are options mispriced? greg orosi. outline option calibration: two methods consistency problem two...
TRANSCRIPT
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!
Survey of Local Volatility Models Lunch at the lab Greg Orosi University of Calgary November 1, 2006
Infection Control in hospital settings. Influenza. Piroska Orosi M.D., Ph.D Faculty of Public Health