Eachway Tunnel Gamma

European Binary OptionsEachway TunnelEachway Tunnel DeltaEachway Tunnel ThetaEachway Tunnel Vega

Eachway tunnel gamma is the metric that describes the change in the delta of an eachway tunnel due to a change in the underlying price. Eachway tunnel gamma is the first derivative of the eachway tunnel delta with respect to a change in underlying price. It is depicted as:

\Gamma = \frac{d\Delta}{dS}

where  \Gamma is gamma and \Delta is the tunnel delta.

Evaluating Eachway Tunnel Gamma

Eachway Tunnel Gamma = R1 x Binary Call Gamma(K1) + R2 x Binary Call Gamma(K2)

― R2 x Binary Call Gamma(K3) ― R1 x Binary Call Gamma(K4)

where K1, K2, K3 & K4 are lowest strike to highest strike, and

where R1 + R2 = 1 and R1 < R2.

Eachway Tunnel Gamma Over Time

The gamma profile of an individual binary call option is positive below the strike, zero at the strike and negative above the strike. In Figure 1a the 0.5 day one can make out the profiles of the gammas of the individual binary calls. In contrast the 25-day gamma is flat at zero as the variations are ironed out through averaging.

Eachway Tunnel Gamma w.r.t. Days to Expiry 0.40.100
Figure 1a – Eachway Tunnel Gamma w.r.t. Days to Expiry 0.40.100

Figure 1b illustrates the 0.1-day eachway tunnel gamma.On adding this profile to Figure 1a  one couldn’t see the wood from the trees, a confusing mess. A look at the scale says a lot. The metric has expended its life usefulness and this is a common theme amongst ultra short term binary options Greeks. Indeed, one can say the same of ultra short-term conventional gamma and theta.

Eachway Tunnel Gamma w.r.t. 0.1 Days to Expiry 0.40.100
Figure 1b – Eachway Tunnel Gamma w.r.t. 0.1 Days to Expiry 0.40.100

Gamma is always lower the longer the time to expiry but the eachway tunnel gamma is lower still as the gamma is in effect the arithmetic mean of the individual gammas.

Eachway Tunnel Gamma and Volatility

Figures 2a and 2b illustrate the gamma over a range of volatilities. Figure 2a excludes the 2% gamma which can be seen in Figure 2b. Yet again the scale of both illustrations explains the logic of showing them separately.

Figure 2a with 5 days to expiry provides reasonably smooth curves which enable the binary options risk manager to transfer risk away. There are no unruly gyrations that would require constant portfolio readjustments.

Eachway Tunnel Gamma w.r.t. Volatility without 2% 0.40.100
Figure 2a – Eachway Tunnel Gamma w.r.t. Volatility 0.40.100

Figure 2b provides a profile that swings sufficiently to cause major gamma hedging issues when volatility is ultra low at 2%.

Eachway Tunnel Gamma w.r.t. Volatility 0.40.100
Figure 2b – Eachway Tunnel Gamma w.r.t. Volatility 0.40.100

At 100.00 there is a small hump in the profile. This is because with still five days to go to expiry a ultra low 2% implied volatility means the eachway tunnel is already worth 98.04 at 100.00. Any change in volatility will have a maximum effect (1.96) on the option’s upside. The upshot is that the absolute vega therefore decreases and is now heading up to zero.

In summary, the volatile nature of this Greek, oscillating as it does between negative and positive, becomes a far less useful metric that its conventional cousin.

By: Hamish Raw

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