estimation of stochastic volatility models with implied
TRANSCRIPT
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Estimation of Stochastic Volatility Models with Implied Volatility
Indices and Pricing ofStraddle OptionStraddle Option
Yue Peng and Steven C. J. Simon
University of EssexCentre for Computational Finance and
Economic Agents
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Outline
• Introduction : Stochastic Volatility Models, Volatility Index and Straddle Option
• Methodology• Methodology• Data and Results• Conclusion
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Stochastic Volatility Models
• Asset variance is assumed to follow a stochastic process
• Constant Elasticity of Variance (CEV)• Heston• GARCH(1,1)
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Constant Elasticity of Variance (CEV) model (Chan et al (1992), Chacko et al (2000) and Jones (2003))
• Under risk-aversion measure P, the logarithm of the asset price and its variance have the following process with CEV model
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Heston and GARCH Model
• The model of Heston is obtained when = 0.5
• The model of GARCH is obtained when= 1
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Volatility Index
• It is a market expectation of short-term future volatility, based on prices of short-term options.
• The volatility index is used as a proxy for the true but unobserved instantaneous volatility.
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ATM Forward Straddle
• A straddle -- a call option and a put option, with the same strike price and the same maturity time on the same underlying asset.
• An at-the-money forward straddle -- the strike price is equal to the corresponding forward price, K =Ft, T
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ATM Forward Straddle Option (STO)
ATMF Straddle, entering at
ATMF STO, entering at T0 , maturity at T1, KSTO is pre-
specified.
T1 T2T0
ATMF Straddle, entering at T1, maturity at T2,
K = FT1, T2 .
At T1, the buyer of ATMF STO has the option to buy a ATMF straddle with price equal to KST 0 . He receives a call and a put with strike price equal to the
forward price FT1, T2.
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ATM Forward Straddle Option (STO)
• Brenner et al (2006) give the closed form solution of ATMF STO with Stein and Stein(1991) model.
• We use Monte Carlo method to price ATMF STO with three stochastic volatility models in 5 stock markets.
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Outline
• Introduction• Methodology: Maximum Likelihood • Methodology: Maximum Likelihood
Expansion • Data and Results• Conclusion
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Maximum-Likelihood Estimation
• The state variables:s(t)--the logarithm of the level of stock indexstock indexv(t)– instantaneous variance (the square of the volatility index)
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The close-form expansion of stochastic volatility models
• The logarithm of the conditional density is
• The true joint likelihood function • The true joint likelihood function x(t) = [s(t), v(t)]’ is unknown. We use closed form expansion instead of the true one
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The close-form expansion of stochastic volatility models
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Outline
• Introduction• Methodology• Data and Results : • Data and Results :
Summary StatisticsParameter Estimation Option Pricing
• Conclusion
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Outline
• Introduction• Methodology• Data and Results : • Data and Results :
Summary StatisticsParameter Estimation Option Pricing
• Conclusion
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• The values of log likelihood function for GARCH and the CEV are quite close. Heston gives much lower values.
• The estimated values of mean reversion speed have large standard errors. The speed have large standard errors. The estimated values for the diffusion terms of variance are always significant.
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• The correlation between stock price and variance of BEL 20 market is much lower and less significant than the other markets. It seems there is not a clear leverage effect for the Belgian stock leverage effect for the Belgian stock index.
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• Heston is rejected for all five markets.
• GARCH is rejected for the BEL 20 and S&P 500 indices.
• These two markets are the least correlated with other markets. Their volatility indices have a much lower kurtosis and skewness.
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• So the reason for this is most likely a reflection of the difference across the volatility indices, but rather an artifact of the methodology.
• For the S&P 500 index the results are in line with Ait-Sahalia and Kimmel (2007), who find that both the Heston and the GARCH model are rejected.
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Outline
• Introduction• Methodology• Data and Results : • Data and Results :
Summary StatisticsParameter Estimation Option Pricing
• Conclusion
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ATMF STO under CEV model (FTSE 100)
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ATMF STO under Heston model (FTSE 100)
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Difference between Heston and CEV (FTSE 100)
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ATMF STO under GARCH model (FTSE 100)
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Difference between GARCH and CEV (FTSE 100)
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ATMF STO under CEV model (S&P 500)
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ATMF STO under Heston model (S&P 500)
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Difference between Heston and CEV (S&P 500)
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ATMF STO under GARCH model (S&P 500)
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Difference between GARCH and CEV (S&P 500)
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Outline
• Introduction• Methodology• Methodology• Data and Results• Conclusion
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Conclusion
• Heston is rejected for all markets. But GARCH model is not rejected for the FTSE 100, AEX and CAC 40.FTSE 100, AEX and CAC 40.
• For the BEL 20 index we only find very little evidence of the leverage effect.
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Conclusion
• The results of ATMF STO confirm the outcomes of hypothesis.
• The difference in parameter estimation across the different stock index markets translates in non-trivial difference in ATMF STO prices.
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Thank youfor your attention
!!