# Binary Put Option Gamma

## Binary Put Option Gamma measures the change in the delta of an option owing to a change in the underlying price and is the gradient of the slope of the binary put option delta against the underlying.

The binary put option gamma indicates how much the delta of an option or portfolio of options will change over a one point move.

A market maker will generally always trade as a long or short gamma player since it reflects a style of trading as opposed to a view on the market. So a market-maker more comfortable being long gamma will quite likely be more aggressive making markets since the long gamma will generally mean losing time value which needs to be covered. Alternatively if the market is not moving the short gamma player may be far more passive since they’re making money anyway; trading may involve being stuffed full of unwanted premium and hence becoming long gamma and long theta.

Either way, 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 Put 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 put

S          =          price of the underlying

δS        =          a change in the price of the underlying

δΔ        =          a change in the value of the delta

The gamma is therefore the ratio of the change in the put 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 put option price with respect to a change in the underlying it follows that the gamma is the second derivative of a change in the put price with respect to a change in the underlying. So the gamma can also be written as:

Γ          =          δ²P/δS²

where:

P          =          the price of the binary put option

Figure 1 shows the 1 day delta profile of a binary put with Figure 2 showing (in black) the same delta profile between the underlying prices of 99.20 and 99.70.

Fig.1 – Binary Put Option Delta

Fig.2 – Slope of the Gamma at $99.90 plus approximating Gamma ‘chords’ The blue ’44 tick chord’ in Figure 2 travels between the point on the delta profile 22 ticks below the price of 99.44 to 22 ticks above where the delta of the binary put option is provided in the bottom row of Table 1. The gradient of this chord is defined by: Gradient = (Δ2 – Δ1 ) / ( S2 – S1 ) where: S2 = S + δS S1 = S – δS Δ2 = Delta at S2 Δ1 = Delta at S1 SInc = Minimum Underlying Price Change i.e. Gradient = −0.6548−(−0.0174) / (99.66−99.22) = -0.02897 as indicated in the bottom row of the central column of Table 1. The gradients of the ‘32 tick chord’ and ‘16 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 Put Gamma Asset Price99.2299.2899.3699.4499.5299.6099.66 Gradient δS = 0.00-0.0252 δS = 0.08-0.0754-0.0258-0.2819 δS = 0.16-0.0338-0.0274-0.4728 δS = 0.22-0.0174-0.0290-0.6548 As the underlying price difference narrows (as reflected by δS = 0.06 and δS = 0.03) the gradient tends to the gamma of -0.0252 at 99.44. The gamma is therefore the first differential of the binary put option delta with respect to the underlying and can be stated mathematically as: δS → 0, Γ = dΔ / 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 Put Option Gamma w.r.t. Implied Volatility Figure 3 illustrates 5-day binary put 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 negative while above the strike it is always positive: this leads directly to the first observation that binary put 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 so absolutely high that as a risk management tool they become bordering on worthless. This is nothing new to the conventional options trader since at-the-money conventional options gamma goes to infinity when time to expiry approaches zero. Since the trough 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. Fig.3 – Binary Put Option Delta Profiles w.r.t. Implied Volatility Fig.4 – Binary Put Option Gamma Profiles w.r.t. Implied Volatility ### Binary Put Option Gammas 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. The above Table 1 is based on 2% implied volatility while Figure 6 is based on 5% implied volatility. The red one-day profile has a value of −2.0658 at the underlying of 99.90 so it becomes very apparent the influence of implied volatility adjacent to the strike as the 2% gamma has collapsed to −22.0569. Fig.5 – Binary Put Options Delta w.r.t. Time to Expiry Fig.6 – Binary Put Options Gamma w.r.t. Time to Expiry ### Binary Put Option Gamma Application Table 2 shows Table 2 of Binary Put Option Delta with gamma added. The table is for 10 days to expiry and 5% implied volatility. Table 2 - Binary Put Option Fair Value with associated Delta and Gamma Asset Price99.8199.8499.8799.9099.9399.9699.99 Fair Value59.248257.832956.407954.975053.535952.092750.6471 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 -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 low 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 high 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 put 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 Put Option Gamma v Conventional Put Option Gamma

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

Fig.7a – Binary Put Option Gamma v Conventional Put Option Gamma – Expiry 25-Days

Fig.7b – Binary Put Option Gamma v Conventional Put Option Gamma – Expiry 10-Days

Fig.7c – Binary Put Option Gamma v Conventional Put Option Gamma – Expiry 4-Days

Fig.7d – Binary Put Option Gamma v Conventional Put Option Gamma – Expiry 1-Days

Fig.7e – Binary Put Option Gamma v Conventional Put Option Gamma – Expiry 0.2-Days

Points of note are:

1)      The change of scale to accommodate the gamma of the binary Put 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

$\textup{Binary&space;Put&space;Option&space;Gamma}=\frac{e^{-rt}d_{1}{N}'\left&space;(&space;d_{2}&space;\right&space;)}{\sigma&space;^{2}S^{2}t}$

where:

$d_{1}=\frac{log\left&space;(&space;\frac{S}{E}&space;\right&space;)+\left&space;(&space;r-D+\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

$d_{2}=\frac{log\left&space;(&space;\frac{S}{E}&space;\right&space;)+\left&space;(&space;r-D-\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

${N}'\left&space;(&space;x&space;\right&space;)=\frac{1}{\sqrt{2\pi&space;}}.e^{-0.5x^{2}}$

and:

$\textup{S}=\textup{price&space;of&space;the&space;underlying}$

$\textup{E}=\textup{strike/exercise&space;price}$

$\textup{r}=\textup{risk&space;free&space;rate&space;of&space;interest}$

$\textup{D}=\textup{continuous&space;dividend&space;yield&space;of&space;underlying}$

$\textup{t}=\textup{time&space;in&space;years&space;to&space;expiry}$

$\sigma&space;=\textup{annualised&space;standard&space;deviation&space;of&space;asset&space;returns}$

### 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.

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