Research ArticleOpen AccessGoogle Scholar indexed
Pricing and Hedging Options Conditional on Market Activity
Department of Mathematics, Florida State University, Tallahassee, FL, USA
Department of Mathematics, University of New Orleans, LA, USA
Department of Mathematics and Statistics, Louisiana Tech, Ruston, LA, USA
- 1 Department of Mathematics, Florida State University, Tallahassee, FL, USA
- 2 Department of Mathematics, University of New Orleans, LA, USA
- 3 Department of Mathematics and Statistics, Louisiana Tech, Ruston, LA, USA
Journal of Mathematical Finance·Volume 12 (2021)·Pages 1–19·Published 28 December 2021·DOI10.4236/jmf.2022.121001
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Abstract
We replicate and price European options on stocks modeled by time-changed geometric Brownian motion. The time change is obtained as the integrated intensity of random arrival times of price changes of the underlier over the life of the option. For European call options we obtain explicit hedging and pricing formulas. This approach is motivated by the need to connect option prices directly to the microstructure properties of the limit order book that determines tick-by-tick stock price changes. The continuous time model is obtained as an appropriate limit of discrete time random walks with random jump times, in the limit of infinitely many independent representative agents.
KeywordsRandom WalksTime-Changed Brownian MotionOption PricingLimit Order Book
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