Variance and Volatility Swaps in the Heston Model

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.

More exotic products include conditional variance swaps, options on variance swaps and covariance and correlation swaps. Using conditional variance swaps, and options on realized variance or volatility, investors can trade volatility while protecting themselves from large spikes. Covariance and correlation swaps can be used to take views on the correlation between two or more assets and are popular in FX markets.

In the Heston model [4], arguably the most popular model of stochastic volatility, the fair variance of a variance swap and the fair volatility of a volatility swap may be computed analytically. In this two-factor model, both the price and the variance are assumed to be stochastic. The price process resembles a geometric Brownian motion and the variance process is a mean reverting square-root process first introduced in short term interest models [2].

The variability of the volatility then allows for a description of the market's implied volatility surface (the volatility smile) in the Heston model. The model is described in detail in the FINCAD Math Reference document Option Pricing with the Heston Model of Stochastic Volatility.

The functions described in this document provide valuation of variance and volatility swaps in the Heston model. Model independent valuation of these vanilla swaps is covered by FINCAD functions as described in the FINCAD Math Reference document Variance and Volatility Swaps.

Analysis Supported

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

·         calibration of the Heston model parameters to market data on variance or volatility swaps

·         calculation of fair variance for a variance swap and fair volatility for a volatility swap in the Heston model; calculation of risk statistics for fair variance and fair volatility respectively.

·         calculation of fair value for both variance and volatility swaps in the Heston model, given the realized variance or volatility to date; calculation of risk statistics for variance swaps 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.

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

Equation 3

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 4

where

= realized variance up to the value date

 = fair variance for the remaining life of the swap

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 5

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 6

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 7

where now

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

Heston Model

In the Heston model, the underlying asset price  follows a standard lognormal process and the instantaneous variance  follows a mean reverting square-root process:

Equation 8

where

 = price of the underlying asset

 = variance of the asset price

 = rate of return of the asset

 and  = Brownian motions correlated via , ρ being the correlation.

, , ,  = parameters of the Heston model: speed of mean reversion, initial volatility, long-term volatility and volatility of volatility.

By integrating the stochastic differential equation of the instantaneous variance and taking expectations, the expectation value of instantaneous variance at time  is:

.

Equation 9

The annualized expectation value for the total realized variance (in the time interval ) is then

.

Equation 10

This expectation value is the fair variance of a variance swap in the Heston model.

The volatility expectation value which is equivalent to the fair volatility of a volatility swap cannot be computed as easily in the Heston model. A second order Taylor expansion of leads to the following approximation for the volatility expectation value [1]:

Equation 11

where

 = the variance of variance given by

Equation 12

Note that the second order approximation of the fair variance given in Equation 11 can become negative, although the fair variance is always positive.

Alternatively, the volatility expectation value can be computed to all orders by numerically solving the integral [1], [3]

Equation 13

where

with

.

In the Heston model, the fair volatility of a volatility swap can therefore be approximated with a first or second order expression or computed to all orders by numerically solving an integral.

Calibration

Based on the expectation values for variance and volatility, it is possible to calibrate the Heston model to market data on variance and volatility swaps, respectively. For the calibration to variance swaps the user enters fair variances of swaps with different maturities and a method of weighting these market data. Here, uncertainties for each swap can be entered or the swaps can be weighted equally or relative to the values of fair variance.

In the case of volatility swaps the user provides market data on volatility swaps and a choice of weighting these data with associated uncertainties, equal weights or relative weights. The user also chooses a method for the calculation of fair volatility. As described above the fair volatility can be calculated using a first or second order approximation or a numerical integration to all orders.

The search for the parameters that best describe these input data starts at initial parameter values and extends over a given range of parameters, both entered by the user. Based on these inputs the calibration is carried out by the FINCAD calibration engine.

This engine provides the user with a choice of three optimization algorithms: Levenberg Marquardt, downhill simplex and differential evolution. Based on one of three error metrics (, weighted , or weighted ) these algorithms search for the model parameters that best describe the input data using the equations of fair variance and fair volatility respectively, as given above.

For more details on the calibration engine the reader is referred to the FINCAD Math Reference documents Option Pricing with the Heston Model of Stochastic Volatility and Calibration of Interest Rate Models..

Functions

 

Valuation

Calibration

Variance Swaps

aaVarianceSwap_Heston_iv

aaVarianceSwap_Heston_p

aaCalibrateVarianceSwap_Heston

Volatility Swaps

aaVolatilitySwap_Heston_iv

aaVolatilitySwap_Heston_p

aaCalibrateVolatilitySwap_Heston

 

Naming Conventions

Suffix/ Prefix

Description

_iv

computes the fair variance (volatility) and corresponding Greeks for a variance (volatility) swap

_p

computes the fair value and corresponding Greeks

Calibrate

calibrates the Heston model parameters to market data on variance swaps or volatility swaps

 

aaVarianceSwap_Heston_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

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

sensitivity to initial volatility

rate of change in fair variance per 1% change in initial annual volatility.  This is the derivative of the fair variance with respect to the initial annual volatility, divided by 100

sensitivity to long-term volatility

rate of change in fair variance per 1% change in long-term annual volatility.  This is the derivative of the fair variance with respect to the long-term annual volatility, divided by 100

sensitivity to speed of mean reversion

rate of change in fair variance per change in the speed of mean reversion.  This is the derivative of the fair variance with respect to the speed of mean reversion

number of market days

the number of days that the swap is active; calculated based on the input accrual method

 

aaVarianceSwap_Heston_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 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

sensitivity to initial volatility

rate of change in fair value per 1% change in initial annual volatility.  This is the derivative of the fair value with respect to the initial annual volatility, divided by 100

sensitivity to long-term volatility

rate of change in fair value per 1% change in long-term annual volatility.  This is the derivative of the fair value with respect to the long-term annual volatility, divided by 100

sensitivity to speed of mean reversion

rate of change in fair value per change in the speed of mean reversion.  This is the derivative of the fair value with respect to the speed of mean reversion

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_Heston_iv: Outputs

Output Statistic

Description

fair volatility

the fair volatility

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

sensitivity to initial volatility

rate of change in fair volatility per 1% change in initial annual volatility. This is the derivative of  the fair volatility with respect to the initial annual volatility, divided by 100

sensitivity to long-term volatility

rate of change in fair volatility per 1% change in long-term annual volatility. This is the derivative of the fair volatility with respect to the long-term annual volatility, divided by 100

sensitivity to speed of mean reversion

rate of change in fair volatility per change in the speed of mean reversion. This is the derivative of the fair volatility with respect to the speed of mean reversion

sensitivity to volatility of volatility

rate of change in fair volatility per change in the volatility of volatility. This is the derivative of the fair volatility with respect to the volatility of volatility

number of market days

the number of days that the swap is active; calculated based on the input accrual method

 

aaVolatilitySwap_Heston_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.

sensitivity to initial volatility

rate of change in fair value per 1% change in initial annual volatility.  This is the derivative of the fair value with respect to the initial annual volatility, divided by 100

sensitivity to long-term volatility

rate of change in fair value per 1% change in long-term annual volatility.  This is the derivative of the fair value with respect to the long-term annual volatility, divided by 100

sensitivity to speed of mean reversion

rate of change in fair value per change in the speed of mean reversion.  This is the derivative of the fair value with respect to the speed of mean reversion

sensitivity to volatility of volatility

rate of change in fair value per change in the volatility of volatility. This is the derivative of the fair value with respect to the volatility of volatility

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 valuation functions require effective and terminating date of the swap and the value date as well as a list of applicable holiday dates. The pricing functions also require a discount factor curve. In addition the Heston model parameters are required.

The calibration functions require as input market data on variance or volatility swaps. The calibrated parameters that these functions output can be used as input to the valuation functions.

Further details for these functions are provided in the following data sources and FINCAD Math Reference documents:

·         FINCAD Math Reference document: Option Pricing with the Heston Model of Stochastic Volatility

·         FINCAD Math Reference document: Calibration of Interest Rate Models

·         FINCAD Math Reference document: Variance and Volatility Swaps

·         Chicago Board Options Exchange: Introduction to VIX Futures and Options

 

Example

Example 1: Valuation of a variance swap in the Heston model

On January 2, 2008, we seek to value a variance swap that came into effect on November 1, 2007 and expires on February 1, 2008. We have a calibrated Heston model available, which we would like to use for this valuation. The example workbook shows how to compute the fair value of the variance swap on the value date and how to compute the fair variance for the remainder of the swap’s life.  

  Valuation of a Variance Swap in the Heston Model Example

Related Functions

Model-Independent Pricing

 

Valuation

Variance Swaps

aaVarianceSwap2_iv

aaVarianceSwap2_p

aaVarianceSwap_port

Volatility

aaVolatilitySwap_iv

aaVolatilitySwap_p

aaVolatilitySwap_port

Heston Model

 

Valuation

Calibration

Options

aaOption_Heston_eu_p

aaOption_Heston_payoff_eu_p

aaOption_Heston_iv

aaCalibrateOptions_Heston

 

References

[1]          Brockhaus, O. (2000) 'Volatility swaps made simple', Risk, January 2000, 92.

[2]          Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985) 'A Theory of the Term Structure of Interest Rate', Econometrica 53: 385-407.

[3]          Gatheral, J. (2006) The Volatility Surface, 1st ed., Hoboken, NJ, Wiley Finance.

[4]          Heston, S. (1993) ‘A closed-form solution for options with stochastic volatility’, Review of Financial Studies, 6, 327-43.

'

 

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.

Copyright

Copyright © FinancialCAD Corporation 2008. All rights reserved.