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Abstract

We discuss the probabilistic properties of the variation based third and fourth moments of financial returns as estimators of the actual moments of the return distributions. The moment variations are defined under non-parametric assumptions with quadratic variation method but for the computational tractability, we use a square root stochastic volatility model for the derivations of moment conditions for estimations. Using the S&P 500 index high frequency data, the realized versions of the moment variations is used for the estimation of a stochastic volatility model. We propose a simple estimation method of a stochastic volatility model using the sample averages of the variations and ARMA estimation. In addition, we compare the results with a generalized method of moments estimation based on the successive relation between realized moments and their lagged values.

τ τ τ (15) − = − −∫ * ( 1) 1( ) ( ) ( ) kT s ws k k T C w t s t dt T τ τ τ (16) = +( ) ( ) ( )rs ss wsC C Cτ τ τ (17) Based on (14)–(17), we can deduce that: 2 2 2 2 * * 2 * 2 |( ( ) ( )) (0)|{| ( ) | } { 2 ( ( ) ( )) (0) 4 ( ( ) ( )) (0) } 4 rs rs rs rs rs rs rs rs rs C C CE e E C C C C C C ξ Δ − −Δ = Δ − −Δ + Δ − −Δ + (18) * /2 2 /2 [ ( )] [ ( ) ( )] (0) 4 sin(2 ) ( ) r r ss ss ss s s E e C C C C f f G f df β β ξ π π τ − = Δ − −Δ ≈ Δ ⋅∫ (19) Reference [12] has given out the expansion of high-order moment, focusing on the gaussian process with a

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