Variance swap

Variance Swap: Definition, Formula, Example, and FAQs

A variance swap is a financial derivative contract that allows two parties to exchange payments based on the future realized variance of an underlying asset against a predetermined fixed rate, known as the variance strike⁴⁵. As a core component of derivatives within quantitative finance, these financial instruments provide pure exposure to volatility without being directly influenced by the direction of the underlying asset's price movement⁴⁴. This characteristic makes variance swaps valuable tools for both hedging and speculation in financial markets.

History and Origin

While the concept of trading volatility has long been of interest to market participants, the formal emergence of variance swaps as tradable instruments began in the late 1990s. Their growth was particularly spurred around 1998, a period characterized by historically high implied volatility ⁴³. Initially, liquid markets for variance swaps did not fully develop due to the absence of a universally accepted pricing methodology. It was only after robust pricing models, often based on replicating the payout using a static portfolio of options and dynamic hedging of the underlying, were introduced that the market for these instruments began to flourish⁴¹, ⁴². Early adoption focused on major equity indices, such as the S&P 500, EURO STOXX 50, DAX, and FTSE, which remain common underlying assets for variance swaps today⁴⁰.

Key Takeaways

A variance swap is a derivative contract where parties exchange a fixed variance rate for the realized variance of an underlying asset over a set period.
It offers pure exposure to volatility, meaning its payoff is independent of the underlying asset's price direction.
Market participants use variance swaps for hedging against volatility risk, speculating on future volatility levels, and engaging in relative value trades³⁸, ³⁹.
The payoff of a variance swap is linear in variance, which differentiates it from other volatility products and implies a convexity effect when viewed in terms of volatility³⁷.
While historically traded over-the-counter (OTC), exchange-traded variance futures are increasingly being introduced to provide greater transparency and capital efficiency³⁵, ³⁶.

Formula and Calculation

The payoff of a variance swap is determined by the difference between the realized variance of the underlying asset and a pre-agreed variance strike, multiplied by a notional amount.

The realized variance ((RV)) over a specific period is typically calculated as the sum of squared logarithmic daily returns, annualized:

RV = \frac{A}{N} \sum_{i=1}^{N} \left( \ln\left(\frac{S_i}{S_{i-1}}\right) \right)^2

Where:

(S_i) = Closing price of the underlying asset on day (i)
(S_{i-1}) = Closing price of the underlying asset on the previous day
(N) = Number of trading days in the observation period
(A) = Annualization factor (e.g., 252 for equity trading days in a year)
(\ln) = Natural logarithm

The payoff to the long party at maturity is calculated as:

\text{Payoff} = \text{Variance Notional} \times (RV - K_{var})

Where:

(\text{Variance Notional}) = The agreed-upon notional amount in currency per variance point³⁴. Note that sometimes a "vega notional" is quoted, which requires conversion to a variance notional.
(RV) = The realized volatility squared, or realized variance, calculated over the contract's term.
(K_{var}) = The fixed variance strike price agreed upon at the inception of the contract³³.

Interpreting the Variance Swap

A variance swap is interpreted as a direct bet on the future magnitude of price movements of an underlying asset. If a party believes that the actual volatility of an asset will be higher than what the market implies (represented by the variance strike), they would take a long position in a variance swap. Conversely, if they expect lower volatility, they would take a short position³¹, ³².

A positive payoff for the long party occurs if the realized volatility (and thus realized variance) exceeds the variance strike, indicating that the market was less volatile than anticipated by the long party. Conversely, if the realized variance is below the strike, the long party will incur a loss, and the short party will profit. The sensitivity of a variance swap to changes in volatility is often quoted in terms of its "vega"³⁰.

Hypothetical Example

Consider an investor, ABC Fund, who believes that the S&P 500 index will experience significantly higher volatility over the next quarter due to upcoming economic announcements, even though its current implied volatility is low. ABC Fund enters into a long variance swap with XYZ Bank with the following terms:

Underlying Asset: S&P 500 Index
Variance Notional: $1,000,000 per variance point
Variance Strike ((K_{var})): 0.0225 (which corresponds to a 15% annualized volatility, since (0.15^2 = 0.0225))
Maturity: 3 months (approximately 63 trading days)

Over the next 3 months, the S&P 500 experiences sharp price swings. At maturity, the realized variance ((RV)) calculated from the daily logarithmic returns of the S&P 500 over the observation period turns out to be 0.0361 (corresponding to 19% annualized volatility, (0.19^2 = 0.0361)).

The payoff for ABC Fund (the long party) is calculated as:

(\text{Payoff} = \text{Variance Notional} \times (RV - K_{var}))
(\text{Payoff} = $1,000,000 \times (0.0361 - 0.0225))
(\text{Payoff} = $1,000,000 \times 0.0136)
(\text{Payoff} = $13,600)

In this scenario, ABC Fund receives $13,600 from XYZ Bank because the realized variance exceeded the agreed-upon variance strike, validating ABC Fund's speculation on increased volatility.

Practical Applications

Variance swaps are critical tools in modern risk management and investment strategies for sophisticated market participants:

Hedging Volatility Risk: Institutional investors, such as pension funds or insurance companies, often use variance swaps to protect their portfolios from adverse market movements caused by heightened volatility. For instance, a life assurance company with products offering guaranteed benefits (e.g., variable annuities) may be exposed to short volatility positions, which can be offset using variance swaps²⁹.
Speculating on Volatility: Traders and hedge funds utilize variance swaps to take directional views on future volatility. If they anticipate a surge in volatility (e.g., due to an upcoming geopolitical event or earnings announcement), they can go long on a variance swap to profit from that expectation²⁷, ²⁸.
Volatility Arbitrage: This strategy involves exploiting discrepancies between implied volatility (derived from options prices) and expected realized volatility. Traders might sell options while simultaneously buying variance swaps to create a market-neutral position and capture pricing inefficiencies²⁶.
Portfolio Diversification: Adding volatility exposure to an equity portfolio can improve overall portfolio diversification because volatility shocks are often negatively correlated with stock index returns²⁵.
Transition from OTC to Exchange-Traded Products: Historically, variance swaps were primarily over-the-counter (OTC) products, customized and traded directly between parties. However, major exchanges are increasingly offering exchange-traded variance futures contracts. For example, Cboe Global Markets announced the launch of its S&P 500 Variance Futures in September 2024, aiming to provide a more accessible and capital-efficient way to replicate OTC variance swap exposures with benefits like central clearing and price discovery²⁴. This development reflects a market trend towards standardized, listed products for volatility trading.

Limitations and Criticisms

While variance swaps offer focused exposure to volatility, they are not without limitations and risks:

Jump Risk: Variance calculations can be significantly skewed by large, sudden price movements ("jumps") in the underlying asset, which might lead to unexpected results or losses for one of the parties²³. This is particularly relevant for short positions in variance swaps, as realized variances can exhibit positive skewness and leptokurtosis, potentially leading to substantial losses²².
Model Risk and Liquidity: The pricing and hedging of variance swaps often rely on complex models and the assumption of continuous trading and sufficient liquidity in the option market for replication²¹. In practice, deep out-of-the-money options may have low liquidity and high transaction costs, making perfect replication challenging²⁰.
Market Amplification: There have been concerns that the dynamic hedging required by market makers for variance swap positions could, in certain scenarios, amplify asset price changes, thereby increasing volatility during large daily moves¹⁹. This "feedback" effect has been cited as a factor in past market volatility events.
Variance Risk Premium: Historically, implied volatility (and thus implied variance) has tended to be higher than realized volatility, a phenomenon known as the "variance risk premium." This means that, on average, taking a long position in a variance swap may result in a negative payoff, making shorting variance swaps profitable on average but also exposing the short seller to significant tail risks¹⁷, ¹⁸.

Variance Swap vs. Volatility Swap

Although often discussed together, variance swaps and volatility swaps have a crucial difference relating to their payoff structure and sensitivity to large market moves:

Feature	Variance Swap	Volatility Swap
Payoff Basis	Realized variance (volatility squared)	Realized volatility (standard deviation)¹⁶
Payoff Linearity	Linear with variance	Linear with volatility
Convexity	Convex with volatility¹⁵	Does not have this specific convexity
Replication	Easier to replicate using a static strip of options and dynamic delta hedging of the underlying¹⁴	More complex to replicate; requires dynamic hedging of the underlying variance swap itself¹³
Payout in Extremes	Payout grows more rapidly with higher volatility due to squaring effect¹²	Payout increases proportionally with volatility

The key distinction lies in convexity. Because the payoff of a variance swap is linear in variance, but variance is the square of volatility ((\sigma^2)), the variance swap's payoff is convex with respect to volatility ¹¹. This convexity means that a variance swap will always outperform a volatility swap with the same strike in situations of large volatility swings¹⁰. This characteristic can be particularly attractive to investors who anticipate significant price movements but are indifferent to the direction of those movements.

FAQs

Q1: What is the primary benefit of a variance swap over options for trading volatility?
A1: A variance swap provides pure exposure to volatility without the directional risk inherent in traditional option contracts. Options are sensitive to the underlying asset's price direction (delta), time decay (theta), and changes in implied volatility. A variance swap, conversely, pays out solely based on the magnitude of price movements, regardless of whether the asset goes up or down⁹.

Q2: Are variance swaps typically traded on exchanges or over-the-counter?
A2: Historically, variance swaps have been primarily over-the-counter (OTC)⁸, meaning they are customized bilateral agreements between two parties. However, there is a growing trend for exchanges like Cboe to introduce standardized, exchange-traded variance futures contracts to offer benefits like central clearing, greater transparency, and improved liquidity⁶, ⁷.

Q3: Can individual investors trade variance swaps?
A3: Variance swaps are complex financial instruments typically used by institutional investors, hedge funds, and sophisticated traders due to their bespoke nature, pricing complexities, and associated risk management requirements⁴, ⁵. While exchange-traded variance futures are becoming available, direct participation in OTC variance swaps usually requires significant capital and expertise.

Q4: What is "realized variance" in the context of a variance swap?
A4: Realized variance is the actual measure of how much an underlying asset's price has moved, or its volatility, over the life of the variance swap contract², ³. It is calculated by taking the sum of the squared daily logarithmic returns of the asset and then annualizing that sum¹. This value is then compared to the predetermined variance strike to determine the swap's payoff.