Valuing a Volatility Swap

Volatility

Volatility (standard deviation) is the square root of the variance (the amount of "noise", risk or variability in the stock price)

(1)
\begin{align} Volatility = \sqrt {Variance} \end{align}

Variance is a measure of the uncertainty of a stock price.

A stock’s volatility is the simplest measure of its riskiness or uncertainty. Formally, the volatility $\Sigma_R$ is the annualized standard deviation of the stock’s returns during the period of interest, where the subscript R denotes the observed or “realized” volatility.

Why trade volatility? Just as stock investors think they know something about the direction of the stock market, or bond investors think they can foresee the probable direction of interest rates, so you may think you have insight into the level of future volatility. If you think current volatility is low, for the right price you might want to take a position that profits if volatility increases. 

Volatility Swaps

A volatility swap is a forward contract on future realized price volatility. In comparison, variance swaps are forward contracts on future realized stock variance.

At expiration, the receiver of the “floating leg” pays (or owes) the difference between the realized volatility and the agreed upon strike. At inception the strike is generally chosen such that the fair value of the swap is zero. This strike is referred to as fair volatility.

Here is how a Volatility Swap works:



Volatility swaps provide pure exposure to volatility alone, unlike vanilla options in which the volatility exposure depends on the price of the underlying asset. This swap can thus be used to speculate on future realized volatility, to trade the spread between realized and implied volatility, or to hedge the volatility exposure of other positions.

At maturity, the payoff of a volatility swap is

(2)
\begin{align} 100*N(\Sigma_R - K_{vol}) \end{align}

where

$\Sigma_R$ = realized volatility in the underlying asset during the life of the swap
$K_{vol}$ = the strike on volatility
$N$ = the notional amount quoted in $per volatility point (hence the factor of 100) The holder of a volatility swap at expiration receives$N$dollars for every point by which the stock’s realized volatility$\Sigma_R$has exceeded the volatility delivery price$K_{vol}$. He or she is swapping a fixed volatility$K_{vol}$for the actual (“floating”) future volatility$\Sigma_R$. Example: 1) Volatility increased to 21% Strike Price$K_{vol}$= 18% Realized Volatility$\Sigma_R$= 21% Notional Amount$N$=$50,000

Payment (HF to D) = 100 * $50,000 (21% - 18%) =$150,000

2) Volatility decreased to 12%

Payment (D to HF) = 100 * $50,000 (18% - 12%) =$300,000 

Bibliography
1. "Variance and Volatility Swaps." FinCad. 13 Sept. 2008 <http://http://www.fincad.com/support/developerfunc/mathref/varianceswaps.htm>.
2. Demeterfi, Kresimir, Emanuel Derman, Michael Kamal, and Joseph Zou. "More than you ever wanted to know* about volatility swaps." Mar. 1999. Goldman Sachs. 12 Sept. 2008 <http://www.ederman.com/new/docs/gs-volatility_swaps.pdf>.
3. Swishchuk, Anatoliy. "Financial Markets with Stochastic Volatilities." 28 Oct. 2004. 14 Sept. 2008 <http://math.ucalgary.ca/~aswish/finlabtalk_28_10_04.ppt>.
page revision: 16, last edited: 28 Mar 2010 02:32
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