Abstract. We derive Godunov-type semidiscrete central schemes for Hamilton-Jacobi equations on triangular meshes. High-order schemes are then obtained by combining our new numerical fluxes with high-order WENO reconstructions on triangular meshes. The numerical fluxes are shown to be monotone in certain cases. The accuracy and high-resolution properties of our scheme are demonstrated in a variety of numerical examples.
We study the feedback effects induced by portfolio optimizers on the underlying asset prices.Through their interaction with reference traders, who trade based on some aggregate incomes process, they are assumed to move asset prices away from the standard log-normal model. With market clearing as our main constraint, we solve analytically for the approximate dynamics of the asset price assuming that the wealth of the portfolio optimizers is small relative to the total market capitalization of the stock. We also calculate numerically the influence of portfolio optimizers when their wealth is not so small. There is good agreement between the numerical and analytical results when the wealth of the optimizers is small. We find that portfolio optimizers influence the price of the risky asset so as to decrease its volatility. The optimal allocation to the risky asset also changes as a result of the portfolio optimizers' actions. In general, it is advantageous to hold more of the risky asset, relative to the log normal Merton model.
Abstract. This paper extends a result obtained by Wigner and von Neumann. We prove that a non-constant real-valued function, f (x), in C 3 (I) where I is an interval of the real line, is a monotone matrix function of order n + 1 on I if and only if a related, modified function gx 0 (x) is a monotone matrix function of order n for every value of x 0 in I, assuming that f is strictly positive on I.
We consider a model of the economy that splits investors into two groups. One group (the reference traders) trades an underlying asset according to the difference in realized returns between that asset and some evolving consensus estimate of those returns; the other group (hedgers) hedge options, namely straddles, on the underlying asset. We consider the cases when hedgers are long the straddle and when the hedgers are short the straddle. We numerically simulate the terminal distribution of the underlying asset price and find that hedgers that are long the straddle tend to push the underlying toward the strike, while hedgers that are short the straddle cause the underlying security to have a bimodal terminal probability distribution with a local minimum at the strike.
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