The eachway call gamma describes the change in the delta of an eachway call due to a change in the underlying price. This gamma is the first derivative of the eachway call delta with respect to a change in underlying price. It is depicted as:
where Δ is the eachway call delta and S is the underlying asset price.
Evaluating Eachway Call Gamma
Eachway Call Gamma = (0.4 x Digital Call Gamma(K1) + 0.6 x Digital Call Gamma(K2))
where the terms to the right are digital call gamma with strikes K1 and K2 respectively.
Gamma Over Time
Below the digital call strike the digital call has positive gamma which turns negative above the strike.
Figure 1’s 0.5-day (red) profile clearly illustrates the gamma rising immediately above the underlying price of 98.00. It then peaks and falls through zero at the lower strike. It subsequently rises to pass through the midpoint of the strikes at 100.00. The gamma continues to rise until the upper strikes influence takes the gamma once again back down through the asset price at 101.00 an into negative territory. A trough is reached again after which the gamma rises back again to zero.
With 8 days to expiry (yellow) the gamma moves from 0.011 to approximately -0.0092 in a very smooth and shallow decline. At this time to expiry both strikes are exerting influence over the overall gamma. At 8 days to expiry the individual digital call gammas are nowhere to be seen. They have cancelled each other out.
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Gamma and Volatility
Figure 2 shows the wild swings in gamma as implied volatility becomes very low.
The higher the implied volatility shallower the decline in gamma. The 18% implied volatility profile being almost a straight line.
On the other hand, the (unrealistic) 2% volatility shows horrendous swings that make the the option portfolio manager’s job an nightmare. Luckily the implied volatility will rarely see such swings around a 10% middle volatility.
The author has been involved in trading 90-day LIBOR where options with 90-days to expiry have gone from 14% to 40% in just two days. That requires a monumental upheaval in the cash money market. That extraordinary day did take place in September 1992 as the UK got ejected from the EMS. While market-making 90-day UK LIBOR in the summer and autumn of 2008 there were a series of days like that. But then I sat at a desk; not the same pandemonium as the floor in 1992.
The eachway call gamma oscillates so often through zero it is not a metric that a risk manager can use except in the aggregation of a portfolio of options on the underlying asset.