Variance in Gamma Driven Markets: Understanding the Dynamics

Published by First Growth Capital, June 2023

In the ever-evolving landscape of financial markets, it is crucial for investors and market-makers to grasp the intricacies and implications of various trading instruments. One such instrument that has gained significant attention is the variance swap. Variance swaps offer investors a unique opportunity to gain exposure to the volatility of an underlying asset without the complexity associated with delta-hedged options. Understanding the nuances of variance swaps and their role in gamma-driven markets is essential for investors looking to navigate these dynamic environments.

The Basics of Variance Swaps

Variance swaps are swap contracts where parties agree to exchange a pre-agreed variance level for the actual amount of variance realized over a specific period. These instruments provide direct exposure to volatility, allowing investors to profit from market movements beyond the implied strike level. One distinguishing feature of variance swaps is their convexity in volatility. A long position in a variance swap benefits more from an increase in volatility than it loses from a corresponding decrease. This convexity makes variance swaps particularly attractive for investors seeking to capitalize on volatility fluctuations.

Liquidity Considerations

Before delving deeper into the applications of variance swaps, it is essential to understand the liquidity landscape surrounding these instruments. Bid/offer spreads for variance swaps have significantly decreased in recent years. For indices, spreads typically range from 0.5 vegas, while single-stocks exhibit spreads of 1-2 vegas in Europe and 2-2.5 vegas in the US and Japan. Although emerging markets may have higher spreads, they are also becoming more liquid.

The most liquid variance swap maturities fall within the 3-month to 2-year range. These maturities align with quarterly options expiry dates, allowing for efficient hedging with exchange-traded options of the same maturity. Volatility indices, such as VIX, VSTOXX, and VDAX, provide benchmark measures for equity market risk. While these indices are not directly tradable, they are widely used in the industry.

With the advent in 2022 of day to expiration on major US stock indices such as the S&P 500 and the Nasdaq 100, as well as ETFs that track these indices, liquid variance swaps can be utilized by investors in need or want of hedging with or arbitraging exposure to markets.

Utilizing Variance Swaps

Investors and market participants employ variance swaps for various purposes, ranging from exploiting volatility views to hedging specific exposures. Let us explore some common use cases for variance swaps:

1. Exploiting a Volatility View

Variance swaps are ideal for investors looking to take a direct view on the volatility of an underlying asset. Unlike delta-hedged options, variance swaps offer straightforward exposure to volatility without the complexities associated with path-dependency.

2. Specific Hedging Purposes

Variance swaps can be utilized for macro-hedging and hedging specific volatility exposures, such as those resulting from structured products. These instruments provide a tailored solution to mitigate the risks associated with specific investment scenarios.

3. Rolling Short Variance

Short variance swaps are employed to capture the observed equity index volatility risk premium. This strategy involves rolling short index variance positions to generate alpha and diversify returns. Returns from short volatility trades tend to be uncorrelated with the underlying, making them suitable as overlay strategies within a well-diversified portfolio.

4. Diversification

Volatility can be treated as an asset class, offering diversification benefits within a portfolio. Rolling short volatility index returns share similarities with bond index returns, providing an alternative source of diversification. Short variance swaps can be considered as a low-correlation substitute for bonds within a well-allocated portfolio.

5. Index Variance Spreads

Variance swaps can be used to trade the spread of volatilities between two indices. These "volatility-beta" trades aim to profit from a widening spread of volatilities as volatility increases. By utilizing a correlated index with a lower beta, investors can mitigate the carry associated with these trades. However, it is important to consider potential regime changes in the composition or behavior of index constituents, as they can impact the effectiveness of this strategy.

Constructing Portfolios for Constant Gamma Exposure

To achieve constant exposure to variance, portfolios need to maintain a constant gamma. There are three possible approaches to constructing such a portfolio:

1. Dynamic Trading of a Single Vanilla Option

One approach involves using a single vanilla option and buying or selling additional amounts of it to ensure a constant gamma over time. While this method allows for the use of a single option strike, it requires dynamic trading and may result in a large accumulation of options as gamma decreases.

2. Re-striking the Option

Another approach is to re-strike the option to maintain a constant gamma. Starting with an at-the-money (ATM) option, investors can sell or hold the option and buy a new ATM option at each re-hedging step. Although this method reduces the need for dynamic trading, it still requires continuous adjustment of options.

3. Portfolio of Options with Constant Gamma

The third approach involves constructing a portfolio of options that maintains a constant gamma across both price (S) and time (t) movements. While this method requires a strip of options across a continuum of strikes, it eliminates the need for dynamic trading of options. This approach is relatively independent of the volatility process driving the underlying asset.

To achieve a constant dollar gamma across strikes, the weights of low-delta options need to be increased, while the weights of high-strike options should be decreased. Naively weighting options by 1/K, where K represents the strike, may achieve constant dollar gamma, but it fails to account for the spreading out of dollar gammas for higher strike options. Weighting each option by 1/K^2, however, ensures a constant dollar gamma exposure.

The Mathematics behind Constant Dollar Gamma

To understand why a portfolio weighted by 1/K^2 provides constant exposure to volatility, we need to examine the relationship between gamma and the underlying asset's price (S). Gamma represents the second derivative of a portfolio's value with respect to the underlying asset. To achieve constant exposure, the second derivative must be proportional to 1/S^2.

A negative natural logarithm of S, represented as -a ln S + bS + c, satisfies this requirement. By integrating 1/S^2 twice, any function of S in this form will have a constant dollar gamma. The action of delta-hedging this combined portfolio will allow for the recovery of the realized variance of the underlying asset.

While log contracts are not actively traded in the market, they can be replicated using a combination of calls and puts weighted by the inverse square of the strike. Integrating the value of this portfolio at expiry demonstrates that the non-linear part of the payoff resembles a negative log contract. This replication approach allows investors to create a variance swap by delta-hedging the constructed portfolio.

Conclusion

Variance swaps provide investors with a direct and straightforward exposure to the volatility of an underlying asset. By understanding the dynamics of variance swaps and their applications in gamma-driven markets, investors can capitalize on volatility fluctuations and hedge specific exposures. Constructing portfolios with constant gamma exposure is a key component of variance swap strategies, ensuring consistent exposure to volatility changes. As market dynamics continue to evolve, the ability to navigate these gamma-driven environments will be a valuable skill for investors and market-makers alike.

Investors seeking to optimize their portfolios and maximize returns should consider incorporating variance swaps into their investment strategies. These instruments offer unique opportunities for diversification, hedging, and capturing volatility risk premiums. At First Growth Capital, we specialize in long biased equity investing with a listed, and liquid, variance swap overlay which gives investors a constant exposure to volatility risk premia as well as underlying market and macro trends.

Disclaimer: The information provided in this article is for educational purposes only and should not be considered as financial advice. Investing in financial markets carries risks, and individuals should conduct thorough research and consult with a qualified financial advisor before making any investment decisions.