This post was published by Hamish Raw of https://hamishraw.com/

Binary put options vega is the metric that describes the change in the fair value of a binary put option due to a change in implied volatility, i.e. it is the first derivative of the binary put option fair value with respect to a change in implied volatility and is depicted as:

V=dP/dV

Figure 1 illustrates profiles of the gold $1700 binary put options vega for a selection of days to expiry and what immediately becomes apparent is that, as with the binary call options vega, the out-of-the-money options have a positive vega while the in-the-money options have negative binary put options vega. This feature is because a higher implied volatility generally goes hand-in-hand with a higher volatility of the underlying asset, in this case gold. Therefore, if the binary put option is out-of-the-money then an increase in volatility is will increase the chances that the gold price will fall below the strike and settle as a winner. The alternative scenario would be that if the underlying gold 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 options vega is positive.

When the binary put option is in-the-money, a static underlying gold price will mean that the option will remain in-the-money and will subsequently be a winner. A rise in volatility will therefore increase the chances that the underlying gold price will rise above the strike and the bet will be a loser, which in turn leads to a lower binary options price. The case will be reversed in the case of a fall in the implied volatility as this will signify less movement in the underlying gold price thereby increasing the probability of the strategy being a winner, in turn making the binary put option price worth more.

The 0.2-day to expiry profile is zero apart from a narrow range around the strike which 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 1-day, 5-days etc. profiles have progressively more time to expiry and consequently the peaks and troughs of the profiles progressively move away from the strike although the absolute maximum value of the binary put options vega remains fairly constant across the number of days.

Figure 2 provides binary put options vega profiles over a range of different implied volatilities.

Binary put options vega is zero when at-the-money so 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 options theta, binary call options vega and binary put options theta, the binary put option is not a good choice for taking a view on implied volatility owing to the risk reversal characteristic at the strike. If one were to take the view that implied volatility will fall then selling an out-of-the-money put would initially involve the directional risk of the underlying price falling along with the risk associated with a rise in implied volatility. If the underlying fell through the strike the risk of the direction of implied volatility is reversed so that if the speculator was correct and implied volatility fell, then this in itself would now cause a rise in the value of the option so increasing the loss. So even if the underlying was sold to hedge the directional risk in a delta-neutral manner, then initially the short gamma position would lose money, and then the risk of the speculator being right is now also a negative factor. The only way this strategy could be used profitably is if the speculator fancied a rise in volatility and bought the put delta-neutral, but even then there are risks in the underlying falling so far to create a loss that the long put cannot match in profits, along with the implied volatility rising once the underlying is below the strike. Long out-of-the-money puts/long underlying will almost undoubtedly work should the underlying rise but even this draws on the assumption that the negative theta doesn’t cause too much damage

Finite Vega

Figure 3of the Binary Put Options page shows a 5-day 25% implied volatility $1700 binary put option price profile. At the underlying gold price of $1725 this put is worth 31.4087. If 24.5% and 25.5% profiles were included then their values would be 29.4185 and 33.0882 respectively. Using the finite difference method:

**Binary Put Options Vega = (P _{1}―P_{2})/(σ_{1}―σ_{2})**

where:

σ_{1} = The higher implied volatility

σ_{2} = The lower implied volatility

P_{1} = Binary put option price with implied volatility of σ_{1}

P_{2} = Binary put option price with implied volatility of σ_{2}

so that the above numbers provide a 5-day, 25% binary put options vega of:

Binary Put Options Vega = (33.0882‒29.4185)/(25.5‒24.5) = 0.7290

If the implied volatility increment was reduced from 0.5 to 0.00001 then:

σ_{1} = 25.00001

σ_{2} = 24.99999

P_{1} = 31.408704

P_{2} = 31.408690

so that the 5-day 25% vega becomes:

Binary Put Options Vega = (31.408704‒31.408690)/(25.00001‒24.99999) = 0.7288

which is the actual tangent of the profile of the price with the horizontal axis being the underlying gold price and the vertical axis being implied volatility.