Beginners first learn to price stock options with a simple binomial tree model for random price changes. It is well known that this classical one-dimensional random walk converges weakly to Brownian motion in the proper space-time scaling limit. Actual stock prices changes occur not at regular times but at random times according to the order flow in an electronic limit order book (LOB), and these are observed to have heteroscedastic and self-exciting characteristics. In this talk, we consider random walks in which jumps occur at random times described by an independent general point...
Mutual funds that invest in private securities value those securities at stale prices. Prices change on average every 2.5 quarters, vary across fund families, and are revised upward dramatically at follow-on funding events. The infrequent, but dramatic price changes yield predictably large fund returns. Fund investors can exploit the stale pricing by buying (selling) before (after) the follow-on funding events (though we find little evidence of this behavior to date). Fund families can opportunistically save up and unleash dry powder (unused markup of private securities) when doing so...
This paper provides a comprehensive analysis of portfolios of active mutual funds, ETFs and hedge funds through the lens of risk (anomaly) factors. We show that these funds do not systematically tilt their portfolios towards profitable factors, such as high book-to-market (BM) ratios, high momentum, small size, high profitability and low investment growth. Strikingly, there are almost no high-BM funds in our sample while there are many low-BM “growth” funds. Portfolios of “growth” funds are concentrated in low BM-stocks but “value” funds hold stocks across the entire BM spectrum. In fact,...
Stochastic gradient descent (SGD) is almost ubiquitously used in training non-convex optimization tasks. Recently, a hypothesis by Keskar et al. (2017) that large batch SGD tends to converge to sharp minima has received increasing attention. We justify this hypothesis by providing new properties of SGD in both finite-time and asymptotic regimes, using tools from Partial Differential Equations. In particular, we give an explicit escaping time of SGD from a local minimum in the finite-time regime. We prove that SGD tends to converge to flatter minima in the asymptotic regime (although it may...
The leverage effect refers to the generally negative correlation between the return of an asset and the changes in its volatility. There is broad agreement in the literature that the effect should be present, and it has been consistently found in empirical work. However, a few papers have pointed out a puzzle: the return distribution of many assets do not appear to be affected by the leverage effect. We analyze the determinants of the return distribution and find that it is driven...