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 model  There are two distinct approaches: cross-sectional based and time-series based  Cross-sectional: minimize deviation between observed market prices and theoretical prices  Time-series: determine parameters from historical asset price

The solution can also be written as: where Under Risk Neutral Pricing: Example: volatility parameter in Black Scholes: Time Series Black-Scholes

Cross Sectional: Black Scholes Example: Calibrating the (volatility of the) Black-Scholes model  Let C T1,K1,..., C TN,KN be market prices of European calls on a stock with maturities and strikes of (T i, K i )  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:

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

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

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 by Joost Driessen, Pascal Maenhout Analyzed options from 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!