Implied Volatility Surface

PRMIA meeting - Calgary

Greg OrosiDepartment 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

Assumptions of the Black-Scholes model:

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

• Black-Scholes formula:

tt

t

dSdt dW

S

Tσdd

Tσ

/2)Tσ(r/X)ln(Sd

where

)N(dXe)N(dSC

12

20

1

2rT

10

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

0 0( , ) ( , ; , ; )obs BS impC T K C S t T K

Theoretical and Actual IV Surfaces

40

60

80

100

120

140

160

180

200 S1

S6

S11

S16

S21

0,00%

5,00%

10,00%

15,00%

20,00%

25,00%

30,00%

35,00%

40,00%

45,00%

50,00%

Strike %

Maturity

Impl Vola S&P500 29May2002 

Skewness – asymmetry

Kurtosis

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

KTTKKTK 432

210),(

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:

N

iiiKiTi TKKTSCC

1

2, ),(,,;min

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

Surface Example

Surface Example 2

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

)( 1dNS

CBS

Smile Adjusted Hedge Ratios

• Following Coleman (2001), using multivariate chain rule:

• VÄHÄMAA (2003) examines performance

KK

C

S

C

K

C

S

C

S

C

S

C

BSBSAdjusted

Adjusted

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

Applications of the IVS Surface

• Application 2: extracting probabilities

• A binary option pays $1 if asset price exceeds strike at expiry:

K

CKSP

)(

)]([

)(

)(

KEeK

C

KEeK

C

KSEeC

rT

rT

rT

Extracting probabilities

• Binary option based on BS-model:

• Smile adjusted Binary:

• Adjustment will be positive

)()( 2dNeK

CKSP rT

K

C

K

C

K

CKSP BSBS

)(

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

Example - S&P500

BS Adjusted

1 month 50% 58%

3 month 49% 62%

6 month 49% 64%

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

Example - S&P500

BS Adjusted

1 month 11% 13%

3 month 24% 29%

6 month 31% 38%

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

Thank You!

Questions and comments!

• E-mail: gorosi@ucalgary.ca