. All theta profiles remain above 0 above the strike price of 101.00.

The scale depicts the amount of price movement over a day so the theta is increased in absolute value to above the actual possible price change of the option. This is because in contrast to all the other Greeks the time to expiry only moves in one direction, it decreases. More can be found on this under ‘Finite Greeks’.

Eachway Call Theta and Volatility

Figure 4 provides eachway calls over a range of implied volatilities and Figure 5 offers the theta over the same range of volatilities.

Eachway Call w.r.t. Volatility
Figure 4 – Eachway Call w.r.t. Volatility

Figure 5 looks more like a spaghetti chart! What does it mean?

The theta is a function of volatility. For example, at 98.40 and with 5 days to expiry the 18.00 volatility eachway call is worth 25.82 (or 0.2582). If  the asset price does not change but volatility falls to 14% then the option is worth less, 21.36 in fact. This means that over 5 days the 18% option can lose 4.46 more than the 14% option. The 18% option therefore has to decay at a faster rate than the 14% option and that is what Figure 5 depicts.

Eachway Call Theta. w.r.t. Volatility
Figure 5 – Eachway Call Theta. w.r.t. Volatility

When volatility is as low as 2% it is clear the eachway profile of Figure 5 is just the profiles of the two eachway call vega of the 99.00 strike and the 101.00 call strike side-by-side.

At 100.80 the 18% volatility the eachway call is 56.39. Theta is positive at this point but at expiry the eachway call would only be worth 40, 16.39 less than the current value. At 2% vol the option is still worth 49.43 but now theta is -0.009, the lowest point of the 2% profile. The eachway call theta is then oscillating either side of zero owing to a changing asset price and changing volatility. Yet now it is clear that time to expiry is changing the theta from positive to negative although the asset price has not changed.

By: Hamish Raw

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