Variance Swaps and Volatility Swaps

A dealer that sells a variance swap can hedge it by replicating it with a portfolio of options¹. Volatility swaps come up in directional views on realized volatility and in hedging positions that tend to lose when volatility rises².

What the two parties exchange

A variance swap is a trade in which two parties exchange, once at maturity, the difference between realized variance and a strike. Volatility is written as a percentage number, so 20% is 20. Variance is the square of annualized volatility, so the variance of 20% volatility is 400. One variance point is a change of 1 in this number. The buyer receives the notional amount for each point by which realized variance exceeds the strike³. As a formula, the buyer receives:

$$N_{\text{var}} \times \left(\sigma_R^2 - K_{\text{var}}\right)$$

Here $\sigma_R^2$ is the annualized variance actually observed over the life of the contract, $K_{\text{var}}$ is the strike set at the start (the delivery price), and $N_{\text{var}}$ is the variance notional. When the amount is negative, the buyer pays the seller. A volatility swap uses volatility without squaring it, and the buyer receives $N_{\text{vol}} \times (\sigma_R - K_{\text{vol}})$. Here $N_{\text{vol}}$ is the notional per volatility point⁴.

The strike is also quoted as variance. For example, 30% volatility is $30^2 = 900$. The variance notional can be expressed as an amount per volatility point squared⁵.

The contract defines how realized variance is counted. It sets the source of the stock prices (such as closing prices) and how often they are observed, the factor that annualizes daily moves (for example, 260 business days a year), and whether the average return is subtracted from each move. Not subtracting the average comes closest to a contract that can be replicated with options⁶.

Expressing the notional and the payments at maturity

Volatility swaps and variance swaps are easier to compare when the notional is expressed as an amount per volatility point (the vega notional). The variance notional $N_{\text{var}}$ follows from the vega notional $N_{\text{vega}}$ as follows:

$$N_{\text{var}} = \frac{N_{\text{vega}}}{2K_{\text{vol}}}$$

With this notional, if realized volatility $\sigma_R$ equals the strike $K_{\text{vol}}$, the payments of the variance swap and the volatility swap match, and so does their change when volatility moves by one point. When $\sigma_R$ is near $K_{\text{vol}}$, the match works as an approximation, and it gets worse as the two move apart⁷. The 25% row of the table below shows an example of that gap.

Consider a hypothetical example in which Company O buys a contract on a stock index with a 20% strike (400 in variance) and a vega notional of 1 million yen per volatility point. The variance notional is 1 million yen ÷ (2 × 20) = 25,000 yen. The volatility swap is assumed to have the same 20% strike, for comparison only.

The seller's receipt has the opposite sign. For example, if realized volatility is 30%, the seller of the variance swap pays 12.5 million yen. As realized volatility rises above 20%, the buyer's receipt on the variance swap grows faster than on the volatility swap. When it falls, the buyer's payment on the variance swap is smaller than on the volatility swap. The payment when realized volatility is 0 is the limit of the buyer's loss, which in this example is 25,000 yen × 400 = 10 million yen⁸. The seller's loss, on the other hand, has no upper limit in the formula.

The difference from a volatility swap is convexity

The buyer's receipt on a variance swap is a quadratic function of volatility, while that on a volatility swap is linear. This difference in shape is called convexity. It is why the two receipts in the table above drift apart.

Figure description

The curves show the buyer's receipt on the two swaps for each level of realized volatility, with the notional set up as above. The two lines cross at the strike. The volatility swap is a straight line and the variance swap is a curve. Above the strike, the variance swap bends upward and receives more, and below it, the payment is smaller.

With the notional set up as above and the strikes aligned, the receipt on the variance swap always exceeds that on the volatility swap. The difference is proportional to the square of the gap between realized volatility and the strike, and in a formula it is $\frac{(\sigma_R - K_{\text{vol}})^2}{2K_{\text{vol}}}$. If this difference were always positive, buying a variance swap and selling a volatility swap would always make a profit. To avoid this, the fair strike of a volatility swap is lower than the square root of the fair strike of a variance swap. How much lower cannot be determined without assumptions about the level of future realized volatility and how much it varies⁹.

A Goldman Sachs paper states that a volatility swap has no simple replication method. One approach treats variance as the underlying asset and volatility as a derivative on its square root, and replicates by trading variance swaps continuously, but that requires a model that includes the volatility of volatility¹⁰. Around 1999, there was no variance swap market liquid enough to use for replication¹⁰. A paper by Sugihara at the Bank of Japan's Institute for Monetary and Economic Studies, by contrast, states that in theory both variance swaps and volatility swaps can be synthesized with a combination of options, with no dynamic hedging needed¹¹.

Replication with a portfolio of options

When the stock price moves continuously, variance can be reproduced by holding a contract that pays the logarithm (a log contract) and adjusting it by trading the stock¹². A log contract is not traded, so it is replaced by puts and calls at every strike from 0 to infinity, each held in an amount inversely proportional to the square of the strike¹². With weights of $1/K^2$ on the strikes, the sensitivity to variance (vega) stays constant regardless of the stock price level¹³.

The cost of this replication is the fair strike. Using market option prices, the fair strike can be found even in a market with a volatility skew, where the Black-Scholes formula does not hold¹². Replication also serves as the hedge when a dealer sells a variance swap¹. Why it is not a perfect hedge is explained in the section "Jumps, the strike range, and caps". The assumptions behind replication are covered in No-Arbitrage and Replication.

Deep diveFormula for replication with a log contract

When the stock price $S_t$ moves continuously, the daily variance $\sigma^2 dt$ can be split into the stock's movement and a logarithm with the following identity.

$$\frac{2}{T}\left[\int_0^T \frac{dS_t}{S_t} - \ln\frac{S_T}{S_0}\right] = \frac{1}{T}\int_0^T \sigma^2 dt$$

The first term on the left is the result of a strategy that keeps trading to hold 1 yen of the stock at all times, and the second term is a short position in a log contract. If the price moves continuously, this relationship holds along any price path¹². Taking the risk-neutral expectation, the first term becomes $rT$ ($r$ is the risk-free rate and $T$ is the number of years to maturity)¹². This derivation is written for the case with no dividends, and including dividends does not change it much¹⁴.

A log contract can be split, at a boundary $S^*$, into a forward and a portfolio of puts and calls. Puts are used for strikes below $S^*$ and calls for strikes above it. $P(K)$ and $C(K)$ are the current prices of the put and call with strike $K$. The fair strike can then be written as follows¹².

$$K_{\text{var}} = \frac{2}{T}\left[rT - \left(\frac{S_0}{S^*}e^{rT} - 1\right) - \ln\frac{S^*}{S_0} + e^{rT}\int_0^{S^*}\frac{P(K)}{K^2}dK + e^{rT}\int_{S^*}^{\infty}\frac{C(K)}{K^2}dK\right]$$

$S_0$ is the stock price at the start. Only a finite set of strikes can actually be traded, so the integrals are replaced by sums over strikes. The amount at each strike is chosen so that the payoff never falls below that of the log contract¹⁵.

The strike and implied volatility

The fair strike of a volatility swap is close to the implied volatility of options with the same maturity². For a variance swap, the strike is determined by the sum of the prices of the options used for replication, weighted in inverse proportion to the square of the strike. If there is no skew and implied volatility is the same at every strike, the fair strike equals the square of that implied volatility.

With a skew, the two do not match. If implied volatility is higher at lower strikes, the fair strike is higher than the square. Consider a hypothetical example in which a stock index has a 3-month maturity, an at-the-money implied volatility of 20%, and implied volatility that rises 1 percentage point for every 5 the strike falls. With the index at 100 at the start and a risk-free rate of 5% a year, estimating from options at strikes from 50 to 150 in steps of 5 gives a strike of 20.467 squared (about 419). As the strike spacing is made finer, the variance strike approaches about 402 with the skew, against 400 ($20^2$) without it¹⁶. When the spacing is coarse, the log contract is overvalued, so the estimate comes out higher than the theoretical value¹⁵.

This relationship is also used in indexes of expected volatility on stock indexes. In calculating the Cboe volatility index, the component options are weighted in inverse proportion to the square of the strike. This weight is the same as the weight used to replicate the payoff of a variance swap with options¹⁷. The variance swap rate equals the square of the model-free implied volatility (MFIVar) obtained from a combination of options¹⁸. How the VIX and the Nikkei Stock Average Volatility Index are calculated is covered in Volatility Products such as VIX Futures.

Jumps, the strike range, and caps

Replication holds when the stock price moves continuously and strikes can be traded from 0 to infinity. In practice, two things break down.

One is that the range of tradable strikes is limited. If the range is narrow, then when the stock price moves outside it, the vega and gamma of the replicating portfolio approach 0, and it fails to accumulate realized variance at the right rate. As a result, a fair variance calculated from a narrow range of strikes is lower than the true value¹⁹.

The other is jumps in the stock price. A replication strategy that keeps adjusting by trading the stock accumulates an amount different from realized variance when there is a jump. If a variance swap is sold and hedged with this strategy, the gap in profit and loss is first proportional to the cube of the jump size $J$ (with a fall counted as positive). The hedged seller gains when the price falls sharply and loses when it rises sharply²⁰. If the jump moves the price outside the range of strikes, the gain or loss from the jump is joined by the effect of failing to accumulate realized variance afterward²⁰.

To limit losses, some participants choose variance swaps or volatility swaps with a cap²¹. A capped variance swap embeds an option on realized variance, and valuing it requires a stochastic model of volatility²¹. There are also volatility swaps that pay on volatility calculated after excluding returns above a certain size (a corridor type)²².

What they are used for and what remains with the dealer

The paper from around 1999 lists three uses of volatility swaps². The first is a directional trade, buying or selling according to a view on the level of realized volatility. The second is closing out before maturity to trade the gap between realized volatility and implied volatility. The third is hedging a position that tends to lose when volatility rises. For example, in a trade that bets on the gap between the stock prices of two companies narrowing because a merger is expected to close, a rise in overall market volatility can make the merger less likely to close, widen the gap, and cause a loss. Equity funds are also probably in a position that tends to lose, because volatility tends to rise when an index falls².

In every use, profit and loss can go in both directions. For example, a volatility buyer receives if realized volatility is above the strike and pays if it is below. Volatility tends to rise when a stock index falls, and tends to stay high after a large fall. If a holder of a stock portfolio buys volatility, then when the index falls sharply and realized volatility exceeds the strike, the swap pays in the direction opposite to the loss on the assets. If the market calms and realized volatility falls below the strike, the swap requires a payment².

The selling dealer is left with replication error. Because of the strike range and spacing and the effect of jumps, the result of replication does not match realized variance. When a volatility swap is sold, assumptions about the volatility of volatility also drive the model's result¹⁰. The hedging error from approximating strikes in discrete steps tends to appear especially near the money, and it is reduced by adjusting the amounts of calls and puts¹.

Notes

1. 杉原慶彦, "わが国株式市場のモデルフリー・インプライド・ボラティリティ" (Model-free implied volatility in the Japanese stock market), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2010), page 118, Appendix 3 (2) イ, "バリアンス・スワップのヘッジ" (Hedging a variance swap) (equations A-23 to A-25) (in Japanese). How to split the amounts of calls and puts near the forward at-the-money point is in equation A-25. Page 46 of the PDF. ↩ ↩² ↩³

2. The same paper, pages 2–3 ("Who Can Use Volatility Swaps?"). The fair value of a volatility swap strike is close to the implied volatility of options with the same maturity; volatility is often negatively correlated with a stock index and tends to stay high after large falls; and the three uses. The paper is from around 1999 and describes volatility swaps. Pages 6–7 of the PDF. ↩ ↩² ↩³ ↩⁴ ↩⁵

3. Goldman Sachs, "More Than You Ever Wanted To Know About Volatility Swaps" (March 1999), page 3 (equation 2 and its explanation). The payment is the difference between realized variance and the delivery price, times an amount per volatility point squared (the notional). Page 7 of the PDF. ↩

4. The same paper, pages 1–2 (equation 1 and its explanation). Pages 5–6 of the PDF. ↩

5. The same paper, page 4. The paper states that the delivery price of a variance swap is quoted as the square of volatility, such as (30%)², and that the notional can be quoted "per volatility point squared". Page 8 of the PDF. ↩

6. The same paper, page 2. The three points to specify in the contract when calculating volatility (the source and frequency of stock prices, the annualization factor, and whether the average is subtracted), and the fact that not subtracting the average comes closest to replication. Page 6 of the PDF. ↩

7. The same paper, page 33 (equation 43 and its explanation). The condition under which a variance swap with variance notional 1/(2 K_vol) approximates a volatility swap. Page 37 of the PDF. ↩

8. Realized volatility cannot fall below 0, so the lower limit of the buyer's receipt is 25,000 yen × (0 − 400) = −10 million yen. The seller's payment has no upper limit in the formula, because realized variance has no upper limit. ↩

9. Goldman Sachs, "More Than You Ever Wanted To Know About Volatility Swaps", pages 33–34. The gap in the approximation of equation 43 (the convexity bias) is (σ_R − K_vol)²/(2 K_vol) and is always positive, and K_vol < √K_var. Pages 37–38 of the PDF. ↩

10. The same paper, pages 33–35. A volatility swap has no simple replication strategy; replicating it by trading variance swaps dynamically requires a model of the volatility of volatility; and around 1999 there was no variance swap market liquid enough to use for replication. Pages 37–39 of the PDF. ↩ ↩² ↩³

11. 杉原慶彦, "わが国株式市場のモデルフリー・インプライド・ボラティリティ" (Model-free implied volatility in the Japanese stock market), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2010), page 118, Appendix 3 (2) (in Japanese). The paper states that no dynamic hedging is needed and that the swaps can be synthesized with a basket of options. Page 46 of the PDF. ↩

12. The same Goldman Sachs paper, pages 17–19 (equations 20–26 and Figure 4). Pages 21–23 of the PDF. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶

13. The same paper, page 13 (explanation of Figure 3) and Appendix A, "Constant Vega Requires Options Weighted Inversely Proportional to the Square of the Strike" (page 37). Page 17 and page 41 of the PDF. ↩

14. Goldman Sachs, "More Than You Ever Wanted To Know About Volatility Swaps", page 15. Page 19 of the PDF. ↩

15. The same paper, pages 20–22 ("An Example of a Variance Swap", equations 27–29 and Table 1). The amounts at each strike found by the procedure in Appendix A give a payoff that is never below the log contract, and coarse strike spacing overstates the value of the log contract. Pages 24–26 of the PDF. ↩ ↩²

16. The same paper, pages 21–22 (Table 1 and the explanation of Figure 5). In the example with at-the-money implied volatility of 20%, a skew of 1 point for every 5 the strike falls, a 3-month maturity, and strikes in steps of 5, the strike is (20.467)². As the strike spacing approaches 0, the strike approaches (20)² = 400 without the skew and about 402 with it. Pages 25–26 of the PDF. ↩

17. Cboe, "Cboe Volatility Index Mathematics Methodology", "1 Introduction". The component options are weighted in inverse proportion to the square of the strike, and this weight matches the weight for replicating the payoff of a variance swap with a portfolio of options. ↩

18. 杉原慶彦, "わが国株式市場のモデルフリー・インプライド・ボラティリティ" (Model-free implied volatility in the Japanese stock market), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2010), pages 117–118, Appendix 3 (1) (equations A-20 to A-22) (in Japanese). The variance swap rate equals the square of MFIVar. Pages 45–46 of the PDF. ↩

19. Goldman Sachs, "More Than You Ever Wanted To Know About Volatility Swaps", pages 27–28 ("Imperfect Replication Due to Limited Strike Range", Table 4 and Figure 9). Pages 31–32 of the PDF. ↩

20. The same paper, pages 28–32 (equations 35–42). The first term of the gap in profit and loss from a jump is proportional to J³, and a fall (J > 0) makes a profit for a seller that hedges with the replication strategy. Pages 32–36 of the PDF. ↩ ↩²

21. The same paper, page 36 ("Conclusions and Future Innovations"). Capped variance swaps and volatility swaps limit losses, and a capped variance swap embeds an option on realized variance whose valuation requires a stochastic model of volatility. Page 40 of the PDF. ↩ ↩²

22. 杉原慶彦, "わが国株式市場のモデルフリー・インプライド・ボラティリティ" (Model-free implied volatility in the Japanese stock market), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2010), page 117, Appendix 3 main text and footnote 28 (in Japanese). A corridor-type swap exchanges historical volatility calculated after excluding returns above a certain size. Page 45 of the PDF. ↩