The following is an excerpt from Managing Expectations by Tony Saliba

# Option Vega

Compared to a stock or bond, options are contracts with a shelf- life and are exposed to a range of unique risks – greeks (i.e. delta, gamma, theta, vega, rho) – each of which measures the sensitivity to some variable including time, volatility, and movement.

Experienced derivative traders know that option prices actually boil down to the market’s expectancy of future volatility of the underlying asset, since all the other determinants of an option’s price—the underlying price, time to maturity, interest rate, and strike price—are objective. Volatility is the subjective unknown, and seldom does an option’s actual, realized volatility replicate the implied volatility reflected in its current traded valuation.

## Vega Defined

Interestingly enough, option vega is the one option greeks not represented by a formal Greek letter – it represents the sensitivity of an option to the changes in implied volatility for a term equivalent to its expiration date. Vega is an estimate of much the theoretical value of an option changes when implied volatility changes one percent.

• Vega is a number that expresses in what direction and to what extent the option price will move if there is a 1% change in the options implied volatility.

• Vega is the first mathematical derivative of an option price with respect to the underlying asset volatility.

• Option vega is equal for both call and put options with the same month and strike price (e.g., if the SPY August 190 call has a vega of .35, the August 190 put will also have a vega of .35).

• Options with less time to expiration have a lower option vega comparatively to options with a long time to expiration.

• Options are most sensitive to changes in the options implied volatility of the underlying asset when they are at-the-money calls or puts.

• Out of the money and in the money options are not nearly as affected by volatility (relative to at-the-money options).

• Option vega can be hedged with another option only. The best vega hedge is a nearby strike of the same expiration month. This relationship is reduced the further the hedged long month is from the hedged short month.

Typically, options professionals express vega as a distinct measure. For the sake of simplicity, professionals multiply vega by the current level of volatility in an effort to make it correspond to a standard percentage move in asset volatility level. If S&P 500 (SPX) volatility is 28% and the option vega is .2, the option will theoretically gain or lose 20 cents when the volatility rises (falls) by one percentage point to 27%.

Long (purchased) calls and puts always have positive vega. Short (sold) calls and puts always have negative vega. A call and put with the same strike price and month will have the same vega. Underlying futures and securities have zero vega as their values are linear and thus not affected by changes in implied volatility.

Vega risk is the risk due to changes in volatility, or the “volatility of volatility”. A strict understanding of vega risk is important in any options strategy or position, as it can generate unforeseen risk, even if all the other greeks are hedged perfectly.

Options vega is similar in shape to both options gamma and theta as an options vega reaches its plateau when it’s at the money. An option’s vega differs from both gamma and theta whereas vega generally increases with time while it generally decreases (with time) for both gamma and theta. That being said, the vega of an at the money option is fairly dependable to volatility changes. However, options further away from the money are not as stable given the complexity of changes in the volatility structure.

## Practicality & Reality of Vega

A good trader needs to understand his risk and be ready to address that risk when personal and prearranged risk limits are breached. The trader’s ability to interpret vega is of utmost importance as its purpose is to gauge the trader’s position sensitivity to changes in the assets implied volatility – one of the largest and most unpredictable risks the trader will face.

Vega can be used to evaluate risk across products, strike prices, and time frames although greater caution needs to be taken here as complexity begets complexity. Consider an options portfolio with random long and short positions spread out amongst various strike prices along with different times to expiration. A seasoned trader will quickly recognize that the “net vega” published by a trading model is at-best “raw vega”. Like the other greeks mentioned, vega is an estimate and subject to wise discernment. Variables including options skew (see Chapter 11) and term structure (see Chapter 12) can and do alter the true value of an option’s vega.

Additionally, the majority of an option price “fair-value” is derived by either a Black-Scholes model or iteration thereof. The options model produces a “fair-value” of an option based on five variables including – the underlying price, the strike price, the term of the option, the interest rate, and the current volatility. A skilled trader will realize (usually through a bad experience) that an options model is using a static volatility input and thus, the “fair-value” option price produced will then assume a static volatility environment. Consequently, option traders realize that options vega is both an estimate and subject to interpretation. It’s difficult – if not impossible – to measure the real vega sensitivity of an options portfolio with a static, human recorded asset volatility.

Under the conventions of the Black-Scholes model, underlying asset return volatility is constant, which is rarely supported by research nor is it realistic! Furthermore, option implied volatility fluctuates over any given period of time – thus the importance of vega.

## Options Vega and Options Strike Price

In theory it’s easy to conceptualize. An option vega – it’s sensitivity to changes in implied volatility – is at its greatest point with an at- the-money option. An option’s vega becomes less and less the further your option is from the at the money strike. What that means is a 1.00-delta call will have almost no vega, a .70-delta call will have more vega compared to a 1.00-delta call and less than a .50-delta call. Similar would hold true for puts.

In a perfect world – one that uses a static volatility – equidistant strike prices would have the same vega. Other words, mathematically speaking a 5% out of the money call should have the same exact vega as an equidistant 5% out of the money put. Yet, due to options implied skew, it could be precisely true that an equidistant 5% out of the money call could have more/less vega than a 5% out of the money put.

A successful trader will have a solid understanding of both the textbook and realistic definition of options vega. Most products have some sort of implied volatility skew and that skew is not static – it changes with sentiment, time, and supply versus demand. As options trades evolve to a strategy and then a position, it’s essential that a trader understand vega on an intuitive level.

## Options Vega and Time

• Vega increases with longer time to expiration and decreases with less time to expiration.

• The term structure of implied volatility describes, for a given exercise strike price at a given date in time, the relationship between implied volatility and option maturity.

It should be of little surprise that the less time to expiration the more reasonably accurate “the market” can be in assessing where the underlying will land at expiry. Similarly, more time to expiration equals less precision – more unknowns – on where the underlying will land. Thus, the more time to an option’s expiration – the more vega an option will have. This makes sense as time value makes up a larger percentage of the premium for longer-term options and it is the time value that is sensitive to changes in volatility. This results in a higher vega for options with longer time to expiration in order to compensate for the additional risk assumed by the seller.

A consistent options trader will not compare, net, add, or subtract the vegas of an options position resultant of various maturities. They will first know precisely where the long and short vega is BEFORE netting vega.

## Vega and Volatility

Vega and implied volatility are certainly not the same thing but they are correlated. Vega tells us an option’s (or an option strategy or positions) sensitivity to implied volatility. Implied volatility is the premium – or extrinsic value paid for the option.

Theoretically speaking, if you isolate – underlying movement, time, and options skew – changes in volatility will affect option prices but change them differently. Option vega is greatest for options at-the- money options, and it is smaller for options completely out of the money or very deep in the money.

Said differently, if an option you’re long (purchased) is far out of the money and near worthless, it matters little how much the underlying’s implied volatility shifts, because the odds the option will suddenly become at the money or in the money are still considered relatively small.

If the option is deeply in the money, the chance that the option will suddenly become worthless with increased volatility is also relatively small. But if the option is at the money, which is on the edge of being worthless or valued, then even a relatively fractional change in the implied volatility in the price of the underlying asset can change the position. Thus, the reason why vega is at its highest point for at the money options.

This is an excerpt from *Managing Expectations* by Tony Saliba.

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### Michael

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