Emergence of long memory in stock volatility from a modified Mike-Farmer model

Abstract

Abstract. -The Mike-Farmer (MF) model was constructed empirically based on the continuous double auction mechanism in an order-driven market, which can successfully reproduce the cubic law of returns and the diffusive behavior of stock prices at the transaction level. However, the volatility (defined by absolute return) in the MF model does not show sound long memory. We propose a modified version of the MF model by including a new ingredient, that is, long memory in the aggressiveness (quantified by the relative prices) of incoming orders, which is an important stylized fact identified by analyzing the order flows of 23 liquid Chinese stocks. Long memory emerges in the volatility synthesized from the modified MF model with the DFA scaling exponent close to 0.76, and the cubic law of returns and the diffusive behavior of prices are also produced at the same time. We also find that the long memory of order signs has no impact on the long memory property of volatility, and the memory effect of order aggressiveness has little impact on the diffusiveness of stock prices.Introduction. -The continuous double auction mechanism is adopted in the electronic trading systems in many stock markets worldwide. In particular, most emerging stock markets are order-driven markets. In a pure order-driven market, there are no market makers or specialists, and market participants submit and cancel orders, which may result in transactions based on price-time priority. Different from quote-driven markets where market makers are liquidity providers, the same trader in an order-driven market can act as either a liquidity taker or a liquidity provider depending on the aggressiveness of her submitted orders. The behaviors of market makers are very complicated, since they have the obligation to maintain the liquidity of stocks and in the meanwhile want to maximize their profits. It is thus natural to argue that it is easier to construct microscopic models for order-driven markets than for quotedriven markets in order to understand the macroscopic regularities of stock markets from a microscopic angle of view.Indeed, a lot of efforts have been made to construct orderdriven models [1], which can be dated back to the 1960's [2].

Econophysics of Agent-Based Models

“…Also, the original model does not focus on volatility clustering. Gu and Zhou (2009) propose a variant that tackles this feature. Another important drawback of the model is the way order signs are simulated.…”

New approaches in agent-based modeling of complex financial systems

“…More recently, the concept for determining the key parameters of the agent-based models from empirical data instead of setting them artificially was suggested [20]. Similar concept has also been applied to the order-driven models, which were first proposed by Mike and Farmer [46] and improved by Gu and Zhou [47,48,49,50]. In this family of order-driven models, the parameters of order submissions and order cancellations are determined using real order book data.…”

“…In this family of order-driven models, the parameters of order submissions and order cancellations are determined using real order book data. For comparison, the agent-based models focus more on the behaviors of agents [40,41,42,43,44,45], while the order-driven models are mainly intended to explore the dynamics of the order flows [46,47,48,49,50]. In section 2, we review several agent-based models that are based on the agents' behaviors with heterogenous personal preferences and interactions.…”

Ising model of financial markets with many assets

“…Another promising approach is followed by the young family of empirical order-driven models. These models use empirical order-flow data to generate transactions via the continuous double auction mechanism and are able to reproduce statistical price features without freely adjustable parameters in a quantitative way [6,7,8,9,10].…”