# 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 gamma’s sensitivity to implied volatility and time to expiry, application of the binary call option gamma, comparisons 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

where:

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:

Γ = δ^{2}P / δS^{2}

where:

P = the price of the binary call

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.

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:

Gradient = (D_{2} – D_{1} ) / ( S_{2} – S_{1} ) x SInc

where:

S_{2} = S + δS

S_{1} = S – δS

Δ_{2} = Delta at S_{2}

Δ_{1} = Delta at S_{1}

SInc = Minimum Underlying Price Change

i.e. Gradient = (45.1746-1.0770) / (99.99-99.81) x 0.01 = 2.4499

as indicated in the bottom row of the central column of Table 1.

The gradients of the ‘12 tick chord’ and ‘6 tick chord’ are calculated in the same manner and are also presented in the central column of Table 1.

Table 1 - From Gradient of Chord to Call Gamma | |||||||
---|---|---|---|---|---|---|---|

99.81 | 99.84 | 99.87 | 99.90 | 99.93 | 99.96 | 99.99 | |

Gradient | |||||||

δS = 0.00 | 22.0569 | ||||||

δS = 0.03 | 1.7620 | 21.4338 | 3.0480 | ||||

δS = 0.06 | 1.1839 | 19.6612 | 3.5432 | ||||

δS = 0.09 | 0.7324 | 17.0072 | 3.7937 |

As the underlying price difference narrows (as reflected by δS = 0.06 and δS = 0.03) the gradient tends to the gamma of 22.0569 at 99.90. The gamma is therefore the first differential of the binary call option delta with respect to the underlying and can be stated mathematically as:

δS → 0, Δ = dP / dS

which means that as δS falls to zero the gradient approaches the tangent (gamma) of the delta profile of Figure 2 at 99.90.

### Binary Call Option Gamma w.r.t. Implied Volatility

Figure 3 illustrates 5-day binary call option delta profiles with Figure 4 providing the associated gammas over a range of implied volatilities as in the legend.

The delta gradient below the strike is always positive while above the strike it is always negative: this leads directly to the first observation that binary call options gamma is always positive when out-of-the-money, always negative when in-the-money.

Where implied volatility falls to as low as 1% both the delta and gamma generate numbers that are so absolutely high that as a risk management tool they become bordering on worthless. This is nothing new to at-the-money conventional options gamma when time to expiry approaches zero.

Since the peak of the delta dictates a zero gradient, the gamma always travels through zero when at-the-money.

Finally, as the implied volatility increases the delta profile flattens, which in turn means that the absolute values of the gamma also decrease.

### Binary Call Option Gamma w.r.t. Time to Expiry

Figures 5 & 6 provide delta and associated gamma profiles over a range of times to expiry.

Pretty much the same observations regarding the relationship between the delta and gamma which were noted over a range of implied volatilities apply to a range of time to expiry.

### Binary Call Option Gamma Application

Table 2 shows the Table 2 of Binary Call Option Delta with the gamma added. The table is for 10 days to expiry and 5% implied volatility.

Table 2 - Binary Call Option Fair Value with associated Delta and Gamma | |||||||
---|---|---|---|---|---|---|---|

Asset Price | $99.81 | $99.84 | $99.87 | $99.90 | $99.93 | $99.96 | $99.99 |

Fair Value | 40.7518 | 42.1671 | 43.5921 | 45.0250 | 46.4641 | 47.9073 | 49.3529 |

Delta | 0.4699 | 0.4735 | 0.4764 | 0.4788 | 0.4805 | 0.4816 | 0.4820 |

Gamma | 0.1284 | 0.1085 | 0.0882 | 0.0676 | 0.0468 | 0.0257 | 0.0046 |

At $99.87 the delta is worth 0.4764 and has a gamma of 0.0882. Therefore, if the underlying rises three ticks from $99.87 to $99.90 the delta will change to:

0.4764 + 0.03 x 0.0882 = 0.47905

If the underlying fell 3 ticks from $99.93 to $99.90 the delta would change to:

0.4805 + (-0.03) x 0.0468 = 0.4791

At $99.90 the delta in Table 2 is 0.4788 so there is a slight discrepancy between the values calculated above and true value in the table. This is because the gammas of 0.0882 and 0.0468 are the gammas for just the two underlying levels of $99.87 and $99.93 respectively, i.e. the gammas change with the underlying.

At $99.90 the gamma is 0.0676 so the value of 0.0882 is too high when assessing the change in delta on an upward move from $99.87 to $99.90, while similarly the gamma of 0.0468 is too low when evaluating the change in delta when the underlying falls from $99.93 to $99.90.

The average of the two gammas at $99.87 and $99.90 is ( 0.0882 + 0.0676 ) / 2 = 0.0779 and should this number be used in the first calculation above then the binary call at $99.90 would be estimated as:

0.4764 + 0.03 x 0.0779/100 = 0.4787

an error of 0.0001.

The average gamma between $99.90 and $99.93 is:

( 0.0676 + 0.0468 ) / 2 = 0.0572

The second calculation above would now generate a price at $99.90 of:

0.4805 + (-0.03) x 0.0572/100 = 0.4788

an error of just zero.

### Binary Call Option Gamma v Conventional Call Option Gamma

Figures 7a-e illustrate the difference over time to expiry between the binary call option gammas and conventional call option gammas.

Points of note are:

1) The change of scale to accommodate the gamma of the binary call as time decreases.

2) Conventional gamma remains positive while the binary gamma is both positive and negative dependent on whether ‘out-of’ or ‘in-the-money’.

### Formula

where:

and:

### Summary

The gamma is probably of greater use to the options portfolio manager and, as such, is a Greek for the specialist.

Some options traders define themselves by their willingness to be long or short gamma, and certainly the author would be amongst that ilk being himself a religiously ‘long gamma’ player.