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Acquired variance swap

Acquired Variance Swap: Definition, Formula, Example, and FAQs

An Acquired Variance Swap is an over-the-counter (OTC) financial derivative contract that allows participants to speculate on or hedge against the future magnitude of price movements, or volatility, of an underlying asset. The "acquired" aspect refers to the realized variance calculated from the observed returns of the underlying asset over the contract's term. As a tool within financial derivatives, it offers pure exposure to volatility, insulating the trade from the directional price movements of the underlying asset itself.

In a typical Acquired Variance Swap, two parties agree to exchange payments based on the difference between the observed, or "acquired," variance of the underlying asset and a predetermined strike price (referred to as the variance strike). The payoff is linear in variance, meaning the profit or loss scales directly with the difference between the realized variance and the strike variance.

History and Origin

The concept of volatility as a tradable asset gained traction with the advent of sophisticated pricing models for options, such as the Black-Scholes model developed in the 1970s. Initially, market makers would delta hedge their options positions to neutralize directional exposure, leaving them with residual risk related to implied volatility. However, delta-hedged options still exhibited significant path dependency, meaning their profit and loss (P&L) was influenced by the specific price trajectory of the underlying asset, not just its overall volatility.22

To address this limitation and create a more direct bet on volatility, financial innovators developed the variance swap. Analysts at J.P. Morgan Securities, in a 2006 research paper, detailed how a portfolio of call options and put options weighted by the inverse square of their strikes could effectively eliminate path dependency and provide a P&L profile linear in variance.20, 21 This theoretical replication strategy formed the foundation for the over-the-counter variance swap market. The instrument emerged as a response to market participants' need for cleaner exposure to volatility, allowing them to express views on future price fluctuations without taking a directional stance on the asset's price. While variance swaps were traded as early as the late 1990s, the market for these instruments developed more fully once robust and universally accepted pricing methodologies were established.19

Key Takeaways

  • An Acquired Variance Swap is an OTC derivative that facilitates speculation or hedging on the future realized volatility of an underlying asset.
  • Its payoff is linear in variance, offering pure exposure to volatility distinct from the underlying asset's price direction.
  • The contract's value is determined by the difference between the observed, or "acquired," variance and a predefined variance strike.
  • Acquired Variance Swaps are widely used by institutional investors and hedge funds for risk management and expressing specific volatility views.
  • They are theoretically replicable using a static portfolio of options, though practical implementation involves certain assumptions and potential limitations.

Formula and Calculation

The payoff of an Acquired Variance Swap at maturity is determined by the difference between the realized variance of the underlying asset over the observation period and the pre-agreed variance strike, multiplied by a notional amount.

The formula for the payoff of an Acquired Variance Swap for the variance buyer is:

Payoff=Nvar×(σrealized2−Kvariance2)\text{Payoff} = N_{\text{var}} \times (\sigma_{\text{realized}}^2 - K_{\text{variance}}^2)

Where:

  • ( N_{\text{var}} ) = Variance Notional (also known as "variance amount" or "vega notional squared")
  • ( \sigma_{\text{realized}}^2 ) = Realized variance of the underlying asset over the observation period
  • ( K_{\text{variance}}^2 ) = Variance strike, agreed upon at the inception of the contract. This is typically quoted in volatility terms (e.g., as a percentage) and then squared.

The realized variance (( \sigma_{\text{realized}}^2 )) is typically calculated as the sum of squared daily logarithmic returns of the underlying asset, annualized. If ( R_i ) represents the daily logarithmic return on day ( i ), and there are ( T ) trading days in the observation period, the realized variance is calculated as:

σrealized2=AT∑i=1T(ln⁡(Si/Si−1))2\sigma_{\text{realized}}^2 = \frac{A}{T} \sum_{i=1}^{T} (\ln(S_i/S_{i-1}))^2

Where:

  • ( A ) = Annualization factor (e.g., 252 for daily trading days in a year)
  • ( T ) = Number of observation days
  • ( S_i ) = Closing price of the underlying asset on day ( i )
  • ( S_{i-1} ) = Closing price of the underlying asset on day ( i-1 )

The contract terms usually set the mean return at zero for simplicity, as its impact on the price is minimal and allows for additive payoffs.18

Interpreting the Acquired Variance Swap

Interpreting an Acquired Variance Swap involves understanding the relationship between the agreed-upon variance strike and the market's expectation of future volatility. When a party buys an Acquired Variance Swap, they are taking a long position in variance. This means they profit if the actual realized volatility of the underlying asset over the contract's life exceeds the variance strike. Conversely, if the realized volatility is lower than the strike, the buyer will incur a loss.

For example, if the variance strike is set at 20% annualized volatility, a buyer anticipates that the underlying asset will experience greater price fluctuations than 20% over the contract period. If the realized volatility turns out to be 25%, the buyer profits. If it is 15%, the buyer loses. This direct relationship makes the Acquired Variance Swap a powerful tool for expressing a view purely on the level of market movement, without being exposed to whether the underlying asset's price goes up or down. This "pure play" on volatility differentiates it from traditional options which embed directional risk.

Hypothetical Example

Consider an investor, ABC Corp., who believes the S&P 500 index will experience higher than anticipated volatility over the next three months due to upcoming economic announcements. ABC Corp. enters into an Acquired Variance Swap as the variance buyer, with a variance notional of $50,000 and a variance strike (annualized volatility) of 18%. The contract will be settled in three months.

Over the three-month observation period, daily logarithmic returns of the S&P 500 are collected. At maturity, the realized variance is calculated from these daily returns.

Suppose the actual realized volatility over the three months, when annualized, turns out to be 22%.

First, convert the volatility figures into variance:

  • Variance Strike (( K_{\text{variance}}2 )) = ( (0.18)2 = 0.0324 )
  • Realized Variance (( \sigma_{\text{realized}}2 )) = ( (0.22)2 = 0.0484 )

Now, calculate the payoff:

Payoff=$50,000×(0.0484−0.0324)\text{Payoff} = \$50,000 \times (0.0484 - 0.0324) Payoff=$50,000×0.0160\text{Payoff} = \$50,000 \times 0.0160 Payoff=$800\text{Payoff} = \$800

In this scenario, ABC Corp. would receive $800 from the variance seller, as the realized volatility exceeded the strike price. If the realized volatility had been, say, 16%, the payoff would be negative, and ABC Corp. would pay the seller. This example demonstrates how the Acquired Variance Swap provides a clear and direct way to profit from, or hedge against, changes in market volatility.

Practical Applications

Acquired Variance Swaps are versatile instruments within the broader field of financial derivatives, used by sophisticated market participants for various purposes:

  • Speculation on Volatility: Traders who have a directional view on future market volatility can use Acquired Variance Swaps to express that view directly. If they expect volatility to rise, they buy the swap; if they expect it to fall, they sell.
  • Hedging Volatility Exposure: Market makers and financial institutions with large portfolios of options often have significant exposure to volatility changes. Acquired Variance Swaps allow them to offset this risk, acting as a crucial risk management tool. For instance, life assurance companies offering products with guaranteed benefits might use variance swaps to offset their short volatility positions.17
  • Portfolio Diversification: Adding Acquired Variance Swaps can enhance a portfolio's risk-return profile by providing exposure to an asset class (volatility) that may behave differently from traditional assets like equities or bonds.
  • Arbitrage Strategies: Discrepancies between implied volatility (derived from options prices) and expected realized volatility can create opportunities for volatility arbitrage, where traders simultaneously buy or sell a variance swap and a replicating portfolio of options.
  • Regulatory Compliance: The growth of the global derivatives market, including instruments like Acquired Variance Swaps, has led to increased regulatory scrutiny, particularly after the 2008 Financial Crisis. Authorities like the U.S. Securities and Exchange Commission (SEC) and the Commodity Futures Trading Commission (CFTC) have established regulatory regimes for security-based swaps and other derivatives to enhance market transparency and reduce systemic risk.16 These regulations, stemming from the Dodd-Frank Act, have necessitated changes in how firms manage and report their over-the-counter (OTC) derivative positions.14, 15 The increased market volatility has also boosted demand for derivatives, with volumes expected to hit record levels, indicating their growing role in managing macroeconomic uncertainty.13

Limitations and Criticisms

Despite their utility, Acquired Variance Swaps have several limitations and criticisms:

  • Complexity: Acquired Variance Swaps are complex financial instruments that require a deep understanding of volatility and derivative pricing. Their theoretical replication relies on assumptions such as continuous trading and a full spectrum of options across all strike prices, which are not fully met in real markets. The discrepancy between theoretical models and practical limitations can lead to hedging errors.12
  • Model Dependence: While often presented as model-independent, the practical pricing and hedging of an Acquired Variance Swap, especially when a static replication portfolio is incomplete, can still rely on assumptions about the underlying asset's price process and the behavior of implied volatility across strikes.
  • Limited Liquidity for Bespoke Contracts: While standardized variance swaps on major indices are relatively liquid, customized or single-name Acquired Variance Swaps traded over-the-counter (OTC) may have limited liquidity, making it challenging to enter or exit positions efficiently.
  • Systemic Risk Concerns: The growth of the OTC derivatives market, including variance swaps, has raised concerns about potential systemic risk to the financial system, particularly due to the interconnectedness of large financial institutions and the opacity of some OTC contracts prior to regulatory reforms. Regulators, including the Federal Reserve, have emphasized the need for comprehensive oversight to mitigate such risks.10, 11 Issues like counterparty credit risk and operational weaknesses were highlighted during the 2008 Financial Crisis, prompting global reforms to improve transparency and risk management in these markets.9
  • Asymmetrical Payoff: The payoff profile of a long variance swap position is convex with respect to volatility. This means a buyer benefits from boosted gains when volatility rises significantly but experiences discounted losses when volatility declines by the same amount. While this convexity can be seen as an advantage for long positions, it also means the fair value of variance swaps can be sensitive to volatility skew.7, 8

Acquired Variance Swap vs. Volatility Swap

The terms "Acquired Variance Swap" and "Volatility Swap" are closely related, as both are financial derivative contracts designed to provide pure exposure to the future volatility of an underlying asset. However, a key difference lies in their payoff structure.

An Acquired Variance Swap's payoff is linear in the variance (volatility squared) of the underlying asset. This linearity in variance is a crucial feature because it makes the instrument more readily replicable and hedged using a static portfolio of options across different strike prices. The "acquired" aspect refers to the direct measurement of realized variance from historical price data.

In contrast, a Volatility Swap has a payoff that is linear in volatility (the square root of variance). While seemingly a minor distinction, this makes volatility swaps theoretically more challenging to hedge and replicate with a static portfolio of options. The square root function introduces non-linearity that complicates the hedging process, making them more model-dependent compared to variance swaps. Historically, many market participants transitioned from dealing in volatility swaps to variance swaps due to the latter's easier practical hedging and pricing, leading to variance swaps generally having greater liquidity.6

FAQs

What is the underlying asset of an Acquired Variance Swap?

The underlying asset for an Acquired Variance Swap can be a stock index (like the S&P 500 or EURO STOXX 50), a single stock, an exchange rate, or an interest rate. Equity indices are among the most common underlying assets.5

How is the variance measured in an Acquired Variance Swap?

The variance is measured, or "acquired," by calculating the realized variance of the underlying asset over a specified observation period. This typically involves summing the squared daily logarithmic returns of the asset and annualizing the result.4

Why do investors use Acquired Variance Swaps instead of just buying or selling options?

While options also provide exposure to volatility, they carry directional risk (delta) and are influenced by other factors like time to expiration. Acquired Variance Swaps offer pure exposure to volatility, separating it from directional price movements, making them a more precise tool for expressing a view on volatility alone.

Is an Acquired Variance Swap centrally cleared?

Typically, Acquired Variance Swaps are over-the-counter (OTC) contracts negotiated directly between two parties. However, post-Financial Crisis regulatory reforms (like the Dodd-Frank Act) have pushed for standardized OTC derivatives to be centrally cleared through central counterparties (CCPs) where appropriate, to reduce counterparty risk and increase transparency.2, 3

Can an Acquired Variance Swap result in unlimited losses?

The potential losses for a short position in an Acquired Variance Swap are theoretically unlimited, as realized volatility can reach very high levels during periods of market upheaval. To mitigate this, many contracts include a cap on the maximum payout, limiting potential losses for the variance seller.1