1 / 21

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.

Download Presentation

Implied Volatility Surface PRMIA meeting - Calgary

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related