Binary Call Option Gamma
An Introduction to Binary Call Option Gamma
Binary call option gamma measures the change in the binary call option delta owing to a change in the underlying price and is the gradient of the slope of the binary call options delta profile versus the underlying.
Below find a Finite Gamma evaluation . followed by the gammas sensitivity to implied volatility and time to expiry . application of the binary call option gamma parisons with conventional call option gamma . and finally the closed-end formula .
The gamma is the measure most commonly used by market-makers or ‘structural’ traders when referring to portfolios of options. The gamma indicates how much the delta of an option or portfolio of options will change over a one point move.
Market makers will generally try to hold books that are neutral to movements in the underlying but will more often than not be a long or a short gamma player. The long or short gamma indicates the position’s exposure to swings in the delta and therefore subsequent exposure to the underlying. Gamma provides a very quick, one glance assessment of the position with respect to a change in the underlying and gamma and is subsequently a very important tool to the binary portfolio risk manager.
Binary Call Option Gamma and Finite Gamma
The gamma Γ of a binary option is defined by:
Γ = δΔ / δS
Δ = the delta of the binary call
S = price of the underlying
δS = a change in the value of the underlying
δΔ = a change in the value of the delta
The gamma is therefore the ratio of the change in the option delta given a change in the price of the underlying. Furthermore, since the delta is the first derivative of a change in the binary call price with respect to a change in the underlying it follows that the gamma is the second derivative of a change in the call price with respect to a change in the underlying. So the gamma can also be written as:
Figure 1 shows the 1 day delta profile of a binary call with Figure 2 showing (in black) the same delta profile between the underlying prices of 99.78 and 99.99.
Fig.1 – Binary Call Option Delta profile
The blue ’18 tick chord’ in Figure 2 travels between the point on the delta profile 9 ticks below the price of 99.90 to 9 ticks above where the delta of the binary call option is provided in the bottom row of Table 1. The gradient of this chord is defined by:
S 2 = S + δS
S 1 = S – δS
Δ 2 = Delta at S 2