A STOCHASTIC CONTROL APPROACH TO BID-ASK PRICE MODELLING

Engel John C. Dela Vega and Robert J. Elliott
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Engel John C. Dela Vega: UniSA Business, University of South Australia, Adelaide, SA 5000, Australia
Robert J. Elliott: UniSA Business, University of South Australia, Adelaide, SA 5000, Australia2Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada T2N 1N4, Canada

International Journal of Theoretical and Applied Finance (IJTAF), 2022, vol. 25, issue 04n05, 1-30

Abstract: This paper develops a model for the bid and ask prices of a European-type asset by formulating a stochastic control problem. The state process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. A Girsanov theorem for Markov chains is implemented for the change of coefficients, including the diffusion coefficient which cannot be changed by the usual Girsanov theorem for Brownian motion. The price of a European-type asset is then determined using an Esscher transform and a system of partial differential equations. A dynamic programming principle and a maximum/minimum principle associated with the stochastic control problem are then derived to model bid and ask prices. These prices are not quotes of traders or market makers but represent estimates in our model on which reasonable quantities could be traded.

Keywords: Markov chains; regime-switching; two price finance; European options; dynamic programming principle; maximum principle (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0219024922500212

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