# Volatility arbitrage

Volatility arbitrage

Volatility arbitrage (or vol arb) is a type of statistical arbitrage that is implemented by trading a delta neutral portfolio of an option and its underlier. The objective is to take advantage of differences between the implied volatility of the option, and a forecast of future realized volatility of the option's underlier. In volatility arbitrage, volatility is used as the unit of relative measure rather than price - that is, traders attempt to buy volatility when it is low and sell volatility when it is high. [cite book
last= Javaheri
first= Alireza
title= Inside Volatility Arbitrage, The Secrets of Skewness
url= http://www.amazon.com/Inside-Volatility-Arbitrage-Secrets-Skewness/dp/0471733873/
date= 2005
publisher= Wiley
isbn= 978-0471733874
] [cite book
last= Gatheral
first= Jim
title= The Volatility Surface: A Practitioner's Guide
url= http://www.amazon.com/Volatility-Surface-Practitioners-Guide-Finance/dp/0471792519/
date= 2006
publisher= Wiley
isbn= 978-0471792512
]

Overview

To an option trader engaging in volatility arbitrage, an option contract is a way to speculate in the volatility of the underlying rather than a directional bet on the underlier's price. If a trader buys options as part of a delta-neutral portfolio, he is said to be long volatility. If he sells options, he is said to be short volatility. So long as the trading is done "delta-neutral", buying an option is a bet that the underlier's future realized volatility will be high, while selling an option is a bet that future realized volatility will be low. Because of "put call parity", it doesn't matter if the options traded are calls or puts. This is true because "put-call parity" posits a risk neutral equivalence relationship between a call, a put and some amount of the underlier. Therefore, being long a delta neutral call results in the same returns as being long a delta neutral put.

Forecast volatility

To engage in volatility arbitrage, a trader must first forecast the underlier's future realized volatility. This is typically done by computing the historical daily returns for the underlier for a given past sample such as 252 days, the number of trading days in a year. The trader may also use other factors, such as whether the period was unusually volatile, or if there are going to be unusual events in the near future, to adjust his forecast. For instance, if the current 252-day volatility for the returns on a stock is computed to be 15%, but it is known that an important patent dispute will likely be settled in the next year, the trader may decide that the appropriate forecast volatility for the stock is 18%.

Market (Implied) Volatility

As described in option valuation techniques, there are a number of factors that are used to determine the theoretical value of an option. However, in practice, the only two inputs to the model that change during the day are the price of the underlier and the volatility. Therefore, the theoretical price of an option can be expressed as:

:$C = f\left(S, sigma, cdot\right) ,$

where $S ,$ is the price of the underlier, and $sigma ,$ is the estimate of future volatility. Because the theoretical price function $f\left(cdot\right) ,$ is a monotonically increasing function of $sigma ,$, there must be a corresponding monotonically increasing function $g\left(\right) ,$ that expresses the volatility "implied" by the option's market price , or

:

Or, in other words, when all other inputs including the stock price $S ,$ are held constant, there exists no more than one "implied volatility" for each market price for the option.

Because "implied volatility" of an option can remain constant even as the underlier's value changes, traders use it as a measure of relative value rather than the option's market price. For instance, if a trader can buy an option whose implied volatility is 10%, it's common to say that the trader can "buy the option for 10%". Conversely, if the trader can sell an option whose implied volatility is 20%, it is said the trader can "sell the option at 20%".

For example, assume a call option is trading at \$1.90 with the underlier's price at \$45.50, yielding an implied volatility of 17.5%. A short time later, the same option might trade at \$2.50 with the underlier's price at \$46.36, yielding an implied volatility of 16.8%. Even though the option's price is higher at the second measurement, the option is still considered cheaper because the implied volatility is lower. The reason this is true is because the trader can sell stock needed to hedge the long call at a higher price.

Mechanism

Armed with a forecast volatility, and capable of measuring an option's market price in terms of implied volatility, the trader is ready to begin a volatility arbitrage trade. A trader looks for options where the implied volatility, is either significantly lower than or higher than the forecast realized volatility $sigma ,$, for the underlier. In the first case, the trader buys the option and hedges with the underlier to make a delta neutral portfolio. In the second case, the trader sells the option and then hedges them.

Over the holding period, the trader will realize a profit on the trade if the underlier's realized volatility is closer to his forecast than it is to the market's forecast (i.e. the implied volatility). The profit is extracted from the trade through the continual re-hedging required to keep the portfolio delta neutral.

ee also

*Delta neutral
*Volatility (finance)
*Implied volatility
*Option (finance)
*Statistical arbitrage
*Volatility smile

References

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