Is Mean-Variance Analysis Vacuous: Or was Beta Still Born?

We show in any economy trading options, with investors havingmean-variance preferences, that there are arbitrage opportunitiesresulting from negative prices for out of the money call options.The theoretical implication of this inconsistency is that mean-varianceanalysis is vacuous. The practical implications of this inconsistencyare investigated by developing an option pricing model for aCAPM type economy. It is observed that negative call pricesbegin to appear at strikes that are two standard deviationsout of the money. Such out-of-the money options often trade.For near money options, the CAPM option pricing model is shownto permit estimation of the mean return on the underlying asset,its volatility and the length of the planning horizon. The model is estimated on S&P 500 futures options data coveringthe period January 1992–September 1994. It is found thatthe mean rate of return though positive, is poorly identified.The estimates for the volatility are stable and average 11%,while those for the planning horizon average 0.95. The hypothesisthat the planning horizon is a year can not be rejected. Theone parameter Black–Scholes model also marginally outperformsthe three parameter CAPM model with average percentage errorsbeing respectively, 3.74% and 4.5%. This out performance ofthe Black–Scholes model is taken as evidence consistentwith the mean-variance analysis being vacuous in a practicalsense as well.     

Two price economies in continuous time

Static and discrete time pricing operators for two price economies are reviewed and then generalized to the continuous time setting of an underlying Hunt process. The continuous time operators define nonlinear partial integro–differential equations that are solved numerically for the three valuations of bid, ask and expectation. The operators employ concave distortions by inducing a probability into the infinitesimal generator of a Hunt process. This probability is then distorted. Two nonlinear operators based on different approaches to truncating small jumps are developed and termed $QV$ for quadratic variation and $NL$ for normalized Lévy. Examples illustrate the resulting valuations. A sample book of derivatives on a single underlier is employed to display the gap between the bid and ask values for the book and the sum of comparable values for the components of the book.     

Two sided efficient frontiers at multiple time horizons

Annals of Finance - Two price economy principles motivate measuring risk by the cost of acquiring the opposite of the centered or pure risk position at its upper price. Asymmetry in returns leads...