Bid-ask spread modelling: a perturbation approach

Title Bid-ask spread modelling: a perturbation approach
Author Vathana LY VATH, Simone SCOTTI
Years November 29, 2008

 

Theoretical Basis An important peculiarity of order books, and equally a drawback, is the lack of information. Indeed, order books are not completely public, e.g, in France, the market regulator restricts the publicly known part of a book to the five best prices. As such, it is impossible to define the shape of prices randomness with only ten data. This is another assumption related to error theory which

seems, however, to agree with empirical observations made by Biais et al. [5] and Potters et al. who show a decreasing shape of order books as price goes away from the mid-price. Yet, the maximum is reached near but not always precisely on the first best ask (bid) price. This shifting on the maximum is hard to justify in a gaussian framework, since the maximum for gaussian density is reached at the mean. Error theory provides us with perfect tools to deal with this problem as it foresees a bias with respect to the theoretical mean. This discrepancy can explain the shifting on the maximum.

Methods and Subjects  
Main Result Result 0.1 (Bid-Ask Spread) The uncertainty on Brownian motion is transmitted to the stochastic process Xt, that represents the asset price.

Result 0.2 Risk aversion theory permits to define a supportable risk probability _ < 0:5 such that an agent accepts to buy the stock at price.

Conclusion the two risk aversions become a function of the trading volume, asset prices and other market liquidity information. For instance, tts dependence on the trading volume allows the representative agent or market maker to adjust his bid and ask prices by increasing them if he runs short of stock, or cutting them if he starts accumulating excessive stocks.

 

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