# Binary Put Options Vega

Binary put options vega is the metric that describes the change in the fair value of a binary put options due to a change in implied volatility.

#### $V = \frac{dP}{d\sigma}$

where $V$ is vega, P is put  value and $\sigma$ is volatility.

Binary put options vega is the first derivative of the binary put option value with respect to a change in implied volatility.

### Binary Put Vega Over Time

Figure 1 illustrates profiles of the $100 binary put vega for a selection of days to expiry. What immediately becomes apparent is that, as with the binary call vega, the in-the-money put options have negative vega. The out-of-the-money binary puts have a positive vega. This is because a higher implied volatility goes hand-in-hand with a higher volatility of the underlying asset. ##### Out-of-the-Money Put If the binary put option is out-of-the-money then an increase in volatility will increase the chance that the asset price will fall below the strike. This would generate a winning bet as the price rises to the 100 settlement price. Figure 1 – Binary Put Options Vega w.r.t. Time to Expiry An alternative scenario would be that if the underlying price has a volatility of zero. Then the price would simply not move meaning that an out-of-the-money option is destined to remain a loser. Therefore an out-of-the-money binary put option’s fair value will rise in concert with a rise in implied volatility and therefore the binary put vega is positive. ##### In-the-Money Put When the binary put option is in-the-money, a static underlying price will mean that the option will remain in-the-money. It will subsequently be a winner. A rise in volatility will therefore increase the chances that the underlying price will rise above the strike and the bet will become a loser. This would in turn lead to a lower binary options price. The in-the-money case will be reversed if there is a fall in the implied volatility. A fall in ‘vol’ will signify less movement in the underlying price thereby increasing the probability of the strategy being a winner. This in turn will make the binary put price worth more. The 0.1-day to expiry profile is zero apart from a narrow range around the strike. This reflects that, as can be seen from Fig.2 of binary put options, there is only a narrow range around the strike where the options premium does not equal 0 or 100. The 0.5-day, 2-days etc. profiles have progressively more time to expiry. This leads to the peaks and troughs of the profiles progressively move away from the strike. Nevertheless, the absolute maximum value of the binary put options vega remains fairly constant across the number of days. ### Binary Put Vega and Volatility Figure 2 provides binary put vega profiles over a range of different implied volatilities. Figure 2 – Binary Put Options Vega w.r.t. Volatility Binary put vega is zero when at-the-money. This means that as the underlying passes through the strike the position will change from short vega to long vega, or vice versa. Yet again, as with binary call theta, binary call vega and binary put theta, the binary put option is not a good choice for taking a view on implied volatility. Example: the asset is trading at 101.00, there are 5 days to expiry and volatility is 25%. The trader sells the 100.00 put in the belief volatility will fall. The following P&L graph and table offer the P&L at 98.00, 99.00, 100.00, 101.00 & 102.00 with volatility at 15%, 25% and 35%. Figure 3 – Short 101.00 Binary Put P&L $98.00 $99.00$100.00 $101.00$102.00 15% -50.45 -34.70 -13.11 8.40 24.09 25% -37.62 -26.74 -13.34 0.00 11.85 35% -32.38 -23.24 -13.57 -3.96 5.07

If one were to take the view that implied volatility will fall one might consider selling an out-of-the-money put. This would initially create:

• directional risk of the underlying price falling.
• A volatility risk of a rise in implied volatility.
• If the underlying fell through the strike the trader is now at  risk of the volatility falling!

By: Hamish Raw

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