Volatility Arbitrage

Every option's price contains a forecast: the market's guess for how much the underlying will bounce around between now and expiry. That guess is the option's implied volatility. Volatility arbitrage is the trade you put on when you disagree with it, when you think the market has priced options too cheaply or too dearly relative to how much the stock will actually move. You bet on the gap and hedge away everything else.

The word "arbitrage" is generous here, it isn't riskless. A better name is relative-value volatility trading. You're not exploiting a certainty; you're taking a view that one number (implied volatility) is wrong relative to another (realized volatility), and structuring the trade so that view is the only thing you're exposed to.

The two volatilities

Everything hinges on distinguishing two things that share a name:

• Implied volatility is backed out of the option's price today. It's forward-looking, the movement the market is charging for.
• Realized volatility is measured from the underlying's actual price moves. It's backward-looking, the movement that truly happened.

If you own a delta-hedged option, gamma scalping tells you exactly what you earn: your re-hedging profits scale with realized volatility, while the theta you pay was set by the implied volatility you bought at. So the profit and loss of a hedged option position is, to first order, a clean bet on the difference:

$$\text{P\&L} \;\propto\; \sigma_{\text{realized}}^2 - \sigma_{\text{implied}}^2 .$$

Here $\sigma_{\text{realized}}$ is what the stock actually did and $\sigma_{\text{implied}}$ is what you paid for. Buy an option (long the realized, short the implied) and you profit when the stock moves more than priced. Sell one and you profit when it moves less.

Worked example

You think a stock is going to be calm, but its one-month options are pricing an implied volatility of 30% a year. You judge it'll realize only about 20%. So you sell the option and delta-hedge it.

Because the P&L tracks the squared vols, the edge is roughly proportional to $0.30^2 - 0.20^2 = 0.09 - 0.04 = 0.05$. Over the month, if the stock indeed pokes along at 20% realized while you collected theta priced at 30%, your gamma losses stay small and the theta you banked wins, you keep the gap. Say the position was sized so a full month at your forecast nets $8,000; that's your edge for being right about the calm.

But suppose you're wrong and the stock realizes 38%. Now $0.20^2$ becomes $0.38^2 = 0.144$, well above the 0.09 you sold, and every violent move costs you on the re-hedge faster than the theta comes in. The same position that would have made $8,000 can lose multiples of it. That asymmetry, short volatility loses badly when you're wrong, is the defining risk of the trade.

The persistent edge, and why it isn't free money

On average, across markets and time, implied volatility tends to sit above realized, options are, on the whole, a little expensive. This gap is the variance risk premium: buyers pay up for insurance and for protection against jumps, so sellers of volatility earn a premium for providing it. Systematically selling hedged options harvests it.

The catch is why that premium exists: it's compensation for real, occasional pain. Selling volatility is like selling insurance, steady small income, punctuated by the rare catastrophe when a crash makes realized vol explode past anything implied. The premium is the market paying you to hold that tail. Collect it carelessly and one bad month erases years of it.

What makes it hard in practice

• Hedging isn't perfect. You capture the realized-vs-implied gap only if you re-hedge cleanly and cheaply. Transaction costs and discrete (not continuous) hedging leak away part of the edge, sometimes all of it.
• Volatility isn't one number. Implied vol differs by strike (the smile) and by expiry (the term structure). A trade that's "long vol" can be secretly long one part of the surface and short another.
• Jumps break the model. The tidy squared-vol P&L assumes smooth moves. A gap, an earnings surprise, a halt, delivers a loss no delta hedge catches, and it always seems to hit the short-vol side.
• Your forecast is the whole game. You're betting your estimate of realized volatility against the market's. If your forecast is no better than implied, there's no edge, just costs and risk.

Closely related is Dispersion Trading, where the mispriced quantity isn't the level of volatility but the correlation baked into index options versus their components.