Volatility Trading

In option trading, the profit and loss of a delta-hedged position can be viewed as the difference between the movement the market priced in at the time of the trade and the movement that actually occurs. This view is used when thinking through how profit and loss arises for each type of trade, and when managing the risks of a short position.

Trading the size of moves, not the direction

When you buy an option, your profit and loss depends on both the direction of the underlying price and volatility. Delta hedging removes the sensitivity to price direction, and what remains is the sensitivity to volatility. The hedge is never perfect, though, because real markets depart from the Black-Scholes assumptions. The departures include volatility that cannot be estimated exactly, trading that cannot be continuous, transaction costs, jumps in prices, and a lack of liquidity¹.

The profit and loss of a volatility trade is determined by the difference between implied volatility at the time of the trade and the realized volatility observed afterward. The buyer of volatility receives money if realized volatility ends up above the level traded and pays if it ends up below. The seller is the reverse. The profit and loss formula after delta hedging is covered in Managing Gamma and Vega.

Instruments

The instruments used to trade volatility differ in what they are sensitive to and in the risks their holders take on.

The mechanics and valuation of variance swaps are covered in Variance Swaps and Volatility Swaps. According to a 2004 working paper by Carr and Wu, Cboe changed how the VIX is calculated on September 22, 2003, to bring it closer to the 30-day variance swap rate on the S&P 500. Trading in VIX futures began on March 26, 2004⁵.

One-day profit and loss of a straddle

Consider a hypothetical example in which R Corp. buys an at-the-money straddle and resets its delta to zero every day. The underlying price is 100, the strike is 100, there is 1 year to expiry, the interest rate is 1%, and implied volatility is 20%. The gamma is 0.019724 for one unit of the call, and 0.039448 for one unit of the straddle, the sum of the call and the put⁶.

Using the formula in "Managing Gamma and Vega," the profit and loss for one day (with a year taken as 365 days) is as follows. This is an approximation that omits the interest rate term⁷.

$$\text{P\&L} \approx \frac{1}{2}\,\Gamma S^2 \left[\left(\frac{\Delta S}{S}\right)^2 - \sigma^2 \Delta t\right]$$

Here $\frac{1}{2}\Gamma S^2 = 197.24$ and $\sigma^2 \Delta t = 0.04 / 365 = 0.00010959$. When the daily move is 2%, the profit and loss is $197.24 \times (0.0004 - 0.00010959) = 0.0573$, a gain, whether the price rises or falls. When the move is 0.5%, it is $197.24 \times (0.000025 - 0.00010959) = -0.0167$, a loss. When the move is $0.2 \times \sqrt{1/365} = 1.0468\%$, the profit and loss is about zero. For the side that sold the straddle and hedged, the signs are reversed⁶. The formula is a one-day approximation and does not include the vega profit and loss when volatility changes.

Trading the gap between realized and expected

The volatility risk premium is realized variance minus the variance the market priced in (the variance swap rate). Carr and Wu approximated the variance swap rate with a value built from a combination of options, and measured the variance risk premium as its difference from the realized variance observed afterward⁸.

In a study of option data from January 1996 to February 2003, the average variance risk premium for the S&P 500 index, the S&P 100 index, and the Dow Jones Industrial Average was strongly negative. The Nasdaq 100 index and many individual stocks were also negative, but with smaller absolute values, as the paper reports⁹. A negative value means that average realized variance was below the variance swap rate. The paper interprets this as buyers of variance accepting a loss on average as the price of protection against rises in index variance⁹.

This is the direction of the average. Values for individual periods scatter around the average¹⁰, so in any given period realized variance can exceed the implied level. Consider a hypothetical example in which R Corp. sells a variance swap with a strike of 25% (625 in variance terms) and a vega notional of 2 million yen per volatility point. This shows the shape of the seller's profit and loss. The variance notional is 2 million yen ÷ (2 × 25) = 40,000 yen, and R Corp.'s payoff is $40{,}000\text{ yen} \times (625 - \sigma_R^2)$¹¹.

Figure description

The curve shows the payoff to R Corp., which sold a variance swap with a strike of 25% and a vega notional of 2 million yen per volatility point, for each level of realized volatility. At realized volatility of 0%, the payoff is 25 million yen, and it falls toward the lower right as volatility rises. It is 0 yen at the 25% strike, −11 million yen at 30%, and −39 million yen at 40%. On the loss side the curve gets steeper the further right it goes, and there is no floor at the right end.

R Corp.'s maximum profit is 25 million yen, reached when realized volatility is 0. On the other side, the loss grows with the square of realized volatility as it rises, and on paper it has no upper limit. Even when implied volatility was priced high, a period in which realized volatility exceeds it produces a loss. The error a seller bears when prices jump is covered in the section on jumps.

Trading the smile and the term structure

Some trades focus not on the level of implied volatility itself but on its differences across strikes and maturities.

Risk reversals

A risk reversal combines a long call and a short put whose strikes are about equally far above and below the market level. Using currency options as its example, a Bank of Japan paper explains that the difference between the call and put premiums reflects a skew in market participants' wariness of sharp moves up and down¹². Index options also show a skew in which implied volatility is higher at lower strikes.

Because the position combines a long call and a short put, it gains when the market rises and loses when it falls, and the premium received on the short put pays for part of the cost of the long call¹³.

Calendar spreads

A calendar spread sells a near-maturity option and buys a far-maturity option at the same strike. Because the far-maturity option has the larger vega, the position has positive vega and is positioned for a rise in implied volatility. When set up at the money, however, gamma is negative and theta is positive. If the market moves sharply, the gamma loss can exceed the vega gain¹⁴.

Another caution is that implied volatility moves differently at different maturities. An Osaka Exchange educational column (in Japanese) explains that the nearer maturity tends to move more sensitively. If only the implied volatility of the near maturity you sold rises, the long far-maturity position cannot cover the loss¹⁴. Also, if you set up the spread when far-maturity implied volatility is already high, you sell the cheaper near maturity and buy the more expensive far maturity, so you enter at an unfavorable level from the start¹⁴.

Dispersion trading

Dispersion trading takes opposite positions in options on a stock index and options on its constituents. Dhaene et al. note that this strategy can be built from calls, puts, straddles, and variance swaps¹⁵.

The variance of an index is determined by the variance of each constituent and the correlation among them. When you hold index options and constituent options in opposite directions, the profit and loss therefore depends on the size of each stock's moves and on the level of correlation. The correlation under the risk-neutral measure, backed out of option prices, is called implied correlation, and its difference from realized correlation is the source of profit and loss. The paper states that implied correlation can differ greatly from realized correlation, that this difference is called the correlation risk premium, and that dispersion trading is one way to target it¹⁶.

Consider a hypothetical example in which an index holds two stocks in equal halves and each stock has a volatility of 30% (900 in variance terms). This isolates the effect of correlation. With correlation $\rho$ between the two stocks, the index variance is as follows¹⁷.

$$\sigma_I^2 = 0.25 \times 900 \times 2 + 2 \times 0.25 \times 900 \times \rho = 450 + 450\rho$$

If implied correlation is 0.6, the strike of the index variance is $450 + 450 \times 0.6 = 720$ (about 26.8% in volatility terms). Suppose R Corp. sells a variance swap on the index and buys variance swaps on each stock at 0.25 times the amount. The variance notional is 10,000 yen for the index. This direction was chosen for the calculation and does not indicate the direction of an actual trade. If each stock's realized variance equals its strike of 900, the stock side has zero profit and loss, and R Corp.'s payoff is determined by the index side alone.

R Corp. gains if realized correlation is below implied correlation and loses if it is above¹⁷. In practice each stock's realized variance also departs from its strike, and that profit and loss is added. The paper proposes the dispersion swap, a new contract that trades the correlation difference directly. The procedure shown here is a separate, simple example built from variance swaps¹⁵.

Risks of volatility trading

Gamma and theta

A short straddle receives premium through theta each day and loses through gamma the more the price moves. It is the side where the signs in the one-day formula above are reversed: with small moves it edges toward a profit as time passes, and with sharp moves it takes a loss. For the short straddle, the Options Industry Council explains that volatility can arrive suddenly like a storm and produce an extreme loss, or may not arrive at all, while the passage of time moves the position a little closer to the hoped-for profit each day¹⁸.

Jumps

Suppose you sell a variance swap and hedge it by replicating with a combination of options. If the price jumps, the replication departs from realized variance. To first order, the departure is proportional to the cube of the jump size. It works in the seller's favor when the price jumps sharply down and against the seller when it jumps sharply up¹⁹. Because delta hedging assumes continuous trading, even a short straddle cannot reset its delta in time when the price jumps¹.

Liquidity

A lack of liquidity is one of the reasons given for why delta hedging is inexact¹. When the market falls sharply, liquidity in far-maturity options drops significantly, which can make at-the-money calendar spreads hard to set up²⁰.

Model assumptions and what this article does not cover

This page does not cover the level of margin required for short positions, the quoting convention for risk reversals on index options, how to build trades around changes in skew, or futures on volatility indices. The valuation of a volatility swap depends on the assumption about the volatility of volatility⁴. The sign of the volatility risk premium is the direction of an average measured on data through 2003, and it does not promise the profit and loss of any future period⁹.

Notes

1. Goldman Sachs, "More Than You Ever Wanted To Know About Volatility Swaps" (March 1999), page 1 (Introduction; PDF page 5). It gives as reasons delta hedging is inexact the estimation of volatility, continuous trading, transaction costs, discontinuous market moves, and liquidity. ↩ ↩² ↩³

2. The Options Industry Council, "Short Straddle," the Max Loss and Max Gain entries. The maximum gain of a short straddle is the premium received, and the maximum loss is unlimited. ↩

3. The same paper, page 3 (equation 2 and its explanation; PDF page 7). The payoff of a variance swap is the difference between realized variance and the delivery price, multiplied by the notional. Because realized variance is zero or more, the seller's profit is capped at the product of the notional and the strike, and the loss grows as realized variance rises. The formula for the cap is the calculation in the main text. ↩

4. The same paper, page 33 (PDF page 37). A volatility swap has no simple replication method, and variance is the quantity that arises naturally from hedged option trading. That valuation needs an assumption about the volatility of volatility is based on the explanation from this page onward (PDF page 37 onward). ↩ ↩²

5. Carr and Wu, "Variance Risk Premia" (May 2004 version), page 2 (PDF page 4). On September 22, 2003, Cboe redefined the VIX to bring it closer to the 30-day variance swap rate on the S&P 500, and on March 26, 2004, the Cboe Futures Exchange began trading VIX futures. ↩

6. A hand calculation. $d_1 = 0.15$, and the gamma is $\varphi(0.15) / (100 \times 0.2 \times 1) = 0.019724$ ($\varphi$ is the standard normal density function). The put has the same gamma as the call, so the straddle has twice that, 0.039448. ↩ ↩²

7. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), pages 11–12 (equation 24). The profit and loss is approximated by splitting it into delta, gamma, and vega terms. The formula in the main text removes the delta term by hedging, sets the vega term to zero, and adds the theta term. ↩

8. Carr and Wu, "Variance Risk Premia," Abstract and page 1 (PDF page 3). The variance swap rate can be approximated by the value of a particular combination of options, and apart from the approximation error, the difference between realized variance and this synthetic value represents the variance risk premium. ↩

9. The same paper, page 3 (PDF page 5), and the passage stating the data period (January 1996 to February 2003; PDF page 27). It states the signs of the averages for indices and individual stocks, and interprets the negative sign as buyers of variance accepting a negative excess return in order to hedge rises. ↩ ↩² ↩³

10. Carr and Wu, "Variance Risk Premia," Table 5 (printed page 53; PDF page 55). The difference between realized variance and the variance swap rate for the S&P 500 (as a value multiplied by 100) has a mean of −2.279 and a standard deviation of 3.339, so the scatter is larger than the mean. ↩

11. A hand calculation. The variance notional is $2{,}000{,}000 / (2 \times 25) = 40{,}000$ yen, and the seller's payoff is $40{,}000 \times (625 - \sigma_R^2)$ yen. At realized volatility of 0 the payoff is $40{,}000 \times 625 = 25{,}000{,}000$ yen. The relation for the notional is the same as equation 43 on page 33 of the Goldman Sachs paper (PDF page 37). ↩

12. Bank of Japan, "リスク・リバーサルからみた為替変動へのリスク認識" (Risk Perception of Exchange Rate Fluctuations from Risk Reversals, our translation; Bank of Japan Review 2012-J-14, August 2012) (in Japanese), page 2. It describes the trade of buying a call and selling a put at strikes about equally far above and below the market level, and the view that the premium difference reflects a skew in wariness of sharp moves. The example is currency options. ↩

13. This is the explanation in the main text. It follows from the basic shapes of option payoffs: a long call gains value as the underlying rises, and a short put loses more as the underlying falls. ↩

14. Osaka Exchange, "カレンダースプレッド" (Calendar Spread, our translation; Kitahama Investment School column) (in Japanese). Vega is larger for the far maturity. Set up at the money, gamma is negative and theta is positive, and a large market move turns gamma into a loss. The near maturity tends to be more sensitive in its implied volatility changes, and setting up the spread when far-maturity implied volatility is high is unfavorable from the start. ↩ ↩² ↩³

15. Dhaene, Linders, Ling and Wang, "Understanding the correlation risk premium" (Annals of Actuarial Science, 2025), section 5.2 (journal pages 246–247). It describes examples built from calls, puts, straddles, and variance swaps, and the dispersion swap that the paper newly proposes. ↩ ↩²

16. The same paper, Abstract and Section 1 (pages 233–234). Implied correlation indicates the relative cheapness or richness of options on the index constituents, and its difference from realized correlation points to the existence of a correlation risk premium. The difference between risk-neutral and real-world correlation is called the correlation gap, and dispersion trading can be used to target it. ↩

17. A hand calculation. The index variance is $450 + 450\rho$: 720 at $\rho = 0.6$ ($\sqrt{720} = 26.83$), 630 at 0.4, and 810 at 0.8. The payoff of selling the index is 720 − 630 = 90 (900,000 yen at a notional of 10,000 yen) and 720 − 810 = −90. On the stock side, the profit and loss is zero if realized variance equals the strike. ↩ ↩²

18. The Options Industry Council, "Short Straddle," the Comments entry ("a race between volatility and time decay") and the Volatility and Time Decay entries. ↩

19. Goldman Sachs paper, page 31 (equations 40–42 and their explanation; PDF page 35). When the price jumps over a short interval, the first correction to the difference in profit and loss between a short variance swap and the replicating strategy is proportional to the cube of the size of the jump, giving a gain for a large fall and a loss for a large rise. The loss on a short straddle is the explanation in the main text. ↩

20. Osaka Exchange, "カレンダースプレッド" (Calendar Spread, our translation; Kitahama Investment School column) (in Japanese), the explanation of strategy 2. In a sharp fall, an at-the-money calendar spread is hard to set up because liquidity in the far maturity drops significantly. ↩