Trading Rules Over Fundamentals: A Stock Price Formula for High Frequency Trading, Bubbles and Crashes

Abstract

In this paper we present a simple closed form stock price formula, which captures empirical regularities of high frequency trading (HFT), based on two factors: (1) exposure to hedge factor; and (2) hedge factor volatility. Thus, the parsimonious formula is not based on fundamental valuation. For applications, we first show that in tandem with a cost of carry model, it allows us to use exposure to and volatility of E-mini contracts to estimate dynamic hedge ratios, and mark-to-market capital gains on contracts. Second, we show that for given exposure to hedge factor, and suitable specification of hedge factor volatility, HFT stock price has a closed form double exponential representation. There, in periods of uncertainty, if volatility is above historic average, a relatively small short selling trade strategy is magnified exponentially, and the stock price plummets when such strategies are phased locked for a sufficient large number of traders. Third, we demonstrate how asymmetric response to news is incorporated in the stock price by and through an endogenous EGARCH type volatility process for past returns; and find that intraday returns have a U-shaped pattern inherited from HFT strategies. Fourth, we show that for any given sub-period, capital gains from trading is bounded from below (crash), i.e. flight to quality, but not from above (bubble), i.e. confidence, when phased locked trade strategies violate prerequisites of van der Corput's Lemma for oscillatory integrals. Fifth, we provide a taxonomy of trading strategies which reveal that high HFT Sharpe ratios, and profitability, rests on exposure to hedge factor, trading costs, volatility thresholds, and algorithm ability to predict volatility induced by bid-ask bounce or otherwise. Thus, extant regulatory proposals to control price dynamics of select stocks, i.e., pause rules such as ''limit up/limit down bands over 5-minute rolling windows, may mitigate but not stop future market crashes or price bubbles from manifesting in underlying indexes that exhibit HFT stock price dynamics.