First-order calculus and option pricing

Peter Carr

Journal of Financial Engineering (JFE), 2014, vol. 01, issue 01, 1-19

Abstract: The modern theory of option pricing rests on It? calculus, which is a second-order calculus based on the quadratic variation of a stochastic process. One can instead develop a first-order stochastic calculus, which is based on the running minimum of a stochastic process, rather than its quadratic variation. We focus here on the analog of geometric Brownian motion (GBM) in this alternative stochastic calculus. The resulting stochastic process is a positive continuous martingale whose laws are easy to calculate. We show that this analog behaves locally like a GBM whenever its running minimum decreases, but behaves locally like an arithmetic Brownian motion otherwise. We provide closed form valuation formulas for vanilla and barrier options written on this process. We also develop a reflection principle for the process and use it to show how a barrier option on this process can be hedged by a static postion in vanilla options.

Keywords: First-order stochastic calculus; option pricing theory; barrier options; static hedging (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (4)

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DOI: 10.1142/S2345768614500093

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