Constant horizon implied volatility (Constant horizon implied statistics with the GEV option pricing model)

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Constant horizon implied volatility(Constant
horizon implied statistics with the GEV option
pricing model)
3rd 7th September 2007, CCFEA Summer School,
University of Essex

• Amadeo Alentorn
• Old Mutual Asset Managers
• Papers available at
• http//privatewww.essex.ac.uk/aalent

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Contents

• Implied volatility
• Implied volatility surfaces
• Implied risk neutral densities
• The GEV option pricing model
• Empirical results
• Applications

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Implied volatility

• The implied volatility of an option contract is
  the volatility implied by the market price of the
  option based on an option pricing model.
• Traders often quote options in terms of implied
  volatility rather than price, as its a more
  useful measure of an options relative value.
• Different ways to calculate it
• Parametric models like, Black-Scholes
• non-parametric methods, like the method to
  calculate the implied volatility index (VIX)

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Implied volatility smile

• In general, options on the same underlying but
  with different strikes will yield different
  implied volatilities.
• In the Black-Scholes model Implied volatility
  is the wrong number to put into wrong formulae to
  obtain the correct price.

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Implied volatility term structure

• Implied volatilities also vary across time to
  maturity (Dumas et al, 1998).

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Beyond Black-Scholes

• Most efforts in the literature at addressing
  these Black-Scholes anomalies have centred around
  assuming
• either a more suitable return process, such as
• Introducing jumps
• Stochastic volatility models
• or fitting parametric models to traded option
  prices
• implied volatility surfaces
• Implied risk neutral densities

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Implied volatility surfaces

• An implied volatility surface is a 3-D plot that
  combines volatility smile and term structure of
  volatility into a consolidated view of all
  options for an underlying.
• Dumas, Fleming and Whaley (JoF, 1998) propose
  several parametric models that attempt to capture
  variation attributable to both strike level and
  time.
• A similar approach was taken by Tompkins (2001),
  but using a cubic function of time to maturity,
  instead of quadratic.

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Implied volatility surfaces
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Implied risk neutral densities

• Another approach to option pricing is based on
  assuming a particular distribution for the
  underlying price at maturity.
• There is a large number of models, studying
  different distributional assumptions
• Mixture models
• Mixture of lognormals (Bahra 1996 and 1997)
• Specific distributions,
• Weibull distribution (Savickas 2002, 2005)
• Generalized distributions,
• Generalized Lambda Distribution (Corrado 2001)

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The implied risk neutral density

• It can be interpreted as the markets aggregate
  risk neutral probability distribution for the
  price of the underlying asset at maturity T.
• It can be extracted it using a cross section of
  traded option prices, with different strikes but
  same maturity (Breeden and Litzenberger, 1978).

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Time to maturity effects

• When calculating a time series of implied RNDs or
  RND related statistics we encounter the problem
  with the time to maturity effect.
• This happens due to the fixed maturity of
  options.
• As we approach maturity, the time horizon of the
  implied RND shortens, the degree of uncertainty
  decreases, and thus, densities of consecutive
  days are not directly comparable.
• Clews et al. (2000) propose a method to solve
  this problem, by calculating a constant time
  horizon implied statistics by linear
  interpolating between two maturities that bracket
  the horizon of interest.
• The GEV model we present here is more general, as
  we estimate the implied density across all
  maturities.

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Term structure of implied RNDs

• In most markets, there are option contracts
  trading for different maturities T1, T2, TN.
• For example, for the FTSE 100, there are traded
  options with maturities on the closest 3 months,
  and also quarterly (Mar, Jun, Sep, and Dec).
• At any given day, we can extract a term structure
  of RNDs from traded option prices.

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Term structure of implied RNDs
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The Implied RND surface

• Usually, when estimating implied RNDs, a separate
  density is obtained for each maturity. For a
  given day, we obtained a different set of
  parameters for each of the maturities.
• Here, we will consider an implied RND surface for
  the GEV model. For a given day, we want to obtain
  a unique set of parameters, consistent with
  option prices for all strikes and maturities.
• This is similar to implied volatility surfaces,
  where the aim is to have a parametric model that
  fits option prices across both dimensions.

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The implied GEV RND surface

• The implied RND surface as a function of time and
  price.

Index level
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The GEV option pricing model

• It is an option pricing model based on the
  Generalized Extreme Value (GEV) distribution.
  This model
• Removes pricing biases associated with
  Black-Scholes
• Capture the stylized facts of the implied RNDs
• Left skewness
• Excess kurtosis (fat tails)
• Has a closed form solution for European options
• Delivers constant time horizon implied statistics
• Delivers the market implied tail shape
• We make no assumption about the price process,
  only on the functional form of the terminal
  distribution.

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The GEV distribution

• It is a distribution from Extreme Value Theory,
  and is defined as the limiting distribution of
  block maxima.
• The standardized GEV distribution is given by
• where
• µ is the location parameter
• s is the scale parameter
• ? is the tail shape parameter

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The GEV for different values of ?
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Option pricing approach

• Our option pricing approach is based on the
  estimating the implied Risk Neutral Density (RND)
  using traded option prices.
• Following Harrison and Pliska (1981) there
  exists a risk neutral density (RND) function,
  g(ST), such that the call option price can be
  written as
• where EQ is the risk-neutral expectation
  operator, conditional on all information
  available at time t.

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Moments of the GEV distribution

• The first two moments of the GEV distribution
  are
• In order to model the implied RND surface across
  time, we will allow the first two moments to vary
  with time to maturity T.

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Time scaling of volatility

• Back-Scholes assumes that volatility scales with
  T1/2
• We explicitly model the scaling of implied
  volatility with time to maturity by introducing a
  scaling parameter, similarly to parametric models
  for IV surfaces.
• The volatility of a GEV distribution is given by
• We allow for time scaling of volatility with Tb

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The forward no-arbitrage condition

• The forward no-arbitrage condition states that
• The mean of the GEV distribution is
• We can rewrite the location parameter µ to be a
  function of time T, the other parameters, and the
  known Futures price Ft,T, thus removing one
  degree of freedom

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Solving the pricing equation

• The price density function, (when negative
  returns RT are GEV distributed) is given by
• After some rearranging, the call option pricing
  equation that needs to be solved is

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The GEV closed form solution

• Using the Gamma function, we can obtain a closed
  form solution for the call option pricing
  equation under GEV returns
• where
• Similarly, we obtain an equation for put options.

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Estimation of the RND surface

• For a given day, we have M maturities available,
  with Nj traded option prices each.
• Using non-linear least squares, we can estimate
  the set of parameters that minimizes the sum of
  squared errors.
• We price both calls and puts simultaneously.
• Note how the optimization problem is across both
  strikes and maturities

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Pricing performance

• We estimate an implied GEV RND surface on a daily
  basis from 1997 to 2003, using closing prices for
  all traded option.
• We also estimate the Black-Scholes (BS) and
  Mixture of lognormals (MLN) models.
• The GEV model exhibits lower pricing errors (in
  sample).

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Time series of pricing errors
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Hedging performance

• The GEV model also outperforms the Black-Scholes,
  and IV model in most cases (out of sample
  performance).

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Volatility scaling parameter

• Recall that the parameter b gives the scaling law
  for the implied volatility (assumed to be 0.5 in
  Black-Scholes).
• We find that in 93 of days, H0 b 0.5 can be
  rejected.
• To test the economic significance of having b, we
  re-estimated the GEV model fixing b 0.5 and
  found that the average pricing error doubled.
• Even in that case, the GEV model still
  outperformed the MLN model, despite of the MLN
  model now having two extra degrees of freedom.

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Volatility scaling parameter
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Applications
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Implied tail shape parameter ?
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Implied volatility (VIX)
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Implied skewness (SIX?)
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Implied kurtosis (KIX?)
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Event studies Asian crisis

• We can compare the implied RNDs before and after
  a crisis, for a constant horizon, to try an asses
  the change in market sentiment reflected by
  option prices.

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Event studies statistics

• In all three events, substantial increase in
  implied volatility and in higher moments.
• Note that ? only substantially increased in the
  first two cases (due to 9/11 not being a
  financial event?)

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Constant horizon E-VaR

• Another interesting constant horizon implied
  measure we can obtain from the implied RND
  surface is Ecomic VaR.
• Alentorn and Markose (2006) showed how to obtain
  constant horizon E-VaR using a two step process
• First, estimation of the implied RND term
  structure
• Second, a linear regression on the log-log plot.
• With the implied RND surface we can obtain a
  E-VaR value for any maturity and any confidence
  level.
• E-VaR values from the implied surface were found
  slightly less volatile, probably due to using a
  more robust estimation method.

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Conclusions

• We showed how the flexibility of the GEV
  distribution allows us to capture different
  levels of skewness and kurtosis. Unlike other
  models, we dont have to specify the type of
  distribution a priori (i.e. Weibull, Fréchet,
  Gumbel).
• The GEV option pricing model removes the well
  known Black-Scholes pricing biases. It also
  outperforms the mixture of lognormals method. It
  also exhibits superior hedging performance.
• By estimating an implied RND surface across both
  strikes and maturities, we can easily obtain
  constant horizon implied statistics.
• The model also delivers the implied tail shape
  parameter, which controls the implied skewness
  and the fatness of the tails, and can be used to
  asses (risk neutral) market expectations of
  extreme outcomes. We find it increase after
  crisis events.