Volga and Smile

Volga Profile

The Volga is the sensitivity of the option vega to the implied volatility. In mathematical terms it is the second derivative of the option price with respect to the implied volatility. In simple terms in tells by how much the option vega changes when the implied volatility changes.  Options trading is also sometimes called volatility trading and the Volga is a measure of the convexity in the terms of volatility of an option price.

The Volga of the call and the put is identical and follows this profile:



The Volga of options increases as the maturity increases, although the Volga of at-the-money options is close to zero.

Let’s look at a numerical example:

						Spot		1500
						Strike		1400
	Volatility	Call Price	Vega	Volga		Maturity		0.5
	15%	127.67698	3.160734703	0.101868		Volatility		15.0%
	16%	130.88596	3.254623503			Interest Rates		3.0%
						Dividend yield		2.0%
		Vega Difference	0.093888799

It appears that the Vega exposure of call option move from 3.1607 to 3.2546. The theoretical value given by the Black-Scholes model is equal to 0.1018 which is close to the difference in our example.

						Spot		1400
						Strike		1400
	Volatility	Call Price	Vega	Volga		Maturity		1
	15%	88.642076	5.419936363	-0.00043		Volatility		15.0%
	16%	94.061726	5.419294665			Interest Rates		3.0%
						Dividend yield		2.0%
		Vega Difference	-0.0006417

From this example the At-The-Money call option has a Volga which is very slightly negative.

Volga in Practice

For the individual investor the Volga is one of the least important Greeks as for the individual investor his position will be mostly dominated by the delta.

For the professional  options trader, the Volga is quite important because it is the convexity to the the parameters which option traders take positions on. It is not possible to hedge out the Vega to be left with just the Volga so while the traders keep that exposure in mind, it is mostly a second order risk for them. That being said, when market makers run out f the money long put positions and implied volatility goes up, then their Vega increases as well, so they benefit from some convexity effect.

Spot price, TTE(time to expiry) and Volga  

Volga: sensitivity of vega to change in implied volatility

Vega = S0*SQRT(T)*N'(d1)

where,

d1= [ ln(S0/K)+rt ] / [vol*SQRT(T)] + [0.5*vol*SQRT(T)]

N'(x) is PDF of SND

Take a numeric example:

Say we have a FX option quotes as follows:

Underlying spot price		90.0000
Tenor (days)		90
Tenor (years)		0.247
r1 (base/foreign/denominator)		0.00%	1.000
r2 (counter/domestic/numerator)		0.00%	1.000
Underlying forward price		90.000
Premium included in delta (y/n)		0
Market Instruments:		Market Vol	Equ Vol
DN Straddle		32.00%
25d Risk Reversal		-1.700%	-1.721%
25d Strangle		0.50%	0.511%

With FX Volatility construction method, we can get the strike and Volga as follows:

Volga	Strike
9.458184	83.50
7.320253	84.35
5.355508	85.20
3.615208	86.05
2.146051	86.90
0.988551	87.75
0.175647	88.60
-0.26838	89.44
-0.32863	90.29
5.99E-06	91.14
5.99E-06	91.14
0.513076	91.79
1.246566	92.45
2.19156	93.10
3.336307	93.75
4.666511	94.40
6.165659	95.05
7.815377	95.70
9.595818	96.35
11.48605	97.01

As we increase spot rate to 110, all Volga are positive:

Volga	Strike
37.39092	83.50
38.85654	86.60
37.8378	89.70
34.10818	92.80
27.85596	95.90
19.84933	99.00
11.44894	102.10
4.364568	105.20
0.204116	108.30
3.03E-06	111.40
3.03E-06	111.40
5.154842	115.12
14.88374	118.84
26.75301	122.57
38.1841	126.29
47.23246	130.02
52.92872	133.74
55.20457	137.46
54.58597	141.19
51.85088	144.91

As volga represents the convexity, for the first scenario, there will be a small bump in the smile around ATM area.

The effect get amplified as TTE(Time to Expiry) increases. As time goes by, Volga curve moves up as follows: