The Variance Swap: A Comprehensive Guide

A variance swap is a financial instrument that allows you to bet on the volatility of a specific asset, such as a stock or a commodity.

The main idea behind a variance swap is to transfer the risk of volatility from one party to another, providing a way to manage and hedge against potential losses.

To understand how a variance swap works, let's consider an example: suppose you're an investor who wants to protect your portfolio from potential market fluctuations.

In this case, you could enter into a variance swap agreement with a counterparty, which would allow you to pay a fixed amount of money based on the realized volatility of the underlying asset.

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A variance swap is a type of financial contract that allows you to bet on the level of volatility in the market. It's a two-party agreement where one party receives a variable payment based on the difference between the realized and implied variance.

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The contract has no exchange of notional principal at initiation, and there are no interim settlement periods. This means that the only payment made is at the expiration of the swap.

The value of a swap is zero at initiation because implied volatility is the best estimate of realized volatility. If realized volatility is higher than the strike, the buyer of the swap makes a profit.

The realized variance is calculated by taking the natural log of the daily price relatives, which is the closing price on day t divided by the closing price on day t-1.

Here's a simple breakdown of the profit/loss scenario:

The notional amount of the swap can be expressed as either variance notional or vega notional. Variance notional represents the profit or loss per point difference between implied variance and realized variance.

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A variance swap is a financial instrument that allows investors to bet on the volatility of an underlying asset. The features of a variance swap include the variance strike, the realized variance, and the vega notional.

The vega notional is a key component of a variance swap, as it converts the payoff into dollar terms. This is done by specifying a notional amount in terms of vega.

The annualised realised variance is calculated based on a prespecified set of sampling points over the period. This calculation can be expressed as: σ²_realised = (A/n) * ∑(Ri²), where A is an annualisation factor and Ri is the natural log return.

The annualisation factor is normally chosen to be approximately the number of sampling points in a year, commonly 252. The number of sampling points is determined by the contract life defined by the number n/A.

Here's a breakdown of the key components of a variance swap:

It's worth noting that the contract terms may not always subtract the mean, which can affect the calculation of the annualised realised variance.

Variance swaps offer pure exposure to the volatility of the underlying price, making them an attractive option for traders who want to speculate on volatility without directional risk.

They provide a way to speculate on volatility without the need for constant delta hedging, which can be costly and time-consuming.

The profit and loss from a variance swap depends directly on the difference between realized and implied volatility, creating an opportunity for volatility arbitrage.

Historically, implied variance has been above realized variance, a phenomenon known as the variance risk premium, allowing for a rolling short variance trade.

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Variance swaps offer a pure exposure to the volatility of the underlying price, which is not the case with call and put options that may carry directional risk.

They provide a direct payout on future realized variance, which is unrivaled by other forms of volatility trading.

The profit and loss from a variance swap depends directly on the difference between realized and implied volatility.

Historically, implied variance has been above realized variance, a phenomenon known as the variance risk premium, creating an opportunity for volatility arbitrage.

The quoted strike is determined by the implied volatility smile in the options market, whereas the ultimate payout will be based upon actual realized variance.

Constant delta hedging is required when using options to speculate on volatility, which can be costly and time-consuming.

A replicating portfolio of a variance swap would require an entire strip of options, which would be very costly to execute.

Variance swaps can be used to hedge options on realized variance, taking advantage of the variance risk premium.

In practice, volatility can be a valuable tool for traders, increasing as uncertainty grows and often exhibiting mean-reverting behavior.

High volatility regimes can follow low volatility levels, making it a good idea to get long a Bitcoin Variance Swap position if you expect volatility to rise.

For instance, an investor who initiated a trade on February 1st to get long a Bitcoin Variance Swap expiring on April 26th for a notional amount of $100,000, with a strike volatility of 30%, received a payoff of $6,139.89 at expiry, since the realized volatility was 38.91%.

A Bitcoin Variance Swap can also be used to profit from a decline in volatility, as seen when an investor sold $100,000 worth of a Bitcoin Variance Swap on March 1st, expiring on April 26th with a strike volatility of 30%, and received a profit of $4,107.06 at expiry, since the realized volatility was 22.12%.

There is no premium to be paid to enter a Variance Swap position, making it a costless trade for the investor.

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A variance swap is a financial instrument that allows investors to hedge against or speculate on the volatility of an underlying asset.

In a variance swap, the payout is typically based on the realized variance of the underlying asset, which is the actual volatility of the asset over a specific period.

The key difference between a variance swap and a volatility swap is that a variance swap pays out based on the actual realized volatility, whereas a volatility swap pays out based on the expected volatility.

Variance swaps are often used by investors to hedge against potential losses due to unexpected changes in volatility, while also providing a way to profit from periods of high volatility.

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In the world of derivatives, there are several closely related strategies that are worth mentioning. Straddle is one such strategy that is often used in conjunction with other instruments.

A straddle involves buying a call and a put option with the same strike price and expiration date. This strategy is often used by traders who expect a significant move in the underlying asset.

Volatility swaps are another related instrument that allows traders to bet on the level of volatility in the market. They can be used to hedge against potential losses or to speculate on increased volatility.

Correlation swaps, on the other hand, allow traders to bet on the correlation between two or more assets. This can be useful for traders who want to hedge against potential losses or to speculate on increased correlation.

Gamma swaps are a type of swap that allows traders to bet on the rate of change of the underlying asset's price. This can be useful for traders who want to hedge against potential losses or to speculate on increased volatility.

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Here are some related instruments that are worth mentioning:

These are just a few examples of the many related instruments that are available in the derivatives market.

Pricing and valuation of variance swaps can be a complex task, but fortunately, there are several methods that can be used to derive the fair variance swap rate.

Any volatility smile model that prices vanilla options can be used to price the variance swap, such as the Heston model, which can provide a closed-form solution for the fair variance swap rate.

Care must be taken with the behavior of the smile model in the wings, as it can have a disproportionate effect on the price.

A short log contract position is equal to being short a futures contract and a collection of puts and calls, which can be used to replicate the variance swap.

The fair variance swap strike can be calculated using the formula: S∗=F0=S0e^{rT}, where S∗ is the fair variance swap strike, F0 is the current forward price, S0 is the current stock price, r is the risk-free interest rate, and T is the time to maturity.

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In the risk-neutral world, the underlying asset price S=(St)0≤t≤T solves the following SDE: dS/S=rSdt+σSdWt.

Using a static replication argument, a short log contract position can be replicated using a bond, a future, and infinitely many puts and calls.

A variance swap can be priced using a portfolio of European call and put options with weights inversely proportional to the square of strike.

The payoff of a variance swap can be derived using Ito's Lemma, which shows that the total variance consists of a rebalanced hedge of 1/Sdt and short a log contract.

The fair variance swap strike can be calculated using the formula: σstrike2=σrealized2×Nvar, where σstrike2 is the fair variance swap strike, σrealized2 is the realized variance, and Nvar is the number of variance observations.

The value of a variance swap at time t0 is given by Vt0=E[(σrealized2-σstrike2)×Nvar], where E denotes the expected value.

To avoid arbitrage opportunity, the value of a variance swap at time t0 should be zero, which means that Vt0=0.

The fair variance strike is then given by σstrike2=σrealized2×Nvar, which remains to be calculated using numerical methods, such as Monte Carlo methods.

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Variance swaps are known for their high volatility, which can be both a blessing and a curse. They offer a way to hedge against or speculate on the volatility of an underlying asset.

High volatility can result in large gains or losses, making variance swaps a high-risk investment. The value of a variance swap can change rapidly in response to market fluctuations.

To manage this risk, investors can use various strategies, such as limiting their exposure to the underlying asset or using other hedging instruments.

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Convexity is a key characteristic of variance swaps that sets them apart from volatility swaps. The payoffs on a variance swap are based on variance, while the strike price is expressed in terms of volatility, making the payoffs convex with respect to volatility.

This convexity means that when realised volatility is below the strike, the losses on the variance swap are smaller than the losses on the volatility derivative. For example, if the strike is 20% and realised volatility is 15%, the variance swap will incur smaller losses compared to a volatility swap.

On the other hand, when realised volatility is above the strike, the gains on the variance swap are greater than the gains on the volatility derivative. In other words, the payoffs are increasing at an increasing rate when volatility rises and decreasing at a decreasing rate when volatility falls.

Here's a summary of the benefits of convexity in variance swaps:

A variance swap can be replicated by delta-hedged static vanillas, and implied vols are fixed on the strike day. This means a variance swap has no vol of vol risk.

However, the spread between a variance swap and a volatility swap can have vol of vol exposure. This is because the local volatility exposure of a variance swap is 2σ, with a second derivative of 2.

If you were to hedge this local volatility exposure using options or a volatility swap, the resulting payoff would be convex in volatility. This makes the spread between the variance swap strike and volatility swap strike long vol of vol.

In other words, a variance swap can be locally vega hedged to produce a convex payoff in volatility, making it long vol of vol. This is similar to how you can delta hedge an option to isolate return volatility exposure.

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