Variance swaps

Variance Swaps: Definition, Formula, Example, and FAQs

What Is Variance Swaps?

Variance swaps are a type of over-the-counter derivatives contract that allows two parties to exchange payments based on the realized volatility (specifically, variance, which is the square of volatility) of an underlying asset over a specified period. These financial instruments provide pure exposure to the magnitude of an asset's price movements, irrespective of the direction of those movements. Unlike traditional options, variance swaps are designed to isolate volatility risk, making them a specialized tool within the broader category of financial instruments. The contract typically defines a notional principal that scales the final payout.

History and Origin

The concept of volatility as a tradable asset gained traction in the financial markets during the late 20th century. While theoretical discussions around volatility derivatives existed earlier, variance swap contracts began trading sporadically in the late 1990s and achieved prominence then. Early development in pricing models, particularly those that could replicate variance swaps with a static portfolio of options, played a crucial role in their increased adoption. According to research on the origins of volatility and variance swaps, the first variance swap may have been traded by Union Bank of Switzerland (UBS) in 1993.⁹, ¹⁰ The development of robust pricing methodologies was key to the widespread adoption of these instruments.⁸

Key Takeaways

• Variance swaps are over-the-counter derivative contracts used to speculate on or hedge against the future realized variance of an underlying asset.
• They offer pure exposure to volatility, meaning their payoff is independent of the underlying asset's price direction.
• The final cash settlement is based on the difference between the realized variance and a pre-agreed variance strike.
• Market participants, including institutional investors and hedge funds, use variance swaps for hedging, speculation, and arbitrage strategies.
• While useful for managing volatility risk, variance swaps can expose users to significant losses if realized volatility substantially exceeds expectations, especially in volatile market conditions.

Formula and Calculation

The payoff of a variance swap at maturity is determined by the difference between the realized variance of the underlying asset and a pre-determined expected variance (known as the variance strike), scaled by a notional amount.

The payoff formula for a variance swap is:

$\text{Payoff} = N_{\text{var}} \times (\sigma^2_{\text{realized}} - K^2_{\text{variance}})$

Where:

• ( N_{\text{var}} ) = Variance Notional (a pre-agreed dollar amount per unit of variance).
• ( \sigma^2_{\text{realized}} ) = Realized Variance, typically calculated as the sum of squared daily logarithmic returns of the underlying asset over the contract's term, annualized.
• ( K^2_{\text{variance}} ) = Variance Strike, the pre-agreed level of variance at the inception of the contract.

If the realized variance is greater than the variance strike, the buyer of the swap receives a payment from the seller. Conversely, if the realized variance is less than the variance strike, the buyer pays the seller.

Interpreting the Variance Swaps

Variance swaps provide a direct measure of market participants' expectations and the actual outcome of an asset's price dispersion. A positive payoff to the buyer of a variance swap indicates that the market was more volatile than anticipated at the contract's inception. Conversely, a negative payoff suggests that the market was less volatile than initially expected. This makes variance swaps valuable tools for both hedging against unexpected swings and for pure speculation on future market turbulence.

Hypothetical Example

Consider an institutional investor who believes that the stock market, represented by a major equity index, will experience higher volatility over the next three months than currently priced in the market. The investor enters into a variance swap with a financial institution.

• Underlying Asset: S&P 500 Index
• Term: 3 months
• Variance Notional ( ( N_{\text{var}} ) ): $100,000 per unit of variance
• Variance Strike ( ( K^2_{\text{variance}} ) ): 0.04 (representing an annualized volatility of 20%, since ( 0.20^2 = 0.04 ))

At the end of three months, the realized variance of the S&P 500 Index over the period is calculated to be 0.06 (representing an annualized volatility of approximately 24.5%).

Using the payoff formula:
( \text{Payoff} = $100,000 \times (0.06 - 0.04) )
( \text{Payoff} = $100,000 \times 0.02 )
( \text{Payoff} = $2,000 )

In this scenario, the investor (buyer of the variance swap) receives $2,000 from the financial institution, as the realized variance exceeded the variance strike. This demonstrates how the payoff directly reflects the difference between expected and actual market volatility.

Practical Applications

Variance swaps are integral to the risk management strategies of various financial market participants.⁷ They are commonly employed by institutional investors, such as hedge funds and asset managers, to either hedge existing volatility exposures or to take a direct directional view on future volatility without needing to take a position in the underlying asset itself. For example, a portfolio manager with significant holdings in equity markets might purchase a variance swap to protect against potential portfolio value erosion during periods of heightened market turbulence.⁶

Furthermore, they are used by market makers to manage the volatility risk embedded in complex structured products and large options portfolios. The market for volatility derivatives, including variance swaps, has shown steady growth, indicating increasing recognition of their utility for sophisticated trading and hedging strategies.⁵ According to the Federal Reserve Bank of San Francisco, variance swaps can effectively be used by equity investors to offset the risk of a fall in the value of their holdings if the market declines, as increases in volatility can persist after sharp price movements.⁴ The Financial Times has also reported on the growth of the volatility derivatives market, highlighting their increasing importance.³

Limitations and Criticisms

Despite their utility, variance swaps are not without limitations and criticisms. A primary concern is their potential for theoretically unlimited losses for the seller if realized variance dramatically exceeds the strike, although in practice, contracts often include caps on volatility to limit this exposure.² They also carry counterparty risk, as they are over-the-counter (OTC) instruments, meaning the risk that one party to the contract defaults on its obligations.

Regulatory bodies have expressed concerns about the complexity and potential risks of certain volatility products, including those similar to variance swaps, emphasizing the need for investors to understand these instruments thoroughly before engaging with them.¹ Additionally, while variance swaps offer "pure" volatility exposure, their replication, especially in the presence of market jumps or discrete trading, can be imperfect, leading to hedging errors. The reliance on sophisticated models and dynamic hedging strategies can introduce significant financial engineering challenges and potential for model risk.

Variance Swaps vs. Volatility Swaps

While both variance swaps and volatility swaps are derivatives designed to trade volatility, they differ fundamentally in their payoff structure and replication.

The key distinction lies in the mathematical function of volatility they track. Variance swaps measure the square of volatility, which simplifies their theoretical replication using a static portfolio of options. Volatility swaps, conversely, directly measure volatility (the standard deviation), making their replication more complex and often requiring continuous delta hedging. This difference in replication ease has contributed to variance swaps generally being more widely traded and liquid than volatility swaps.

FAQs

Q1: Are variance swaps traded on an exchange?

No, variance swaps are typically traded over-the-counter (OTC) directly between two parties, such as a financial institution and an institutional investor. This customization allows for flexibility in terms and conditions.

Q2: How do variance swaps differ from traditional options?

Unlike traditional options, which provide exposure to both price direction and volatility, variance swaps offer pure exposure to volatility. Options require constant delta hedging to isolate volatility risk, whereas variance swaps inherently remove directional exposure. Their payoff is solely based on whether the realized volatility (or variance) is higher or lower than the agreed-upon strike price.

Q3: Who typically uses variance swaps?

Variance swaps are primarily used by sophisticated institutional investors, such as hedge funds, proprietary trading desks, and asset managers. They employ these instruments for hedging portfolios against volatility risk, speculation on future market volatility, and exploiting arbitrage opportunities between implied and realized volatility.

Q4: Can individual investors trade variance swaps?

While theoretically possible, variance swaps are generally not suitable for individual investors due to their complex nature, the substantial capital typically involved (notional principal), and the over-the-counter trading environment. Retail investors typically gain volatility exposure through exchange-traded products like futures or exchange-traded funds (ETFs) linked to volatility indices.

Q5: What is "realized variance" in the context of a variance swap?

Realized variance refers to the actual historical variance of the underlying asset's returns measured over the life of the swap contract. It is typically calculated by summing the squared daily logarithmic returns of the asset and then annualizing the result. This value is compared against the pre-agreed variance strike at the contract's maturity to determine the final payoff.