The limitations of estimating implied densities from option prices

Abstract

Bakshi et al. (Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options. Rev. Financ. Stud., 2003, 16(1), 101–143) develop a novel method for estimating a stocks's τ-period risk-neutral return moments from option prices. Their result relies on the integration of all OTM calls and puts to find the prices of a quadratic, cubic, and quartic contract; from which risk-neutral estimates of the 2nd, 3rd, and 4th moments are derived. Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) point out that error is introduced into the BKM estimation in practice since options do not span a continuum and the range of moneyness and distance between strikes varies across firms and maturities. We use Monte Carlo simulation to show that the standard approach of Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) used within Conrad et al. (Ex ante skewness and expected stock returns. J. Finance., 2013, 68(1), 85–124) and other leading empirical works leads to very biased, and in some cases inconsistent, estimates of BKM implied moments. More so, we prove the particularly concerning result that estimation error using the Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) methodology increases dramatically when a stock's actual return distribution is non-normal or the firm's stock price is not exactly at-the-money. Fortunately, the 2-step interpolation method developed by DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) is largely robust to the estimation errors we document. Based on our simulation results, we recommend the use of the DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) methodology to estimate the 3rd and 4th implied moments, and either the DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) methodology or the methodology of Fukasawa et al. (Model-free implied volatility: From Surface to Index. Int. J. Theor. Appl. Finance, 2011, 14(4), 433–463) to estimate the 2nd implied moment (implied volatility). In addition, we argue that the economically large and robust results of our MC simulations and brief empirical study indicate that the findings of prior studies which use the estimation methodology of Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493), followed by Conrad et al. (Ex ante skewness and expected stock returns. J. Finance., 2013, 68(1), 85–124), should be reevaluated.