Volatility Derivatives and Variance Swaps

Volatility as an Asset Class

Volatility is considered a distinct asset class by many market participants due to its characteristics. Research has indicated that volatility tends to increase during periods of market turbulence. There is an inverse relationship between volatility and equity prices. Moreover, volatility typically rises even more during bear markets, making it a commonly used instrument for portfolio hedging. Various assets can be utilized as hedging instruments, including the following options.

Volatility Instruments as “Fear Gauges”

The CBOE Volatility Index (VIX) is a widely used measure of expected volatility in the S&P 500 over the next 30 days. It is calculated based on the prices of S&P 500 Index options. It is noteworthy that the CBOE has published it since 1993. Besides, VIX futures have been available since 2004. Various regions have volatility indexes, such as the VSTOXX for the EURO STOXX 50 index and the VDAX-NEW for the DAX stock index options in Germany.

Investors who take a long volatility position by purchasing futures contracts on the fear index can benefit from increased volatility. This is a common strategy for hedging an extended portfolio of equities. On the other hand, going short volatility can be profitable when volatility is low or trending lower in calm market conditions. However, going short carries the risk of significant losses as bear markets often occur swiftly and sharply.

Variance Swaps

Variance swaps are another tool that investors can use to trade risk. Unlike other swaps, variance swaps do not involve exchanging cash at the beginning or during the contract. This makes them slightly more challenging to value.

The payoff of a long variance position in a variance swap is positive when the realized variance surpasses the variance strike. Conversely, it is negative when the realized variance is below the variance strike. In this case, the swap buyer receives payment from the swap seller if the payment amount is positive. The settlement payoff is calculated according to the following formula:

$$ \begin{align*}
{{\text{Settlement amount}_T}} = & (\text{Variance notional}) \\ & (\text{Realized variance} – \text{Variance strike}) \end{align*} $$

A distinction exists between Vega notional and Variance notional. Vega notional represents the expected profit and loss of a variance swap for a 1% change in volatility from the strike price. It helps traders assess the magnitude of the trade. For example, with a Vega notional of $10,000, a one-point difference in volatility would result in a profit or loss close to $10,000.

On the other hand, Variance notional considers the convex nature of the swap's payoff. It is similar to the relationship between duration and convexity in fixed income. Duration approximates small shifts from the current state, while convexity accounts for the exact measurement of changes in bond present value. The following equation can be used to convert between the two figures.

$$ \text{Variance notional} = \frac {\text{Vega notional}}{ (2 \times \text{strike price}) } $$

When calculating the variance of a swap during its lifetime, it becomes more complex. The formula includes the use of a present value factor:

$$ \begin{align*} & \text{VarSwap}_t \\ & = \text{Variance notional} \times PV_t(T) \\ & \times \left\{\frac {t}{T} \times [\text{RealizedVol}(0,t)]^2 + \left(T-\frac {t}{T} \right) \times [\text{ImpliedVol}(t,T)]^2 – \text{Strike}^2 \right\} \end{align*} $$

Example

Mary Loeffler is a trader who uses an algorithm to sell volatility on the S&P 500 Index. She has sold $50,000 vega notional of a one-year variance swap on the S&P 500 with a strike of 22% (annual volatility). After six months, the S&P 500's realized volatility is 16% (annualized). On the same day, a new six-month variance swap on the S&P 500 has a fair strike of 19%. Now, let's calculate the current value of the swap.

Values for the inputs are as follows:

Volatility strike on existing swap = 22.

Variance strike on existing swap \(= 22^2 = 484\).

Variance notional \(=\frac {\text{Vega notional} }{ (2 \times \text{strike price}) } \Rightarrow \frac {\$50,000 }{ (2 \times 22)} = 1,136.36 \)

\(PV_t(T) =\frac {1}{[1 + (2.75\% \times \frac {6}{12})]} = 0.986436\) ( = Present Value Interest Factor for six months, where the annual rate is 2.75%).

The current value of the swap is:

$$ \text{VarSwap}_t = 1,136.36 \times (0.986436) \times \left[ \left(\frac {6}{12} \right) \times 256 + \left(\frac {6}{12} \right) \times 361 – 484 \right] = $$

$$ \begin{align*} \text{VarSwap}_t & = 1,136.36 \times (0.986436) \times [0.5 \times 256 + 0.5 \times 361 – 484] \\
& = 1,136.36 \times (0.986436) \times [128 + 180.5 – 484] \\
& = 1,107.107 \times -175.5 = -\$194,297.33 \end{align*} $$

Her mark-to-market value is positive, given that Santos is short the variance swap. 

Question

Deepak Krocha engages in systematic volatility-selling strategies on the S&P 500 Index. He has sold a one-year variance swap on the S&P 500 with a notional value of $75,000 vega, structured at a strike level of 25% (expressed as annual volatility). After six months, the S&P 500 has seen a realized volatility of 18% annually. On this same day, the market fair strike for a new six-month variance swap on the S&P 500 stands at 22%.

Determine the current value of the variance swap sold by Krocha (note that the annual interest rate is 3.5%).

1. -$325,798.
2. $325,798.
3. $327,000.

Solution

The correct answer is A.

$$ \begin{align*}
& \text{Var swap}_t \\ & =\text{Variance notional} \times PV_t (T) \\ & \times \left\{\frac {t}{T} \times [\text{Realized Vol}(0,t)]^2+\frac {T-t}{T} \times [\text{Implied Vol}(t,T)]^2-\text{Strike}^2 \right\} \end{align*} $$

Volatility strike on existing swap=25

Variance strike on existing swap\(=25^2=625\)

$$ \text{Variance notional}=\frac {\text{Vega notional}}{2 \times \text{Strike}}=\frac {\$75,000}{2 \times 25}=1,500 $$

\(\text{RealizedVol}(0,t)^2 =18^2 =324\)

\(\text{ImpliedVol}(t,T)^2 = 22^2 = 484\)

\(t= 6\)

\(T= 12\)

Therefore, the Present Value Interest Factor for six months, where the annual rate is 3.5%, is calculated as follows:

$$ PV_t (T)=\frac {1}{ \left[1+ \left(3.5\% \times \frac {6}{12} \right) \right]} =0.9828 $$

The current value of the swap becomes:

$$ \begin{align*} \text{VarSwap}_t & =1,500 \times 0.9828 \times \left[ \frac {6}{12} \times 324+\frac {6}{12} \times 484-625 \right] \\ &=-\$325,798.20 \end{align*} $$

Given that Krocha is short the variance swap, the mark-to-market value is positive for him, and it equals $325,798.

B and C are incorrect. As Krocha holds a short position in the variance swap, the mark-to-market value is in his favor, amounting to $325,798.

Derivatives and Risk Management: Learning Module 2: Swaps, Forwards, and Futures Strategies; Los 2(d) Demonstrate the use of volatility derivatives and variance swaps