This paper provides a “non-extensive” information theoretic perspective on the relationship between risk and incomplete states uncertainty. Theoretically and empirically, we demonstrate that a substitution effect between the latter two may take place. Theoretically, the “non-extensive” volatility measure is concave with respect to the standard (based on normal distribution) volatility measure. With the degree of concavity depending on an incomplete states uncertainty parameter-the Tsallis-q. Empirically, the latter negatively causes the normal measure of volatility, positively affecting the tails of the distribution of realised log-returns.
6 Figures and Tables
Fig. 1. The Tsallis p.d.f. for different values of q (bottom panel) and a comparison between the Tsallis p.d.f. and its escort p.d.f. used in equation (3)
Fig. 2. This figure plots equation (7) for predetermined values of q (black – q=1, red – q=2 and blue – q=3). For a given q > 1, σq is concave with respect to the normal measure of volatility σ.
Fig. 3. Top panel indicates the historical (normal) and non-extensive volatility measures (σ1 and σq , respectively). The bottom panel indicates the estimated q-parameter.
Fig. 5. Information flows from σV IX to q and to σ1,t. While, information also flows from q to σ1,t but not vice-versa.
Fig. 6. Top panel indicates the Tsallis-q associated with the ith portfolio. Bottom panel indicates σp – the risk associated with the ith portfolio.
Fig. 7. Top left corner of this figure plots expected portfolio return µp vs. the Tsallis-q. Bottom left corner plots the portfolio σp,q and σp,1 vs. the Tsallis-q. The right side plot the mean-variance frontier with respect to σp,q and σp,1.
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