A No-arbitrage Model of Liquidity in Financial Markets involving Brownian Sheets by Dr. Henry Schellhorn

Event Date: 

Monday, February 13, 2012 -
3:30pm to 5:00pm

Event Date Details: 

Refreshments served at 3:15 PM

Event Location: 

  • South Hall 5607F

Dr. Henry Schellhorn (Claremont Graduate University)

Title: A No-arbitrage Model of Liquidity in Financial Markets involving Brownian Sheets

Abstract: We consider a dynamic market model where buyers and sellers submit limit orders. If at a given moment in time, the buyer is unable to complete his entire order due to the shortage of sell orders at the required limit price, the unmatched part of the order is recorded in the order book. Subsequently these buy unmatched orders may be matched with new incoming sell orders. The total number of buy orders D(p,t) entered in the system at time t is a collection of stochastic processes indexed by a continuous parameter p, the limit price, and its dynamics are modelled by a stochastic partial differential equation (SPDE) driven by a Brownian sheet. The same type of dynamics hold for the total number of sell orders S(p,t).

The curves D and S are related to but not identical to the demand and supply curves, and thus the exogenous data is not identical to the exogenous data specified in other models (such as Cetin, Jarrow, and Protter 2004, or CJP). The equilibrium price process is the (limit) price at which these two curves are equal at all times. We derive the SPDE of the equilibrium process and then provide necessary and sufficient conditions for the existence of a risk-neutral measure where this process is a local martingale. This result is important in finance because the first fundamental theorem of asset pricing in the CJP model holds if and only if there is a measure where the equilibrium price process is a local martingale. Interestingly, the risk-neutral measure is not unique in our model, which shows that more economic conditions are needed to close it.

The no-arbitrage conditions we obtain are applicable to a wide class of models, in the same way that the Heath-Jarrow-Morton conditions apply to a wide class of interest rate models. We implement and calibrate several parametric models and analyze empirically when arbitrage occurred using real order book data. We also investigate the effectiveness of some arbitrage strategies based on these parametric models.