The eachway put gamma describes the change in the delta of an eachway put due to a change in the underlying price, i.e. it is the first derivative of the eachway put delta with respect to a change in underlying price and is depicted as:
where Δ is the Eachway Put Delta and S is underlying asset price.
Evaluating Eachway Put Gamma
Eachway Put Gamma = R1 x Digital Put Gamma(K1) + R2 x Digital Put Gamma(K2)
where:
- the terms on the right are the digital put gamma with strikes K1 and K2, K1 being the lower of the two, and
- R1 + R2 = 1 and R1 ≥ R2.
Gamma Over Time
As with the digital put the gamma is long above the strike and short below the strike. When there are 0.1 or 0.5 days to expiry this feature stands out in both the below illustrations. The 8 and 25 day to expiry profiles are pretty flat and are influenced by both strikes. The green 2-day profile provides an indication that decreasing time to expiry is more dependent on the individual strike’s gamma as opposed to an average of both.
| European Digitals | Eachway Puts | Eachway Put Delta | Eachway Put Theta | Eachway Put Vega |
Gamma and Volatility
From the scale it is clear that the gamma is not greatly effected by changes in volatility. Figures 3a and 3b of eachway put delta show extremely shallow deltas until the volatility falls to 2%. Since the gamma is the first derivative of the delta it should be no surprise to see such low absolute values for the below gammas.
Generally, whether conventional options or digital options, the rule is that higher the implied volatility the lower the gamma. The same applies to time to expiry, i.e. the more time to expiry the lower the gamma.