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Some Classes of Discrete-Time Stochastic Processes Read chapter Read first chapter

2013 | OriginalPaper | Chapter

14. Extensions

Author : Prof. Stéphane Crépey

Published in: Financial Modeling

Publisher: Springer Berlin Heidelberg

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Abstract

Have you ever seen a financial contract which only pays one single cash-flow at a terminal maturity T? There are some (I heard “vanillas”!), but not so many. Ever seen a continuous-time coupon stream? I’d be surprised. More broadly, to account for many real-life practical features such as discrete dividends or discrete path-dependence, we need to extend the theory a little bit. With this motivation, in this chapter, we provide various extensions to the BSDE and PDE results of Chaps. 12 and 13. First we deal with discrete dividends on a financial derivative or an underlying asset. Then we extend the results of the previous chapters to more general reflected BSDEs that appear in the case of intermittent call protection in Chap. 10.

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previous chapter Analytic Approach
next chapter Technical Proofs (∗∗)
Footnotes
1
Including here the jumps defined by θ in X.
 
2
A superscript t refers, as usual, to a constant initial condition (t,x,i) of \(\mathcal{X}\).
 
3
A variant of that of Definition 14.1.3.
 
Literature
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38.
Bielecki, T. R., Crépey, S., Jeanblanc, M., & Rutkowski, M. (2009). Valuation and hedging of defaultable game options in a hazard process model. Journal of Applied Mathematics and Stochastic Analysis. Article ID 695798 (long preprint version on www.​defaultrisk.​com).
39.
Bielecki, T. R., Crépey, S., Jeanblanc, M., & Rutkowski, M. (2010). Arbitrage pricing of defaultable game options with applications to convertible bonds. Quantitative Finance, 8(8), 795–810. CrossRef
40.
Bielecki, T. R., Crépey, S., Jeanblanc, M., & Rutkowski, M. (2011). Convertible bonds in a defaultable diffusion model. In A. Kohatsu-Higa, N. Privault, & S. Sheu (Eds.), Stochastic analysis with financial applications (pp. 255–298). Basel: Birkhäuser. CrossRef
65.
Chassagneux, J.-F., & Crépey, S. (2012). Doubly reflected BSDEs with call protection and their approximation. ESAIM: Probability and Statistics (in revision).
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Cont, R., & Fournié, D. (2012). Functional Itô calculus and stochastic integral representation of martingales. Annals of Probability, 41(1), 109–133. CrossRef
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Crépey, S., & Matoussi, A. (2008). Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison principle. The Annals of Applied Probability, 18(5), 2041–2069. MathSciNetMATHCrossRef
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Guyon, J., & Henry-Labordère, P. (2012). Nonlinear pricing methods in quantitative finance. London: Chapman & Hall.
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Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probability Theory and Related Fields, 113(4), 473–499. MathSciNetMATHCrossRef
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Peng, S., & Xu, M. (2005). The smallest g-supermartingale and reflected BSDE with single and double L 2 obstacles. Annales de L’I.H.P. Probabilités Et Statistiques, 41(3), 605–630. MathSciNetMATH
Metadata
Title
Extensions
Author
Prof. Stéphane Crépey
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
Financial Modeling

Print ISBN: 978-3-642-37112-7

Electronic ISBN: 978-3-642-37113-4

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-37113-4_14

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