ABSTRACT
We consider the convergence of the solution of a discrete-time utility maximization problem for a sequence of binomial models to the Black-Scholes-Merton model for general utility functions. In previous work by D. Kreps and the second named author a counter-example for positively skewed non-symmetric binomial models has been constructed, while the symmetric case was left as an open problem. In the present article we show that convergence holds for the symmetric case and for negatively skewed binomial models. The proof depends on some rather fine estimates of the tail behaviors of binomial random variables. We also review some general results on the convergence of discrete models to Black-Scholes-Merton as developed in a recent monograph by D. Kreps.
ABSTRACT
We examine Kreps' conjecture that optimal expected utility in the classic Black-Scholes-Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete-time economies that "approach" the BSM economy in a natural sense: The nth discrete-time economy is generated by a scaled n-step random walk, based on an unscaled random variable ζ with mean 0, variance 1, and bounded support. We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that E [ ζ 3 ] > 0 .
ABSTRACT
Understanding the structure of financial markets deals with suitably determining the functional relation between financial variables. In this respect, important variables are the trading activity, defined here as the number of trades N, the traded volume V, the asset price P, the squared volatility σ 2 , the bid-ask spread S and the cost of trading C. Different reasonings result in simple proportionality relations ("scaling laws") between these variables. A basic proportionality is established between the trading activity and the squared volatility, i.e., N â¼ σ 2 . More sophisticated relations are the so called 3/2-law N 3 / 2 â¼ σ P V / C and the intriguing scaling N â¼ ( σ P / S ) 2 . We prove that these "scaling laws" are the only possible relations for considered sets of variables by means of a well-known argument from physics: dimensional analysis. Moreover, we provide empirical evidence based on data from the NASDAQ stock exchange showing that the sophisticated relations hold with a certain degree of universality. Finally, we discuss the time scaling of the volatility σ , which turns out to be more subtle than one might naively expect.
ABSTRACT
Cover's celebrated theorem states that the long-run yield of a properly chosen "universal" portfolio is almost as good as that of the best retrospectively chosen constant rebalanced portfolio. The "universality" refers to the fact that this result is model-free, that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the numéraire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model-free result is complemented by a comparison with the numéraire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time.
ABSTRACT
Every submartingale S of class D has a unique Doob-Meyer decomposition S = M + A , where M is a martingale and A is a predictable increasing process starting at 0. We provide a short proof of the Doob-Meyer decomposition theorem. Several previously known arguments are included to keep the paper self-contained.
