Affine and Polynomial Processes

Ernst Eberlein; Jan Kallsen

doi:10.1007/978-3-030-26106-1_6

Mathematical Finance pp 337-372 | Cite as

Affine and Polynomial Processes

Authors
Authors and affiliations

Ernst Eberlein
Jan Kallsen

Chapter

First Online: 04 December 2019

1.2k Downloads

Part of the Springer Finance book series (FINANCE)

Abstract

Affine processes appear here for two reasons. They are treated in this part on stochastic calculus because they solve linear or more generally affine martingale problems.

This is a preview of subscription content, to check access.

Appendix 1: Problems

The exercises correspond to the section with the same number.

6.1

The Gamma-OU stochastic volatility model is based on the bivariate affine process X = (X₁, X₂) satisfying the SDE

$$\displaystyle \begin{aligned} dX_1(t)&=-\lambda X_1(t-)dt+dZ(t),\\ dX_2(t)&=(\mu+\delta X_1(t-))dt+\sqrt{X_1(t-)}dW(t)+\varrho Z(t), \end{aligned} $$

with parameters μ, δ, ϱ, λ, a Wiener process W, and a subordinator Z with characteristic exponent $\psi ^Z(u)={iu\beta \over \alpha -iu}$ for some α, β > 0. Verify that the functions Ψ₀, Ψ₁, Ψ₂ in the characteristic function (6.29) are of the form

$$\displaystyle \begin{aligned} \Psi_0(t,u)&=\int_0^t\psi^Z(-i\Psi_1(s,u)+\varrho u_2)ds+\mu tiu_2,\\ \Psi_1(t,u)&=e^{-\lambda t}iu_1+{1-e^{-\lambda t}\over\lambda}\psi^L(u_2),\\ \Psi_2(t,u)&=iu_2, \end{aligned} $$

where $\psi ^L(u_2):=\delta iu_2-{1\over 2}u_2^2$.

6.2

Compute the matrix B for n = 4 and the moments E(X(t)^k), k = 1, 2, 3, 4 for the extended square-root process X of Example 6.9.

6.3

The Cox–Ingersoll–Ross model in interest rate theory is based on the extended square-root process X of Example 6.9. Compute the joint characteristic function $E(\exp (iu_1X(t)+iu_2\int _0^tX(s)ds))$, $u_1,u_2\in \mathbb R$ of X(t) and $\int _0^tX(s)ds$.

6.4

The Heston model in Mathematical Finance is based on the bivariate affine process X = (X₁, X₂) solving the SDE

$$\displaystyle \begin{aligned} dX_1(t)&=(\mu+\delta X_2(t))dt+\sqrt{X_2(t)}dW(t), \end{aligned} $$

(6.54)

$$\displaystyle \begin{aligned} dX_2(t)&=(\kappa-\lambda X_2(t))dt+\sigma\sqrt{X_2(t)}dZ(t),{} \end{aligned} $$

(6.55)

where κ ≥ 0, μ, δ, λ, σ denote constants and W, Z Wiener processes with constant correlation ϱ. Determine all measure changes as in Proposition 6.22 such that X is affine and $e^{X_1}$ is a local martingale under the new probability measure. Is X again of the form (6.54, 6.55) up to different parameter values?

6.5

The Hull–White model in interest rate theory is based on the solution to the SDE dX(t) = (κ(t) − λX₂(t))dt + σdW(t) with parameters λ, σ, a continuous function $\kappa :\mathbb R _+\to \mathbb R$, and some Wiener process W. Compute the joint characteristic function $E(\exp (iu_1X(t)+iu_2\int _0^tX(s)ds))$, $u_1,u_2\in \mathbb R$ of X(t) and $\int _0^tX(s)ds$.

6.6 (Pearson Diffusion)

Show that continuous univariate polynomial processes are weak solutions to the SDE

$$\displaystyle \begin{aligned}dX(t)=\big(a_{1,0}+a_{1,1}X(t)\big)dt+\sqrt{a_{2,0}+a_{2,1}X(t)+a_{2,2}X(t)^2}dW(t)\end{aligned}$$

with some parameters a_1,0, a_1,1, a_2,0, a_2,1, a_2,2. Determine the matrices A, B of Proposition 6.27 in terms of these coefficients.

Appendix 2: Notes and Comments

Since textbook treatments of general affine processes are still missing, we refer to [80, 104] for results and references on affine processes and to [63, 106, 107] for the more general class of polynomial processes. Proposition 6.8 is stated in [169, Corollary 3.3], albeit with an incorrect expression for the joint characteristic function.

Theorems 6.13, 6.15, 6.26 are based on the observation of [63] that affine processes are polynomial and hence their moments can be obtained by solving linear ODEs. The first part of Proposition 6.19 is stated in [172, Lemma 2.7]. The models in Examples 6.4, 6.9 and in the problem section are revisited in Chaps. 8 and 14 in Part II of this book.

Concerning polynomial processes we follow here [107] in dispensing with the Markov property. Even if the latter holds in all relevant examples, the present approach allows us to apply the moment formula without verifying it in the first place. On the other hand, we stick to [63] by considering m-polynomial processes for finite m. This allows us to apply the moment formula even if not all moments exist.

As a final comment we remark that it does not seem easy to characterise all symbols or local characteristics that belong to polynomial processes. For results in this direction we refer to [64].

References

63.
C. Cuchiero, M. Keller-Ressel, J. Teichmann, Polynomial processes and their applications to mathematical finance. Finance Stochast. 16(4), 711–740 (2012)MathSciNetCrossRefGoogle Scholar
64.
C. Cuchiero, M. Larsson, S. Svaluto-Ferro, Polynomial jump-diffusions on the unit simplex. Ann. Appl. Probab. 28(4), 2451–2500 (2018)MathSciNetCrossRefGoogle Scholar
80.
D. Duffie, D. Filipović, W. Schachermayer, Affine processes and applications in finance. Ann. Appl. Probab. 13(3), 984–1053 (2003)MathSciNetCrossRefGoogle Scholar
100.
S. Ethier, T. Kurtz, Markov Processes: Characterization and Convergence (Wiley, New York, 1986)CrossRefGoogle Scholar
104.
D. Filipović, Time-inhomogeneous affine processes. Stoch. Process. Appl. 115(4), 639–659 (2005)MathSciNetCrossRefGoogle Scholar
106.
D. Filipović, M. Larsson, Polynomial diffusions and applications in finance. Finance Stochast. 20(4), 931–972 (2016)MathSciNetCrossRefGoogle Scholar
107.
D. Filipović, M. Larsson, Polynomial jump-diffusion models. arXiv preprint arXiv:1711.08043 (2017)Google Scholar
169.
J. Kallsen, A didactic note on affine stochastic volatility models. From Stochastic Calculus to Mathematical Finance (Springer, Berlin, 2006), pp. 343–368CrossRefGoogle Scholar
170.
J. Kallsen, P. Krühner, On uniqueness of solutions to martingale problems—counterexamples and sufficient criteria. arXiv preprint arXiv:1607.02998 (2016)Google Scholar
172.
J. Kallsen, J. Muhle-Karbe, Exponentially affine martingales, affine measure changes and exponential moments of affine processes. Stoch. Process. Appl. 120(2), 163–181 (2010)MathSciNetCrossRefGoogle Scholar
191.
D. Kendall, On the generalized “birth-and-death” process. Ann. Math. Stat. 19, 1–15 (1948)MathSciNetCrossRefGoogle Scholar
241.
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Ernst Eberlein
- 1
Jan Kallsen
- 2

1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
2.Department of MathematicsKiel UniversityKielGermany

Affine and Polynomial Processes

Abstract

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite chapter