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Implied Volatility Surface PRMIA meeting Calgary

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Review of Black-Scholes framework. Description of the Implied Volatility Surface ... Assumptions of the Black-Scholes model: ... Kurtosis. Modeling the IVS Surface ... – PowerPoint PPT presentation

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Title: Implied Volatility Surface PRMIA meeting Calgary


1
Implied Volatility SurfacePRMIA meeting -
Calgary
  • Greg Orosi
  • Department of Mathematics and Statistics
  • University of Calgary
  • October 11, 2007

2
Outline
  • Review of Black-Scholes framework
  • Description of the Implied Volatility Surface
  • Representation of IVS
  • Applications

3
Assumptions of the Black-Scholes model
  • Black-Scholes assumes asset follows a Geometric
    Brownian Motion with constant volatility
  • Black-Scholes formula

4
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

5
Theoretical and Actual IV Surfaces
  •  

6
Skewness asymmetry
7
Kurtosis
8
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

9
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

10
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

11
Surface Example
12
Surface Example 2
13
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

14
Smile Adjusted Hedge Ratios
  • Following Coleman (2001), using multivariate
    chain rule
  • VÄHÄMAA (2003) examines performance

15
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

16
Applications of the IVS Surface
  • Application 2 extracting probabilities
  • A binary option pays 1 if asset price exceeds
    strike at expiry

17
Extracting probabilities
  • Binary option based on BS-model
  • Smile adjusted Binary
  • Adjustment will be positive

18
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

19
Example - SP500
  • Using data from Jan. 2 2004
  • Price at the money binary

20
Example - SP500
  • Using data from Jan. 2 2004
  • Price binary with strike 1.05stock

21
Thank You!
  • Questions and comments!
  • E-mail gorosi_at_ucalgary.ca
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