modeling and pricing of variance swaps for stochastic...
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Modeling and Pricing of Variance Swaps for
Multi-Factor Stochastic Volatilities
with Delay
Anatoliy SwishchukMathematical and Computational Finance Lab
Department of Mathematics and Statistics
University of Calgary, Calgary, AB, Canada
CAIMS/SCMAI 2007May 20-24, Banff Centre, Alberta, Canada
The research is supported by NSERC
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Outline
• Volatility: Types
• Stochastic Volatility (SV): Models
• One-Factor SV with Delay
• SV with Delay: Why?
• Multi-Factor SV with Delay (MFSVD): Two-Factor and Three-Factor
• Swaps: Definition
• Variance Swaps for MFSVD
• Numerical Examples
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Volatility
• Volatility is the standard deviation of the change in value of a financial instrument with specific time horizon
• It is often used to quantify the risk of the instrument over that time period
• The higher volatility, the riskier the security
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Types of Volatilities• Historical V: standard deviation (uses historical
(daily, weekly, monthly, quarterly, yearly)) price data to empirically measure the volatility of a market or instrument in the past
• Implied V: volatility implied by the market priceof the option based on an option pricing model (smile and skew-varying volatility by strike)
• Local V: given the prices of call or put options across all strikes and maturities, we may deduce the volatility which produces those prices via full Black-Scholes equation (Dupire formulae (1994))
• Stochastic V: volatility is not constant, but a stochastic process (explains smile and skew)
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Stochastic Volatility: Models
• Discrete SV: GARCH model (Bollerslev (1986))
• Heston SV model (1993)
• Mean-Reverting SV model(Wilmott, Haug, Javaheri(2000))
• SV With Delay (Kazmerchuk, Swishchuk, Wu (2002))
• ARCH model (Engle (1982))
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One-Factor SV with DelayThe underlying asset S(t) follows the process
The asset volatility is defined as the solution of the following equation
Why this equation?
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From GARCH to SV with Delay
-discrete-time GARCH(1,1)
-discrete-time GARCH(1,1) (l=1)
-continuous-time GARCH (expectation of log-returns is zero)
-continuous-time GARCH (non-zero expectation of log-return)
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Comparison with GARCH (1,1)
GARCH (1,1) Log-returns for S(t) (Ito formula)
Continuous-Time GARCH for SV with Delay
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Main Features of 1-Factor SV with Delay
1) continuous-time analogue of discrete-time GARCH model
2) Mean-reversion
3) Market is complete (W is the same as for the stock price)
4) Incorporate the expectation of log-returns
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Equation for the Expectation of Variance
Equation for the Variance
Equation for the Expectation
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Solution of the Equation for the Expectation of Variance (1FSV)
Equation to be solved
Stationary solution General solution
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General Solution
Integro-differential equation with delay
General solution
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Multi-Factor SV with Delay
• Multi-Factor-Mean SV with Delay-mean-reversion level V is a stochastic process
V->V (t)-stochastic process• V (t) - geometric Brownian motion (GBM) (two-
factor)
• V (t) - Ornstein-Uhlenbeck (UE) process (two-factor)
• V (t) - Pilipovich one-factor (two-factor)
• V(t) – Pilipovich two-factor process (three-factor)
One-Factor SV with Delay----->
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2-Factor SV with Delay: GBM Mean-Reversion (GBMMR)
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2-Factor SV with Delay (GBMMR): Equation for the E and Solution
Equation for the E
Solution
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2-Factor SV with Delay: OU Mean-Reversion (OUMR)
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2-Factor SV with Delay (OUMR): Equation for the E and Solution
Equation for E
Solution
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2-Factor SV with Delay: Pilipovich 1-Factor Mean-Reversion (OFMR)
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2-Factor SV with Delay (Pilipovich 1FMR): Equation for the E and Solution
Equation for E
Solution
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3-Factor SV with Delay: Pilipovich 2-Factor Mean-Reversion (2FMR)
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3-Factor SV with Delay (Pilipovich 2FMR): Equation for the E and Solution
Equation for E
Solution
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Main Features of All the Solutions for MFSVD
• 1) Contains solution of one-factor SV with Delay
• 2) Contains additional terms due to the new parameters (more factors-more parameters)
• Solution (MFSVD)=Solution (1FSVD) + Additional Terms (Due to the stochastic mean-reversion)
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Variance Swaps
Variance swaps are forward contract on future realized stock variance
Forward contract-an agreement to buy or sell something
at a future date for a set price (forward price)
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Realized Continuous Variance
Realized (or Observed) Continuous Variance:
where is a stock volatility,
T is expiration date or maturity.
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Why Trade Volatility (Variance)?
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How does the Volatility Swap Work?
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Payoff of Variance Swaps
A Variance Swap is a forward contract on
realized variance.
Its payoff at expiration is equal to
N is a notional amount ($/variance);
Kvar is a strike price
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Valuing of Variance Swap forStochastic Volatility with Delay
Value of Variance Swap (present value):
To calculate variance swap we need only ,
where EP* is an expectation (or mean value), r is interest rate.
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Valuing of Variance Swap for One-Factor SV with Delay in Stationary Regime
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Valuing of Variance Swap for One-Factor SV with Delay in General Case
In this way
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Valuing of Variance Swap for One-Factor SV with Delay in General Case
We need to find EP*[Var(S)]:
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Delay as a Measure of Risk
Keep delay as low as possible: in any cases
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Comparison of SV in HestonModel with SV with Delay
Heston Model (1993)
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Comparison of SV in Heston Model with SV with Delay II
When \tau=0 (the same expression as above):
Swap for SV in Heston Model
Swap for SV with Delay
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Numerical Example 1: S&P60 Canada Index (1997-2002)
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Dependence of Variance (Realized) Swap for One-Factor SV with Delay on Maturity
(S&P60 Canada Index)
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Variance (Realized) Swap for One-Factor SV with Delay (S&P60 Canada Index)
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Numerical Example 2: S&P500 (1990-1993)
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Dependence of Variance (Realized) Swap for One-Factor SV with Delay on Maturity
(S&P500)
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Variance (Realized) Swap for One-Factor SV with Delay (S&P500 Index)
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E of Variance and the Price of Variance Swap for 1-Factor SV with Delay
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Variance and The Price of Variance Swap for SV with Delay (GBMMR)
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Variance and The Price of Variance Swap SV with Delay (OUMR)
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Variance and The Price of Variance Swap for SV with Delay (Pilipovich 1FMR)
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Variance and The Price of Variance Swap for
SV with Delay (Pilipovich 2FMR)
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2-F(GBMMR)2-F(OUMR)
2-F(Pilipovich OFMR)
2-F(PilipovichTFMR)
One-FactorComparison
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Conclusion
• There is no big difference between One-Factor SV with Delay and Multi-Factor SV with Delay
• One-Factor SV with Delay catches almost all the features of Multi-Factor SV with Delay
• One-Factor SV with Delay is Similar to the SV in Heston Model
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Publications• ‘Continuous-time GARCH model for Stochastic
Volatility with Delay’ (Kazmerchuk, Sw, Wu), CAMQ, 2005, v. 3, No. 2
• ‘Modeling and Pricing of Variance Swap for Stochastic Volatility with Delay’ (Sw), WilmottMagazine, Issue 19, September 2005
• ‘Modeling and Pricing of Variance Swaps for Multi-Factor Stochastic Volatilities with Delay’ (Sw), CAMQ, 2006, v. 14, No. 4
• ‘The Pricing of Options for Security Markets with Delay’ (Kazmerchuk, Sw, Wu), Mathematics and Computers in Simulations, 2007 (to be published)
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The End
Thank you for your time and attention!
Home page:http://www.math.ucalgary.ca/~aswish/
E-mails:
[email protected]@ucalgary.ca