Implied Volatility Surface PRMIA meeting - Calgary

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Implied Volatility SurfacePRMIA meeting -
Calgary

• Greg Orosi
• Department of Mathematics and Statistics
• University of Calgary
• October 11, 2007

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Outline

• Review of Black-Scholes framework
• Description of the Implied Volatility Surface
• Representation of IVS
• Applications

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Assumptions of the Black-Scholes model

• Black-Scholes assumes asset follows a Geometric
  Brownian Motion with constant volatility
• Black-Scholes formula

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Assumptions of the Black-Scholes model

• By inverting the Black-Scholes formula, implied
  volatility can be calculated for each option
• By plotting these IVs we get volatility surface
• Since Geometric Brownian Motion assumptions are
  violated, implied volatilities exhibit a
  dependence on strike price and expiry

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Theoretical and Actual IV Surfaces

•  

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Skewness asymmetry
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Kurtosis
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Modeling the IVS Surface

• Practitioners model the implied volatility
  surface as a linear function of moneyness and
  expiry time
• Parameters of the model are determined by
  computing implied volatilities, and performing an
  OLS regression or NLS minimization

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NLS minimzation

• Given N option prices CT1,K1, ..., CTN,KN on a
• stock with maturities and strikes of (Ti, Ki)
• Determine the values of the parameters by
  solving

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OLS or NLS regression?

• According to Christoffersen, Jacobs and Heston
  (2004) NLS surface significantly outperforms OLS
• Christoffersen, Jacobs and Heston claim (2007)
  NLS surface is the best performing surface in
  current literature
• Two comments
• Tested on SP500, but crude oil is similar
• Nonparametric methods can improve

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Surface Example
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Surface Example 2
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Applications of the IVS Surface

• Application 1 more accurate hedge ratios
• Delta based on BS-model
• However adjustment should be made because
  volatility is dependent on strike

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Smile Adjusted Hedge Ratios

• Following Coleman (2001), using multivariate
  chain rule
• VÄHÄMAA (2003) examines performance

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Smile Adjusted Hedge Ratios - Results

• VÄHÄMAA finds FTSE 100 index option market shows
  that the delta hedging performance of the BS
  model can be substantially improved by adjusting
  the BS delta
• Mean average hedging error of the delta-neutral
  portfolio can be reduced by 20 for a 5-day
  hedging horizon

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Applications of the IVS Surface

• Application 2 extracting probabilities
• A binary option pays 1 if asset price exceeds
  strike at expiry

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Extracting probabilities

• Binary option based on BS-model
• Smile adjusted Binary
• Adjustment will be positive

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Extracting probabilities

• These probabilities are forward looking and hence
  contain information about known future movements
• Known news announcement
• Past prices might not contain this information
• Examples

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Example - SP500

• Using data from Jan. 2 2004
• Price at the money binary

BS Adjusted
1 month 50 58
3 month 49 62
6 month 49 64
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Example - SP500

• Using data from Jan. 2 2004
• Price binary with strike 1.05stock

BS Adjusted
1 month 11 13
3 month 24 29
6 month 31 38
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Thank You!

• Questions and comments!
• E-mail gorosi_at_ucalgary.ca