Volatility arbitrage

Volatility arbitrage, a.k.a. Vol Arb, is a trading strategy in which a delta neutral portfolio of an option and its underlier are traded and held for long periods of time. 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. Over the holding period, the portfolio will be continually re-hedged as the underlier's price changes to keep the portfolio delta neutral.

Overview

To a professional option trader engaging in volatility arbitrage, an option contract is a way to speculate in the volatility of the underlier rather than a directional bet on the underlier's price. 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 an risk neutral equivalence relationship between a call, a put and some amount of the underlier. Therefore, being a long a delta neutral call is the same 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 historic 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 is computed to be 15%, but it's known that an import patent dispute will likely be settled in the next year, the trader may decide that the appropriate forecast volatility for the stock is 18%. That is, based on past movements and upcoming events, the stock is most likely to be 18% higher or lower from it's current price one year from today.

Market (Implied) Volatility

As described in option valuation techniques, there a number of factors that are used to determine the theoretical value of an option. However, it 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 expressed as:

$C=f(S,\sigma ,\cdot )\,$

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

$\sigma _{\bar {C}}=g({\bar {C}},\cdot )\,$

Or, in other words, there exists no more than one implied volatility $\sigma _{\bar {C}}\,$ for each market price ${\bar {C}}\,$ for the option.

Because implied volatility of an option can remain constant even as the underlier's value change, traders often 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 $\sigma _{\bar {C}}\,$ 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 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 need to hedge the long call at a higher price.

Mechanism

To engage in volatility arbitrage, a trader looks for options where the implied volatility, $\sigma _{\bar {C}}\,$ 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 hedges.

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.