Modelling and Pricing of Variance Swaps for Stochastic Volatility with Delay Anatoliy Swishchuk...
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Transcript of Modelling and Pricing of Variance Swaps for Stochastic Volatility with Delay Anatoliy Swishchuk...
Modelling and Pricing of Variance Swaps for
Stochastic Volatility with Delay Anatoliy Swishchuk
Mathematical and Computational Finance Laboratory
Department of Mathematics and Statistics
University of Calgary, Calgary, AB, Canada
MITACS Project Meeting: Modelling Trading and Risk in the Market
BIRS, Banff, AB, Canada November 11-13, 2004
This research is partially supported by MITACS and Start-Up Grant (Faculty of Science, U of C, Calgary, AB)
Swaps
• Stock• Bonds (bank
accounts)
• Option• Forward contract• Swaps-agreements between
two counterparts to exchange cash flows in the future to a prearrange formula
Basic Securities Derivative Securities
Security-a piece of paper representing a promise
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)
Variance is a measure of the uncertainty of a stock price.
Variance=(standard deviation)^2=(volatility)^2
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
Realized Continuous Variance
Realized (or Observed) Continuous Variance:
where is a stock volatility,
T is expiration date or maturity.
Types of Stochastic Volatilities
• Regime-switching stochastic volatility (Elliott & Swishchuk (2004) “Pricing options and variance swaps in Brownian and fractional Brownian markets”, working paper)
• Stochastic volatility itself (CIR process in Heston model)
• Stochastic volatility with delay (Kazmerchuk, Swishchuk & Wu (2002) “Continuous-time GARCH model for stochastic volatility with delay”, working paper)
Figure 2: S&P60 Canada Index Volatility Swap
Realized Continuous Variance for
Stochastic Volatility with Delay
Initial Data
deterministic function
Stock Price
Equation for Stochastic Variance with Delay (Continuous-Time GARCH Model)
Our (Kazmerchuk, Swishchuk, Wu (2002) “The Option Pricing Formula for Security Markets with Delayed Response”) first attempt was:
This is a continuous-time analogue of its discrete-time GARCH(1,1) model
J.-C. Duan remarked that it is important to incorporate the expectation of log-return into the model
The Continuous-Time GARCH Stochastic Volatility Model
This model incorporates the expectation of log-return
Discrete-time GARCH(1,1) Model
Stochastic Volatility with Delay
Main Features of this Model• Continuous-time analogue of discrete-time GARCH
model• Mean-reversion• Does not contain another Wiener process• Complete market
• Incorporates the expectation of log-return
Valuing of Variance Swap forStochastic Volatility with Delay
Value of Variance Swap (present value):
To calculate variance swap we need only EP*{V},
where and
where EP* is an expectation (or mean value), r is interest rate.
Continuous-Time GARCH Model
or
where
Deterministic Equation for Expectation of Variance with Delay
There is no explicit solution for this equation besides stationary solution.
Stationary Solution of the Equation with Delay
Valuing of Variance Swap with Delay in Stationary Regime
Approximate Solution of the Equation with Delay
In this way
Valuing of Variance Swap with Delay in General Case
We need to find EP*[Var(S)]:
Numerical Example 1: S&P60 Canada Index (1997-2002)
Dependence of Variance Swap with Delay
on Maturity (S&P60 Canada Index)
Variance Swap with Delay (S&P60 Canada Index)
Numerical Example 2: S&P500 (1990-1993)
Dependence of Variance Swap with Delay on Maturity (S&P500)
Variance Swap with Delay (S&P500 Index)
Conclusions
• Variance swap for regime-switching stochastic volatility model;
• Variance, volatility, covariance and correlation swaps for Heston model;
• Variance swap for stochastic volatility with delay;
• Numerical examples: S&P60 Canada Index and S&P500 index
Thank you for your attention!