Volatility Arbitrage

Volatility arbitrage is a strategy in which a trader takes a position in options, hedged against movements in the underlying asset, in order to profit from a difference between two measures of volatility. The first is implied volatility, the level built into the option's market price. The second is the trader's forecast of the realized volatility of the underlying over the option's life. If the forecast is higher than the implied level, options look cheap and are bought. If it is lower, they look expensive and are sold.

The strategy is used by hedge funds, proprietary trading firms and the options desks of banks. Despite the name, it is not arbitrage in the strict sense of a riskless profit. Its outcome depends on a forecast that may be wrong, and it is better described as a relative-value trade on volatility.

The trader removes exposure to the direction of the underlying by making the position delta neutral. For a bought option this means holding an offsetting position in the underlying asset and adjusting it as the price moves, a process called delta hedging.

A trader who owns options has positive gamma. Each time the underlying moves, re-hedging means selling after a rise and buying after a fall, which locks in a small gain. Against this, the options lose value each day through time decay. The gains from re-hedging depend on how much the underlying actually moves, and the daily decay reflects the implied volatility paid. If actual movement exceeds what was priced in, the gains outweigh the decay. Collecting these gains is known as gamma scalping. A seller of options has the reverse position: daily income from decay, and losses from re-hedging.

For a delta-hedged option position over a short interval, the profit is approximately:

• $\Gamma$ is the gamma of the position.
• $S$ is the price of the underlying.
• $\sigma_{r}$ is the volatility realized over the interval, and $\sigma_{i}$ the implied volatility at which the option was traded.
• $\Delta t$ is the length of the interval in years.

A simpler rule of thumb multiplies the position's vega by the gap between the two volatilities. Suppose a trader buys a straddle with a vega of $5,000 per volatility point at an implied volatility of 18 percent, and the underlying goes on to realize 22 percent. The expected gain is roughly $5{,}000 \times (22 - 18) = 20{,}000$ dollars. The actual result will differ, because gamma changes as the price moves away from the strike, so the outcome depends on the path the price takes and not only on the average volatility.

• Outright long or short volatility. A single underlying's options are bought or sold against a forecast produced by volatility forecasting models.
• Surface relative value. Options at different strikes or expiries on the same underlying are traded against each other when one part of the volatility surface looks out of line with the rest.
• Dispersion. Options on an index are traded against options on its member shares, a position on correlation known as dispersion trading.
• Variance swaps. These contracts pay the difference between realized and implied variance directly, without the need to hedge.
• Cross-asset trades. Volatility in related markets, such as two share indices, is traded on the expectation that the gap between them will close.

On equity indices, implied volatility has on average exceeded the volatility later realized. The difference is the volatility risk premium. It means that selling hedged options has earned a positive return over long periods, and many volatility arbitrage programs are net sellers. The premium exists because buyers of options are willing to pay for protection, and sellers bear the risk of sudden large losses. A strategy that only sells volatility is therefore collecting a risk premium and not exploiting a mispricing.

• Forecast error. Volatility can differ sharply from any forecast, particularly around unexpected events.
• Jumps. Hedging assumes prices move continuously. A gap in the price cannot be hedged and produces a large loss for an option seller.
• Hedging costs. Frequent re-hedging incurs a transaction cost each time, and hedging less often leaves the position exposed between adjustments.
• Mark-to-market losses. Even if the forecast proves right by expiry, implied volatility can move against the position first, causing losses and margin calls.
• Model risk. Deltas and gammas come from a pricing model, and an unsuitable model produces a poor hedge.

The danger of short-volatility positions was shown on 5 February 2018, when a sudden jump in equity index volatility caused exchange-traded products that were short volatility futures to lose most of their value in a single day.