Are Options Mispriced?

Greg Orosi

Outline

• Option Calibration: two methods

• Consistency Problem

• Two Empirical Observations

• Results

Option Calibration

Calibrating a model: estimating the parameters of a given

theoretical modelThere are two distinct approaches: cross-sectional based and

time-series basedCross-sectional: minimize deviation between observed market

prices and theoretical pricesTime-series: determine parameters from historical asset price

]|),([),( )(tT

tTrt STScEetSc

SdzrSdtdS

The solution can also be written as:

where

Under Risk Neutral Pricing:

Example: volatility parameter in Black Scholes:

Time Series Black-Scholes

1

1

2

T

RRT

tt

Cross Sectional: Black Scholes

Example: Calibrating the (volatility of the) Black-Scholes model

Let CT1,K1, ..., CTN,KN be market prices of European calls on a stock with maturities and strikes of (Ti, Ki)

Let C(0,s;K,T,) be the Black-Scholes price of a European call with strike K, maturity T if the volatility equals

Determine that value solving:

N

iiiKiTi KTsCC

1

2,

0,,;,0min

Crude Oil

Advantages and Disadvantages

Cross-sectional is forward looking – contains more information than time series

Time-series is not forward looking but less likely to misprice options

Implied Parameters

• Consider more complex model than B-S

• We can find “implied parameters” for other models by cross-sectional calibration, and parameters from time-series

• Compare the two sets of parameters

`

Heston model

Implied and Actual Volatility Monthly Jan 1992-Jan 2004

Implied Volatility & Actual Volatility, Monthly, Jan 1992-Jan 2004

0

50

100

150

200

250

300

350

400

1990 1992 1994 1996 1998 2000 2002 2004 2006

Year

0

1

2

3

4

5

6

7

Implied

Actual

Skewness and Kurtosis

Skewness – asymmetry

Kurtosis

Consistency Problem

• Parameters obtained from cross-sectional calibration and time-series calibration are different– Cross sectional values imply higher skewness– Also imply higher kurtosis

• It seems option markets imply significantly different dynamics for asset than historical parameters: consistency problem– Which is right? Are options mispriced?

• If options are mispriced there should be profitable trading strategies

Can options be mispriced?

Yes! Before 1987 crash plot of implied volatilities used to be flat! => Profit by buying OTM puts

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

•  

•  

Option Markets

• Since 1987 crash, σ tends to be low strike price, known as “options smirk”

• So option markets “learned” and incorporated a higher likelihood of a sudden large movement than a model based on GBM

Empirical Observation 1

• Cause of skewness: puts are more expensive than calls, because they can serve as insurance against a crash

Shorting Puts

• Maybe there is excess return by shorting puts– Situation reversed from before 1987 crash– Only for stocks– For commodities we can consider kurtosis trade

• Results later

Possible Cause of Kurtosis

• Option market participants prefer far out of the money options because of large payoffs

• Causes high demand

• Willing to pay large transaction cost

Empirical Observation 2

• Implied volatilities are higher than historical:

Empirical Observation 2

• Called negative implied volatility premium

• Implied volatilities should be higher than historical

• There are various risks in writing an option even if a market maker is vega and delta hedged:– Jump risk

Shorting Straddles

• If the premium is high for writing an option, then shorting at the money straddles could return excess profit:

Results

• An Empirical Portfolio Perspective on Option Pricing Anomalies - 2005

by Joost Driessen, Pascal Maenhout

• Analyzed options from 1987-2001 for S&P500

• Accounted for jump risk and transaction costs

• Assumed power utility

Results

• Montly CEW for different values of RA

• Under transaction cost strategies return:– 10.2% annually for short straddle (RA=2)

– 18.2% (RA=1)

– 11.5% annually for short put (RA=1)

– 19.4% (RA=2)

• Weights are negative in the portfolio for all values of RA

Conclusion

• So based on data stock options ARE mispriced!

• We can use stochastic volatility parameters to identify mispriced options

• It is best to use a mixture of the cross-sectional and time-series for SV parameter estimation

Thank You!

Questions and comments!