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Girsanov, Numeraires, and All That

Published online by Cambridge University Press:  20 October 2022

Patrick S. Hagan
Affiliation:
Gorilla Science
Andrew Lesniewski
Affiliation:
Bernard M. Baruch College, City University of New York

Summary

In this Element the authors review the technique of the change of numeraire in the martingale approach to option pricing. Their intention is to present a reader friendly explanation of the technique itself, and illustrate how it is applied in various fields of quantitative finance as the basis for building option valuation models. They start with an informal review of Girsanov's theorem, followed by a brief summary of the basic concepts of the arbitrage free pricing, and the technique of change of numeraire. This is followed by a number of applications of the change of numeraire technique including interest rate models, FX quanto adjustments, credit risk modeling, mortgage backed securities, and CMS rates.
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Online ISBN: 9781009339278
Publisher: Cambridge University Press
Print publication: 17 November 2022

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References

Andersen, L. Lecture Notes on Credit Models, Courant Institute of Mathematical Sciences (2009).Google Scholar
Björk, T. Arbitrage Theory in Continuous Time, Oxford University Press (2009).Google Scholar
Blanchet-Scalliet, C., El Karoui, N., and Martellini, L. Dynamic asset pricing theory with uncertain time-horizon, J. Econ. Dyn. Contr., 29, 17371764 (2005).CrossRefGoogle Scholar
Brigo, D. Multi-currency credit default swaps, Int. J. Theor. Appl. Finance, 22, 1950018 (2019).Google Scholar
Brigo, D., and Mercurio, F. Interest Rate Models: Theory and Practice, Springer Verlag (2006).Google Scholar
Davidson, A., and Levin, A. Mortgage Valuation Models, Oxford University Press (2014).CrossRefGoogle Scholar
Dehm, J. C. Stochastic Mortality: Modeling and Optimal Investment, University of Ulm (2020).Google Scholar
Fabozzi, F. J. The Handbook of Mortgage-Backed Securities, Oxford University Press (2016).Google Scholar
Felpel, M., Kienitz, J., and McWalter, T. A. Effective stochastic volatility: applications to ZABR-type models, Quantitative Finance, 21, 837852 (2021).CrossRefGoogle Scholar
Geman, H., El Karoui, N., and Rochet, J.-C. Changes of numeraire, changes of probability measure and option pricing, J. Appl. Prob., 32, 443458 (1995).CrossRefGoogle Scholar
Hagan, P. CMS conundrums: pricing CMS swaps, caps, and floors, Wilmott Magazine, March, 3844 (2004).Google Scholar
Hagan, P. S., Kumar, D., Lesniewski, A., and Woodward, D. E. Managing smile risk, Wilmott Magazine, September, 84108 (2002).Google Scholar
Hagan, P. S., and Lesniewski, A. LIBOR market model with SABR style stochastic volatility, preprint (2008).Google Scholar
Hagan, P. S., Lesniewski, A., Skoufis, G. E., and Woodward, D. E. Convexity without replication, Wilmott Magazine, January, 5869 (2020).Google Scholar
Harrison, J. M., and Kreps, and D. M. Martingales and arbitrage in multiperiod securities markets, J. Econ. Theory, 20, 381408 (1979).Google Scholar
Harrison, J. M., and Pliska, S. R. Martingales and stochastic integrals in the theory of continuous trading, Stoch. Proc. Appl., 11, 215260 (1981).Google Scholar
Jamshidian, F. Valuation of credit default swaps and swaptions, Finance and Stochastics, 8, 343371 (2004).CrossRefGoogle Scholar
Lando, D., and Nielsen, A. B. Spreads, Quanto CDS. SSRN 3268890 (2018).Google Scholar
Lyashenko, A., and Mercurio, F. Looking forward to backward-looking rates: completing the generalized forward market model, preprint (2021).Google Scholar
Musiela, M., and Rutkowski, M. Martingale Methods in Financial Modeling, Springer Verlag (2004).Google Scholar
Oksendal, B. Stochastic Differential Equations: An Introduction with Applications, Springer Verlag (2013).Google Scholar
Pelsser, A. Mathematical foundation of convexity correction, Quant. Finance, 3, 5965 (2003).Google Scholar
Rebonato, R. Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond, Princeton University Press (2002).Google Scholar
Rebonato, R., McKay, K., and White, R. The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives, Wiley (2009).Google Scholar
Rudin, W. Real and Complex Analysis, McGraw Hill (1986).Google Scholar
Schönbucher, P. A note on survival measures and the pricing of options on credit default swaps, working paper (2003).Google Scholar
Schweizer, M. A guided tour through quadratic hedging approaches, SFB 373 discussion paper (1991).Google Scholar
Turfus, C. Analytic pricing of quanto CDS, researchgate.net (2019).Google Scholar

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