An Empirical Comparison of Affine and NonAffine Models for Equity Index Options

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An Empirical Comparison of Affine and Non-Affine
Models for Equity Index Options

• Peter Christoffersen,
• Kris Jacobs and Karim Mimouni
• McGill University
• www.christoffersen.ca

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Research Question

• The vast majority of the literature on option
  valuation applies models in the affine class.
• The availability of the conditional
  characteristic function allows for relatively
  easy and quick valuation of options.
• But the affine structure comes at a potential
  cost in terms of accuracy.
• How large is this cost? In and out of sample?
• In continuous time settings?
• In discrete time settings?

3
Models Investigated

• 1- Continuous Time Stochastic Volatility Models
• Affine Heston (1993) Square Root AF-SV
• Non-Affine New Root One NA-SV
• 2- Discrete-Time GARCH Models
• Affine Heston and Nandi (2000) Square Root
  AF-GARCH
• Non-Affine Engle and Ng (1993) Root One
  NA-GARCH

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Preview of Main Results
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Selected Index Option Literature

• Affine Models
• Bakshi, Cao and Chen (1997)
• Bates (2000)
• Chernov and Ghysels (2000)
• Heston and Nandi (2000)
• Pan (2002)
• Eraker (2004)
• Broadie, Chernov and Johannes (2004)
• ...
• Non-affine Models
• Benzoni (2002)
• Jones (2003)

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Contribution

• Affine versus non-affine model comparison using
  new and convenient methodology usable in real
  time and with flexible objective functions.
• Objective function uses rich information in
  option prices while keeping model consistent with
  underlying asset return over time.
• Avoids over-fitting.
• Continuous and discrete time comparisons
• Evenly tightly parameterized models.
• In and out-of-sample experiments.

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Overview

• 1) Implementing AF-SV
• -- Closed-form Pricing, Auxiliary Particle
  Filter
• 2) Implementing NA-SV
• -- Monte Carlo Pricing, Auxiliary Particle
  Filter
• 3) Implementing AF-GARCH
• -- Closed-form Pricing, Return Filter
• 4) Implementing NA-GARCH
• -- Monte Carlo Pricing, Return Filter
• 5) Empirical Results and Discussion
• 6) Conclusion and Future Work

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Empirical Set-Up

• Objective of the form
• Enforce consistency with underlying returns by
  using them in volatility filtering

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1) Heston Stochastic Volatility

• Physical Process (with correlation ?)
• Risk Neutral Process (with correlation ? and
  price of volatility risk ?)

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Affine SV Pricing

• Do Fourier inversion of the conditional
  characteristic function to get the call price

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AF-SV Discretization

• Volatility is a continuous time latent factor in
  this model. There is no easy way to update it
  using only observables.
• We first need to discretize the volatility (Euler
  scheme) keeping it positive (log transformation).
  Using Itos lemma.

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AF-SV Filtering

• In order to do option valuation we need a filter
  on observed returns for the latent volatility
  factor conditional on a given parameter vector.
• We use the Auxiliary Particle Filter (APF).
• Step A Select likely particles
• Step B Simulate the state forward
• Step C Compute new weights. Calculate filtered
  volatility as weighted average.

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APF Step A Select Likely Particles

• On day t we have an initial set of N volatility
  particles and weights
• Compute a conditional summary statistic for
  log(Vt1)
• Simulate auxiliary index variable
• Resample with replacement from N particles using
  the (normalized) index variable. Obtain N
  probability weighted particles.

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APF Step B Simulate State Forward

• Use observed stock return and its dynamic to get
  stock innovations
• Draw new i.i.d. shock to get volatility
  innovation
• Vt1 now easily calculated for each particle
  using the volatility dynamic.

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APF Step C Compute Weights

• We need Wt1. We have Vt1 from which we can
  calculate likelihood of Rt2, which in turn can
  be used as a particle weight
• Normalize weights to sum to 1
• Compute the filtered volatility as the weighted
  average of the particles. Repeat for all t.

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AF-SV Objective Optimization

• Finally do NLS optimization of
• Summary of steps
• 1- Propose starting values for parameters,
• 2- Filter the volatility via APF,
• 3- Evaluate option prices using pricing model
  (easy)
• 4- Compute the SSE using observed market prices,
• 5- Let optimizer update parameters and start
  over.
• Note We could use any well-defined loss function
  here. Christoffersen and Jacobs (2004).

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2) Non-Affine Stochastic Volatility

• Physical Process (with correlation ?)
• Risk Neutral Process (with correlation ? and
  price of volatility risk ?)

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NA-SV Objective Optimization

• NLS optimization of
• Steps
• 1- Propose starting values for parameters,
• 2- Filter the volatility via APF,
• 3- Evaluate option prices according to the model
  by Monte Carlo simulation,
• 4- Compute the SSE relative to observed market
  prices,
• 5- Let the optimizer update parameters and start
  over.

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3) Heston-Nandi AF-GARCH

• Physical Process
• Risk Neutral Process

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AF-GARCH Option Pricing

• Using Fourier inversion of the conditional
  characteristic function,
• we get the call price
• The conditional characteristic function is a set
  of difference equations with terminal conditions.

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AF-GARCH Filtering

• GARCH is by design a filter so easy updating
  using only observable returns.
• Solve for z in the return dynamic and substitute
  into the volatility dynamic to get

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AF-GARCH Objective Optimization

• NLS optimization of
• Steps
• 1- Propose starting values of parameters,
• 2- Filter the volatility on returns only (easy),
• 3- Evaluate option prices according to the model
  (easy),
• 4- Compute the SSE relative to observed market
  prices,
• 5- Let the optimizer update parameters and start
  over.

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AF-GARCH Rewritten

• We can rewrite model in diffusion form
• Where
• And

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4) Engle-Ng Non-Affine GARCH

• Physical Process
• Risk Neutral Process

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NA-GARCH Rewritten

• We can rewrite in NA-SV form
• Where
• And

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NA-GARCH Objective Optimization

• NLS optimization of
• Steps
• 1- Propose some candidate parameters,
• 2- Filter the volatility on returns only (easy),
• 3- Evaluate option prices according to the model
  by Monte Carlo simulation,
• 4- Compute the SSE relative to observed market
  prices,
• 5- Let the algorithm pick new parameters and
  start over.

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5) Empirical Results

• SP500 European style index call options
• Contracts filtered as in Bakshi, Cao and Chen
  (1997). Maturity between 7 and 365 days.
• Estimate the four models on the Wednesdays in
  each of the six years 1990,,1995.
• Report in-sample Wednesday RMSE by year.
• Report out-of-sample Thursday RMSE by year.

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Tables 1 and 2 Option Contracts (In/Out)
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Tables 1 and 2 Average Price (In/Out)
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Tables 1 and 2 Implied Vol. (In/Out)
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Figure 1 Implied Volatility
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Table 3 SV Estimates. 1990-1995
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Table 3 GARCH Estimates
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Figure 2 Spot Volatility PathsVolatility Jumps
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Table 5 RMSE In Sample
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Table 5 RMSE Out-of-Sample
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In-Sample RMSE by Moneyness and Maturity
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Out-of-Sample RMSE by Moneyness and Maturity
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Figure 3 Weekly RMSE
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Figure 4 Weekly Bias
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Discussion

• Main findings at this point
• Affine models seem to clearly dominate non-affine
  models
• NA-GARCH quite close to NA-SV
• AF-GARCH better than AF-SV
• What can be driving these results?

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Conditional Leverage Effect

• In our four models
• Note the power on conditional variance

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Figure 5 Conditional Leverage Paths. (Skewness)
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Conditional Variance of Variance

• In our four models
• Note the power on conditional variance

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Figure 6 Conditional Volatility of Variance
Paths. (Kurtosis)
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NA-SV versus NA-GARCH

• Duan (1997) shows that the NA-SV model we use can
  be viewed as the continuous time limit of the
  NA-GARCH model consider.
• The price of risk parameter, d, in the one-shock
  GARCH option pricing model plays a role very
  similar to the price of volatility risk, ?, in
  the two-shock SV model.
• The volatility risk premium specification could
  be explored further. We have used the most
  standard one which is convenient in the AF-SV
  model but which could easily be generalized in
  the NA-SV model.

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AF-SV versus AF-GARCH

• The superior performance of the AF-GARCH over the
  AF-SV model is puzzling.
• Heston-Nandi provide AF-SV limit of the AF-GARCH
  model but for perfectly correlated shocks (c /-
  infinity) only.
• Perhaps more general limit exists which is not
  the AF-SV model?
• Consider the following shock correlations

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Conditional Correlations

• In our four models
• AF-GARCH has time-varying correlation!

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State Price Densities (1990 Est.)
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State Price Densities (1995 Est.)
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6) Conclusion

• Three Main Results
• Affine models clearly dominate non-affine models
  particularly in the continuous time setting.
• The NA-SV is close to NA-GARCH which can be
  reconciled by the continuous time limit and the
  similar roles played by d and ?.
• AF-GARCH better than AF-SV. Puzzling, but perhaps
  due to AF-GARCHs time-varying shock correlation.
  The general AF-GARCH continuous time limit is not
  the AF-SV.

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Whats Next?

• Try Non-Affine Versions of
• Volatility Component Models
• Bates (2000)
• Christoffersen, Jacobs and Wang (2005)
• Poisson Jumps / Non-normal innovations
• Eraker (2004),
• Broadie, Chernov and Johannes (2004)
• Christoffersen, Heston and Jacobs (2005)
• Levy Processes
• Huang and Wu (2004)