Are Options Mispriced? Greg Orosi. Outline Option Calibration: two methods Consistency Problem Two Empirical Observations Results.
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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!
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