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In this paper, we consider the response of the oil-futures option market to the onset of severe conditions in the aftermath of Feb. 15, 2020. Motivated in part by the decline of the WTI futures contract into negative territory on April 20, 2020, for the derivative market on oil futures we consider an analytical contrast between the traditional Black model and its long-ago predecessor, the Bachelier model. Under 2020 crash conditions, the Bachelier model performs better than Black, displaying a significantly flatter vol smile. Based in part on previous published research for short-dated maturities , the rationale for this difference is built on the contrast between between implied Black and Bachelier volatilities. Other than for extreme strikes and high Black vols, we show that the rapport works well in a wider range of maturities and volatilities. Using options data over the year 2020, we explore a notion of normalized strike to measure quantitatively the vol skew. [ABSTRACT FROM AUTHOR]

We show that the conjugacy class of an eventually expanding continuous piecewise affine interval map is contained in a codimension 1 submanifold of parameter space. In particular conjugacy classes have empty interior. This is based on a study of the relation between induced Markov maps and ergodic theoretical behavior. [ABSTRACT FROM AUTHOR]