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GARCH Option Pricing Model of Duan(1995)

• Asset returns follow the generalized
  autoregressive conditional heteroskedastic
  (GARCH) process.
• GARCH (1,1) model is the most commonly used GARCH
  process.

• St is the stock price at time t
• rf is the constant one-period risk-free rate of
  return
• ? is constant unit risk premium

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GARCH (1,1) Model

• To ensure covariance stationary of the GARCH
  (1,1) process

• The stationarity conditions are important to
  ensure that the moments of the normal
  distribution are finite. See Greene (2003) for
  more explanation.

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GARCH (1,1) Model Estimation

• Maximum likelihood Select likelihood function

• Estimated parameters from GARCH option
  PricesDaily.xls written by Khanthavit (2007).

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From GARCH process to LRNVR

• Duan (1995) used the locally risk-neutral
  valuation relationship (LRNVR) to derive GARCH
  option pricing.

• This is to ensure that the one-period ahead
  conditional variance is invariant with respect to
  a change to the risk-neutralized pricing measure.

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The Terminal Asset Price

• The terminal asset price is as follows

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Option Pricing Model Call Option

• European call option with exercise price X
  maturing at time T has to time-t value equal to

• For GARCH(1,1) model, Xt and ht1 serve as the
  sufficient statistics for Tt.
• The delta of the call option equals to

• is an indicator function, i.e. equals 1if
  ST X and equals 0 otherwise.

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Option Pricing Model Put Option

• European put option with exercise price X
  maturing at time T has to time-t value can be
  derived from put-call parity relationship.

• The delta of the put option equals to

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Advantage and Disadvantage of GARCH Option
Pricing Model

• Volatility is observable from discrete asset
  price data and only a few parameters need to be
  estimated even in a long time series of options
  records.
• Unfortunately, the analytic solution for the
  GARCH option price is not available because the
  conditional distribution over more than one
  period cannot be analytically derived.

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Option Pricing Model Estimation

• Monte Carlo simulation can be used. (GARCH option
  PricesDaily.xls)
• Simulate 10,000 times to get 10,000 values of
  possible terminal asset prices.
• Get 10,000 possible option values.
• Use expected value as the option price.

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Option Pricing Model Estimation Alternative

• Heston, S., and S. Nandi, 2000, A Closed Form
  GARCH Option Pricing Model, The Review of
  Financial Studies, 13, 585-625.

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Points for Consideration

• Duan (1995) pointed out that this model can only
  be used for the valuation of individual equity
  options. The market portfolio is expected to be
  highly correlated with aggregate assumption and
  the returns will not follow a GARCH process.
• How often do we need to estimate the parameters
  for GARCH model?
• When should we estimate the parameters from the
  option price directly?
• What model would be the best?
• Other GARCH-type model such as EGARCH and NGARCH
  could be explored in the future.

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