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Implied Volatility Surface PRMIA meeting - Calgary

Implied Volatility Surface PRMIA meeting - Calgary. Greg Orosi Department of Mathematics and Statistics University of Calgary October 11, 2007. Outline. Review of Black-Scholes framework Description of the Implied Volatility Surface Representation of IVS Applications.

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Implied Volatility Surface PRMIA meeting - Calgary

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  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 S&P500, 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 - S&P500 • Using data from Jan. 2 2004 • Price at the money binary

  20. Example - S&P500 • Using data from Jan. 2 2004 • Price binary with strike = 1.05*stock

  21. Thank You! Questions and comments! • E-mail: gorosi@ucalgary.ca

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