nach oben

Mathematics and Financial Economics

Erschienen in:

27.06.2017

A simple trinomial lattice approach for the skew-extended CIR models

verfasst von: Xiaoyang Zhuo, Guangli Xu, Haoyan Zhang

Erschienen in: Mathematics and Financial Economics | Ausgabe 4/2017

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper, we introduce a simple and efficient trinomial lattice tree approach for the skew Cox-Ingersoll-Ross (CIR) model and the doubly skewed CIR model. Suffering from the terms of local times and non-constant volatility, we apply two transforms to the skew-extended CIR processes. Then we construct a modified trinomial tree for the transformed processes which are piecewise tractable diffusions with constant volatility. As a result, the tree for the original skew-extended CIR processes can be easily obtained by using the inverse transform. Results of applications to zero-coupon bonds, European and American options demonstrate that our simple tree approach is efficient and satisfactory.

Vorheriger Artikel Long-term factorization of affine pricing kernels

Nächster Artikel An analytical study of norms and Banach spaces induced by the entropic value-at-risk

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

A filtration $\{\mathcal {F}_t\}_{t\ge 0}$ is said to satisfy the usual conditions if it is right-continuous and $\mathcal {F}_0$ contains all the P-negligible events in $\mathcal {F}$.

Moreover, if the Feller condition (see [14]) is satisfied, $2k\theta \ge \sigma ^2$, a zero bound cannot be reached. But we do not impose the Feller condition, i.e., zero is an attainable regular boundary in this paper.

The first transform $f(\cdot )$ can be extended to a general skew process $r_t$,

$$\begin{aligned} {\mathrm {d}}r_t=\mu (r_t){\mathrm {d}}t+\sigma (r_t){\mathrm {d}}W_t +(2p-1){\mathrm {d}}\hat{L}_t^r(a), \end{aligned}$$

where $\mu (r_t)$ and $\sigma (r_t)$ are functions of $r_t$ which can ensure the existence and uniqueness of skew process $r_t$ with some constraints. With the above $r_t$, the transform $f(\cdot )$ just need satisfy $f'(\cdot )=\frac{1}{\sigma (\cdot )}$. Then we have

$$\begin{aligned} X_t=f(r_0)+\int _0^t\left( \mu (r_s)f'(r_s)+\frac{1}{2}\sigma ^2(r_s)f''(r_s)\right) {\mathrm {d}}s+W_t+(2p-1)\hat{L}_t^X(f(a)). \end{aligned}$$

When applying to a general skew process $X_t$,

$$\begin{aligned} {\mathrm {d}}X_t=\mu (X_t){\mathrm {d}}t+\sigma (X_t){\mathrm {d}}W_t +(2p-1){\mathrm {d}}\hat{L}_t^X\left( f(a)\right) , \end{aligned}$$

the local time can always be removed if we modify the above transform $g(\cdot )$ as follows

$$\begin{aligned} g(x)=\left\{ \begin{array}{ll} p(x-f(a))+f(a) , &{}\quad 0<x<f(a), \\ (1-p)(x-f(a))+f(a) , &{}\quad x \ge f(a). \end{array} \right. \end{aligned}$$

See the Appendix for more details on the spectral expansion for the skew CIR process as the interest rate model.

As shown in the Appendix, the spectral expansions converge quickly and smoothly to a steady level with respect to the number of roots.

NONOSC includes regular, exist, entrance and NONOSC natural boundary.

Appuhamillage, T., Sheldon, D.: First passage time of skew Brownian motion. J. Appl. Probab. 49, 685–696 (2012)MathSciNetCrossRefMATH

Bass, R.F., Chen, Z.Q.: Brownian motion with singular drift. Ann. Probab. 31, 791–817 (2003)MathSciNetCrossRefMATH

Beliaeva, N., Nawalkha, S.: A simple approach to pricing American options under the Heston stochastic volatility model. J. Deriv. 17, 25–43 (2010)CrossRef

Beliaeva, N., Nawalkha, S.: Pricing American interest rate options under the jump-extended constant-elasticity-of-variance short rate models. J. Bank. Financ. 36, 151–163 (2012)CrossRef

Cantrell, R., Cosner, C.: Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design. Theor. Popul. Biol. 55, 189–207 (1999)CrossRefMATH

Corns, T.R.A., Satchell, S.E.: Skew Brownian motion and pricing European options. Eur. J. Financ. 13, 523–544 (2007)CrossRef

Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econom. J. Econom. Soc. 53, 385–407 (1985)MathSciNetMATH

Decamps, M., Schepper, A.D., Goovaerts, M.: Applications of $\delta $-perturbation to the pricing of derivative securities. Phy. A Stat. Mech. Appl. 342, 677–692 (2004)MathSciNetCrossRef

Decamps, M., Goovaerts, M., Schoutens, W.: Asymmetric skew Bessel processes and their applications to finance. J. Comput. Appl. Math. 186, 130–147 (2006a)MathSciNetCrossRefMATH

10.

Decamps, M., Goovaerts, M., Schoutens, W.: Self exciting threshold interest rates models. Int. J. Theor. Appl. Financ. 9, 1093–1122 (2006b)MathSciNetCrossRefMATH

11.

Decamps, M., Schepper, A.D., Goovaerts, M., Schoutens, W.: A note on some new perpetuities. Scand. Actuar. J. 2005, 261–270 (2005)MathSciNetCrossRefMATH

12.

Duffie, D., Singleton, K.: Credit Risk. Princeton University Press, Princeton (2003)

13.

Engelbert, H.J., Schmidt, W.: Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations (part III). Math. Nachr. 151, 149–197 (1991)MathSciNetCrossRefMATH

14.

Feller, W.: Two singular diffusion problems. Ann. Math. 54, 173–182 (1951)MathSciNetCrossRefMATH

15.

Gairat, A., Shcherbakov, V.: Density of skew Brownian motion and its functionals with application in finance. Math. Financ. pp. 1–20 (2016). doi:10.1111/mafi.12120

16.

Gorovoi, V., Linetsky, V.: Black’s model of interest rates as options, eigenfunction expansions and Japanese interest rates. Math. Financ. 14, 49–78 (2004)MathSciNetCrossRefMATH

17.

Harrison, J.M., Shepp, L.A.: On skew Brownian motion. Ann. Probab. 9, 309–313 (1981)MathSciNetCrossRefMATH

18.

Heston, S.L.: A closed form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993)CrossRef

19.

Itô, K., McKean, H.P.: Diffusion Processes and Their Sample Paths. Springer, Berlin (1965)MATH

20.

Karatzas, I.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991)MATH

21.

Kou, S.C., Kou, S.G.: A diffusion model for growth stocks. Math. Oper. Res. 29, 191–212 (2004)MathSciNetCrossRefMATH

22.

Lang, R.: Effective conductivity and skew Brownian motion. J. Stat. Phys. 80, 125–146 (1995)MathSciNetCrossRefMATH

23.

Le Gall, J.F.: One-dimensional stochastic differential equations involving the local times of the unknown process. In: Truman, A., Williams, D. (eds.) Stochastic Analysis and Applications. Lecture Notes in Mathematics, vol. 1095, Springer, Berlin (1984)

24.

Lejay, A.: Simulating a diffusion on a graph: application to reservoir engineering. Mt. Carlo Methods Appl. 9, 241–255 (2003)MathSciNetCrossRefMATH

25.

Lejay, A.: Monte Carlo methods for fissured porous media: a gridless approach. Mt. Carlo Methods Appl. 10, 385–392 (2004)MathSciNetMATH

26.

Li, A., Ritchken, P., Sankarasubramanian, L.: Lattice models for pricing American interest rate claims. J. Financ. 50, 719–737 (1995)CrossRef

27.

Linetsky, V.: The spectral decomposition of the option value. Int. J. Theor. Appl. Financ. 7, 337–384 (2004)MathSciNetCrossRefMATH

28.

Nawalkha, S.K., Beliaeva, N.A., Soto, G.M.: Dynamic Term Structure Modeling: The Fixed Income Valuation Course. Wiley, Hoboken (2007)

29.

Nelson, D.B., Ramaswamy, K.: Simple binomial processes as diffusion approximations in financial models. Rev. Financ. Stud. 3, 393–430 (1990)CrossRef

30.

Nkiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics: A Unified Introduction with Applications. Birkhäuser, Basel (1988)CrossRef

31.

Ouknine, Y., Rutkowski, M.: Local times of functions of continuous semimartingales. Stoch. Anal. Appl. 13, 211–231 (1995)MathSciNetCrossRefMATH

32.

Protter, P.E.: Stochastic Integration and Differential Equations. Springer, Berlin (1990)CrossRefMATH

33.

Rossello, D.: Arbitrage in skew Brownian motion models. Insur. Math. Econ. 50, 50–56 (2012)MathSciNetCrossRefMATH

34.

Song, S., Xu, G., Wang, Y.: On first hitting times for skew CIR processes. Methodol. Comput. Appl. Probab. 18, 169–180 (2016)MathSciNetCrossRefMATH

35.

Trutnau, G.: Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve. Stoch. Process. Appl. 120, 381–402 (2010)MathSciNetCrossRefMATH

36.

Trutnau, G.: Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve. Stoch. Process. Appl. 121, 1845–1863 (2011)MathSciNetCrossRefMATH

37.

Walsh, J.B.: A diffusion with a discontinuous local time. Astérisque 52, 37–45 (1978)

38.

Weinryb, S.: Homogeneisation pour des processus associes a des frontieres permeables. Ann. lIHP Probab. Stat. 20, 373–407 (1984)MathSciNetMATH

39.

Xu, G., Song, S., Wang, Y.: Some properties of doubly skewed CIR processes. J. Math. Anal. Appl. 434, 1194–1210 (2016)MathSciNetCrossRefMATH

40.

Xu, G., Song, S., Wang, Y.: The valuation of options on foreign exchange rate in a target zone. Int. J. Theor. Appl. Financ. 19, 1–19 (2016)MathSciNetCrossRefMATH

41.

Zhang, M.: Calculation of diffusive shock acceleration of charged particles by skew Brownian motion. Astrophys 541, 428–435 (2000)CrossRef

Titel: A simple trinomial lattice approach for the skew-extended CIR models
verfasst von: Xiaoyang Zhuo
Guangli Xu
Haoyan Zhang
Publikationsdatum: 27.06.2017
Verlag: Springer Berlin Heidelberg
Erschienen in: Mathematics and Financial Economics / Ausgabe 4/2017
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-017-0192-1

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Springer Professional "Wirtschaft+Technik"

Weitere Artikel der Ausgabe 4/2017

Optimal investment with transaction costs under cumulative prospect theory in discrete time

On the existence of competitive equilibrium in frictionless and incomplete stochastic asset markets

An analytical study of norms and Banach spaces induced by the entropic value-at-risk

Long-term factorization of affine pricing kernels

Optimal entry to an irreversible investment plan with non convex costs