# Variance and Volatility Swaps

## Introduction

Rather than gaining exposure to the market's volatility through standard call and put options, investors can take views on the future realized volatility directly by trading derivatives on variance and volatility.  The simplest such instruments are variance and volatility swaps.

A volatility swap is a forward contract on future realized price volatility.  Similarly, a variance swap is a forward contract on future realized price variance, variance being the square of volatility.  At expiry the receiver of the “floating leg” pays (or owes) the difference between the realized variance (or 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 variance (or fair volatility).

Both swaps provide pure exposure to volatility alone, unlike vanilla options in which the volatility exposure depends on the price of the underlying asset.  These swaps 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.

Variance swaps are theoretically simpler than volatility swaps: they can be hedged with a static position in European call and put options (with suitably chosen strikes), together with a dynamic position in the underlying asset .  Volatility swaps, on the other hand, are harder to hedge, although recent work suggests that it is possible .

To the extent that variance and volatility swaps can be hedged using plain vanilla European call and put options, the pricing of such instruments is possible in a model-independent manner: the price can be calculated from market-observed prices of European options of different strikes.  The effects of the volatility smile, for instance, are accounted for by construction.

Of course, the prices of variance and volatility swaps can also be calculated within a model of asset price dynamics, and FINCAD also provides the functionality to price such swaps using the Heston model of stochastic volatility.  This is described in the Volatility and Variance Swaps in the Heston Model FINCAD Math Reference document.

## Analysis Supported

FINCAD variance and volatility swap functions can be used for the following analysis:

·         calculation of fair variance for a variance swap and fair volatility for a volatility swap in a model independent manner, through replication arguments; calculation of the number of European options of each strike needed to build the replicating portfolio; calculation of risk statistics for fair variance and fair volatility respectively.

·         calculation of fair value for both variance and volatility swaps, given the realized variance or volatility to date, and the fair variance or volatility for the remaining life of the swap; calculation of risk statistics for both variance and volatility swaps.

## Technical Details

### Variance Swaps

At maturity, the payoff of a variance swap is Equation 1

where = notional amount quoted in \$ per volatility point squared (hence the factor of = 10,000) = realized variance in the underlying asset during the life of the swap = strike on variance

The (annualized) realized variance over the time period [0, ] measured in years (the remaining life of the swap) is given by Equation 2

where = volatility of the return on the underlying asset.

In practice, the variance is calculated with daily monitoring of the log-return of the underlying asset Equation 3

where = normalization factor, for example 252/(total number of days between 0 and )

At inception, the strike of the swap is chosen such that the expected payoff is zero: Equation 4

This is called the fair variance of a variance swap.

If the swap is already in effect and is being valued at some time such that , then the present value of the swap can be calculated by combining the realized variance up to the value date with the expected realized variance for the remainder of the swap's life.  The present value is then Equation 5

where  = realized variance up to the value date τ = fair variance for the remaining life of the swap

### Variance Swaps: Replication in Theory

To replicate the fair variance given in Equation 4 using European call and put options, we follow the description in .  Assuming that the underlying asset follows a standard lognormal process Equation 6

Ito's lemma implies Equation 7

so by subtraction Equation 8

Denoting the risk-free interest rate and the dividend yield by and respectively, in a risk-neutral world the first term is equal to .  Choosing some arbitrary value , the second term - the log contract - is split according to , and one can show Equation 9

The first term is forward contracts struck at , and the second (third) term is a portfolio of put (call) options, each option being weighted by .

The fair variance can thus be written as Equation 10

replication being achieved through a static position in a portfolio of call and put options, a dynamic position in shares and some cash.  Note that the value of should generally be the forward price or, in the discrete case, a value as close to the forward price as possible.  The portfolio would thus entirely consist of out-of-the-money options.

### Variance Swaps: Replication in Practice

In practice a portfolio of put and call options with continuous strikes cannot be constructed.  The two integrals in Equation 10 can be approximated by a portfolio of options with discrete strikes, which together have the payoff Equation 11

It turns out that this portfolio is given by a sum over strikes Equation 12

of call and put options with weights  .

Equation 13

The functions described here allow the user to enter an implied volatility smile.  The above replicating portfolio can then be constructed from options with the given strikes (and implied volatilities).  Alternatively, the portfolio can be constructed from a given number of options with equally spaced strikes between some minimum and maximum values.

In this case, the implied volatility smile is interpolated and/or extrapolated to the relevant strikes, and there is a choice of interpolating method: linear, cubic spline or smoothed cubic spline.  The latter choice requires the input of a smoothing parameter, giving rise to a cubic spline with a curvature penalty term proportional to the smoothing parameter.  The resulting spline will no longer intersect with the points in the original volatility smile, but will tend to smooth it out.  As the smoothing parameter varies between zero and infinity, the curve varies between a cubic spline and a least-squares fit.

### Volatility Swaps

The definition of the volatility swap is analogous to that of the variance swap.  At maturity, the payoff of a volatility swap is Equation 14

where = notional amount quoted in \$ per volatility point (hence the factor of 100) = realized volatility in the underlying asset during the life of the swap = the strike on volatility

At inception, the strike of the swap is chosen such that the expected payoff is zero: Equation 15

This is called the fair volatility.

If the swap is already in effect and is being valued at some time such that , then the present value of the swap can be calculated by combining the realized volatility up to the value date with the expected realized volatility for the remainder of the swap's life.  The present value is then Equation 16

where now since it is variance, and not volatility, which is additive.

### Volatility Swaps: Replication

It is often said that a volatility swap cannot be replicated in the same way as the variance swap, as the volatility swap is sensitive to the volatility of volatility.  Since Equation 17

the difference between the fair variance and the (square of the) fair volatility is directly related to the volatility of volatility.  This is what is known as the convexity adjustment.

Although recent results suggest that the volatility swap can be replicated , the current FINCAD implementation is to take the simplest strategy, which is to compute the fair volatility as the square root of the expected variance: Equation 18

By Jensen’s inequality, this approximation always overestimates the expected volatility.  It amounts to setting the volatility of volatility to zero in Equation 17, and is thus a sensible approximation to make only when the volatility of volatility is small.

It is then approximately true that one can replicate a volatility swap with strike and notional (in units of \$ per volatility point) with a variance swap of strike and notional (in units of \$ per volatility point squared) where Equation 19

with latter result arising from Equation 20

The weights given in aaVolatilitySwap_port are the weights of the variance swap which approximates the volatility swap.

## Functions

 Valuation Variance Swaps Volatility Swaps

Naming Conventions

 Suffix / Prefix Description _iv computes the fair variance (volatility) and corresponding Greeks for a variance (volatility) swap _port computes the weights for European options of various strikes needed to replicate the payoff of a variance (volatility) swap _p computes the fair value and corresponding Greeks for a variance (volatility) swap

aaVarianceSwap2_ iv: Outputs

 Output Statistic Description fair variance the fair variance in units of variance fair variance as volatility the square root of the fair variance delta rate of change in the fair variance per change in the current value of the underlying stock.  This is the derivative of the fair variance with respect to the current value of the underlying gamma rate of change in the value of delta per change in the current value of the underlying stock.  This is the second derivative of the fair variance with respect to the current value of the underlying theta rate of change in fair variance per one day decrease in time. This is the negative of the derivative of the fair variance with respect to time (in years), divided by 365 vega change in the fair variance per 1% change in annual volatility.  All the data in the implied volatility smile table are bumped by 1% to find this value rho of rate rate of change in fair value per 1% change in the risk free rate. This is the derivative of the fair variance with respect to the risk-free rate, divided by 100 rho of holding cost rate of change in fair value per 1% change in the holding cost. This is the derivative of the fair variance with respect to the holding cost, divided by 100 cost of options portfolio (per unit principal) the total cost per unit principal of the portfolio of options needed to replicate the swap number of market days the number of days that the swap is active; calculated based on the input accrual method

aaVarianceSwap2_ p: Outputs

 Output Statistic Description fair value the fair value of the variance swap at the value date sensitivity to realized variance rate of change in fair value per 1% change in realized volatility. This is the derivative of the fair value with respect to realized variance, divided by 10,000 sensitivity to implied variance rate of change in fair value per 1% change in implied volatility. This is the derivative of the fair value with respect to implied variance, divided by 10,000 theta rate of change in fair value per one day decrease in time. This is the negative of the derivative of the fair value with respect to time (in years), divided by 365 rho of rate rate of change in fair value per 1% change in the risk free rate. This is the derivative of the fair value with respect to the risk-free rate, divided by 100 combined variance the weighted average of the realized variance and the implied variance in units of variance combined variance as volatility the square root of the combined variance number of market days passed the number of days that the swap has been active; calculated based on the input accrual method number of market days remaining the number of days remaining in the life of the swap; calculated based on the input accrual method

aaVolatilitySwap_ iv: Outputs

 Output Statistic Description fair volatility the fair volatility delta rate of change in the fair volatility per change in the current value of the underlying stock.  This is the derivative of the fair volatility with respect to the current value of the underlying gamma rate of change in the value of delta per change in the current value of the underlying stock.  This is the second derivative of the fair  volatility with respect to the current value of the underlying theta rate of change in fair volatility per one day decrease in time. This is the negative of the derivative of the fair volatility with respect to time (in years), divided by 365 vega change in the fair volatility per 1% change in annual volatility.  All the data in the implied volatility smile table are bumped by 1% to find this value rho of rate rate of change in fair volatility per 1% change in the risk free rate. This is the derivative of the fair volatility with respect to the risk-free rate, divided by 100 rho of holding cost rate of change in fair volatility per 1% change in the holding cost. This is the derivative of the fair volatility with respect to the holding cost, divided by 100 cost of options portfolio (per unit principal) the total cost per unit principal of the portfolio of options needed to replicate the swap number of market days the number of days that the swap is active; calculated based on the input accrual method

aaVolatilitySwap_ p: Outputs

 Output Statistic Description fair value the fair value of the volatility swap at the value date sensitivity to realized volatility rate of change in fair value per 1% change in realized volatility. This is the derivative of the fair value with respect to realized volatility, divided by 100 sensitivity to implied volatility rate of change in fair value per 1% change in implied volatility. This is the derivative of the fair value with respect to implied volatility, divided by 100 theta rate of change in fair value per one day decrease in time. This is the negative of the derivative of the fair value with respect to time (in years), divided by 365 rho of rate rate of change in fair value per 1% change in the risk free rate. This is the derivative of the fair value with respect to the risk-free rate, divided by 100. combined volatility the weighted average of the realized volatility and the implied volatility number of market days passed the number of days that the swap has been active; calculated based on the input accrual method number of market days remaining the number of days remaining in the life of the swap; calculated based on the input accrual method

## Market Data Requirements

The _iv and _port functions require the value date, the expiry date of the swap, a list of applicable holiday dates and discount factor curves (or flat rates) for the risk-free rate and for the dividend yield of the underlying asset.

They also require a list of implied volatilities for European options of various strikes (the implied volatility smile).

The _p function requires the value date, the effective and terminating dates of the swap, a list of applicable holiday dates and a discount factor curve (or flat rate) for the risk-free rate.

They also require the realized variance (volatility) to date and the fair variance (volatility) for the remaining life of the swap.  The former can be calculated with the FINCAD functions aaVLT_h or aaVLT_h2, and the latter can be calculated using the respective _iv function.

## Example

### Example 1: Valuation of a Variance Swap using a Portfolio of European Options

On January 2, 2008, we seek to value a variance swap that came into effect on November 1, 2007 and expires on April 2, 2008. The example workbook illustrates how to value this swap using the functions aaVarianceSwap2_iv and aaVarianceSwap2_p.

The workbook also shows how to use aaVarianceSwap2_port to calculate the portfolio of European put and call options required for the replication of the variance swap for the remainder of its life.

## Related Functions

Heston Model Pricing

 Valuation Variance Swaps Volatility Swaps

Realized Variance / Volatility

 Valuation Utility Functions

## References

          Demeterfi, K., Derman, E., Kamal, M. and Zou, J. and (1999) 'A Guide to Volatility and Variance Swaps', Journal of Derivatives, 6, 9-32; 'More Than You Ever Wanted to Know About Volatility Swaps', Goldman Sachs Quantitative Strategies Research Notes, March 8, 1999.

'

Disclaimer

With respect to this document, FinancialCAD Corporation (“FINCAD”) makes no warranty either express or implied, including, but not limited to, any implied warranty of merchantability or fitness for a particular purpose. In no event shall FINCAD be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of this document or the information contained in it. This document should not be relied on as a substitute for your own independent research or the advice of your professional financial, accounting or other advisors.

This information is subject to change without notice. FINCAD assumes no responsibility for any errors in this document or their consequences and reserves the right to make changes to this document without notice.