Fundamentals: Stochastic Analysis and Applications, Interest Rate Dynamics, and Basic Principles of Pricing Interest Rate Derivatives

Gerhard Larcher
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Gerhard Larcher: Johannes Kepler University of Linz

Chapter 3 in The Art of Quantitative Finance Vol.2, 2023, pp 251-353 from Springer

Abstract: Abstract We start this chapter with a crash course in stochastic analysis. We give a heuristic introduction into the basic principles of these powerful techniques, i.e. we explain in an intuitive way the concepts of stochastic processes, stochastic integration, Ito formula, and stochastic differential equations. We then apply these tools for modelling interest rates and for pricing interest rate derivatives like caps, floors, or interest rate swaps. Additionally, we give alternative proofs of the Black-Scholes formula with the help of stochastic analysis, and thereby we gain essential new insights into the dynamics of financial markets. We define complete markets, and we show that the (multidimensional) Black-Scholes market (under certain conditions) is complete. Finally, we consider some examples of incomplete markets, and we analyse possible approaches to the valuation of derivatives in incomplete markets.

Keywords: Heuristic methods from stochastic analysis; Stochastic processes; Interest rate models; Ornstein-Uhlenbeck model; Zero-coupon bonds and interest rates; Interest rate swaps; Valuation of interest rate derivatives; Floors; Caps; Vasicek model; Hull-White model; Complete markets; The Black-Scholes market is complete; Valuation in incomplete markets (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sptchp:978-3-031-23870-3_3

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DOI: 10.1007/978-3-031-23870-3_3

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