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In this paper we investigate (augmented) Dickey±Fuller (DF) and Lagrange multiplier (LM) type unit root tests for autoregressive time series through comprehensive Monte Carlo simulations. We consider two sorts of null and alternative hypotheses: a unit root without drift versus level stationarity and a unit root with drift versus trend stationarity. The DF-type coef®cient tests are found to show the best overall performance in both cases, at least if the sample size is suf®ciently large. However, it is also found that the DF and LM tests are roughly complementary with regard to their ®nite-sample power. We therefore consider combining these two types of unit root tests to obtain (ad hoc but)`robust' test procedures. Critical values for the proposed tests are provided.

We propose a new method for approximating the expected quadratic variation of an asset based on its option prices. The quadratic variation of an asset price is often regarded as a measure of its volatility, and its expected value under pricing measure can be understood as the market's expectation of future volatility. We utilize the relation between the asset variance and the Black-Scholes implied volatility surface, and discuss the merits of this new model-free approach compared to the CBOE procedure underlying the VIX index. The interpolation scheme for the volatility surface we introduce is designed to be consistent with arbitrage bounds. We show numerically under the Heston stochastic volatility model that this approach significantly reduces the approximation errors, and we further provide empirical evidence from the Nikkei 225 options that the new implied volatility index is more accurate in predicting future volatility. Keywords: Model-free implied volatility index; volatility forecasting; volatility surface; variance swaps. 433 Int. J. Theor. Appl. Finan. 2011.14:433-463. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 01/05/15. For personal use only. 434 M. Fukasawa et al.

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