Implied Volatility Smirk

1
Implied Volatility Smirk

• Jin E. Zhang, HKU
• Yi Xiang, HKUST
• 2005 CICF, Kunming

2
Implied Volatility

• Defined as the root that equates the
  Black-Scholes formula to the market price of an
  option
• It is a function of strike and maturity ---
  Implied volatility surface
• It is an alternative way to quote the price of an
  option

3
Implied Volatility Smirk

• Rubinstein (1994) documented pre-crash (before
  1987) smile and post-crash smirk of SPX options
• Dennis and Mayhew (2002) studied the
  cross-sectional behavior of the smirk
• Foresi and Wu (wp) documented smirk for 12
  countries
• Bollen and Whaley (2004) studied the source of
  the smirk

4
Explain the smirk

• Negative slope is due to the negative skewness of
  risk-neutral probability
• Smile is due to excess Kurtosis
• Negative skewness can be produced by stochastic
  volatility and/or jumps in asset return and/or
  volatility, (Bates 2000, Pan 2002,
  Eraker-Johannes-Polson 2003)

5
Main results of this paper

• Quantify the smirk
• Derive an analytical expression for risk-neutral
  probability density
• Derive an analytical relation between
  risk-neutral cumulants and implied volatility
  smirk
• Study the term structure and dynamics of the
  smirk
• Calibrate option pricing model by using the term
  structure of the smirk

6

Concepts and notations

• Implied forward price
• Moneyness
• Implied volatility (IV)
• At-the-money IV
• IV skewness
• IV smileness
• IV smirkness
• Risk-neutral cumulants

7

Implied forward price

• It is a forward price implied in option prices
• It is computed from the nearest the money call
  and put based on put-call parity

8

Moneyness

• Logarithm of strike price over forward price
  normalized by the standard deviation of expected
  return on maturity
• Measures how far the strike is away from the
  implied forward price
• A measure of average volatility, , we use VIX

9
Quantify the smirk

• Fit implied volatility (IV) with a quadratic
  function that passes through the point
  at-the-money
• Minimize volume weighted error
• ATM-IV , IV skewness , IV smileness
• IV smirkness

10

IV smirk on November 4, 2003 for November SPX
options
11

Option price error

• The error is defined as the difference between
  the Black-Scholes formula with some IV function
  and the market price
• Flat IV, volume-weighted error is 78 cents
• Skewed IV, 31 cents
• Smirked IV, 12 cents
• Smallest bid-ask spread is 15 cents

12

Risk-neutral density

• Can be recovered from the Black-Scholes formula
  with smirked IV.
• The result is

13

Relation between IV smirkness and risk-neutral
cumulants

• Stock price model
• Edgeworth expansion
• Convexity adjustment
• Option pricing formula

14

Relation between IV smirkness and risk-neutral
cumulants

• Match two option pricing formulas
• We have the relation

15

Relation between IV smirkness and risk-neutral
cumulants

• Asymptotic relation
• Rule of thumb, if

16

The term structure of smirkness

• For each maturity, fit IV smirk with a quadratic
  function
• Obtain ATM-IV, skewness and smileness
• Smirkness as a function of maturity --- term
  structure
• The three term structures fully describe the
  information of current option market
• Can be and should be used to calibrate
  option-pricing models

17

IV smirk on Nov-04-03
18

IV smirk on Nov-04-03
19

The term structure of ATM-IV
20

The term structure of skewness
21

The term structure of smileness
22

The dynamics of smirkness

• For options that mature on the same date,
  September 16, 1999
• Study the time-change dynamics of ATM-IV,
  skewness and smileness from September 25, 1998 to
  September 3, 1999
• The time to maturity is changing
• It becomes shorter and shorter

23

The dynamics of ATM-IV
24

The dynamics of skewness
25

The dynamics of smileness
26

Calibrating Option-pricing model

• Construct the term structures of smirkness from
  the market data
• Force the term structures of smirkness implied
  from an option-pricing model passing through the
  points in the market term structures from the
  first nearest term to the second nearest term,
  and so on
• Solve the n equations for n parameters

27

Case 1 Constant Elasticity of Variance (CEV)
model

• The CEV model
• Option pricing
• is a complementary non-central
  chi-square distribution with degrees of
  freedom and non-centrality parameter .

28

Case 2 Finite Moment Log Stable (FMLS) process

• Stock price model
• Option pricing

29

Term structure of ATM-IV
30

Term structure of skewness
31

Term structure of smileness
32

Conclusions

• Contributions
• Quantify IV smirk with a quadratic function
• Analytical relation between IV smirkness and
  cumulants of SPD
• A maturity- and liquidity-based procedure to
  calibrate option pricing models
• Further research
• Smirk implied in different models
• The dynamics of smirkness