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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. - PowerPoint PPT Presentation
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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