Interest Rate Models

Ernst Eberlein; Jan Kallsen

doi:10.1007/978-3-030-26106-1_14

Mathematical Finance pp 663-731 | Cite as

Interest Rate Models

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Ernst Eberlein
Jan Kallsen

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First Online: 04 December 2019

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Abstract

From an abstract point of view interest rate markets fall into the reach of the general theory which is discussed in Chaps. 9– 13. They differ from equity markets just in the kind of assets, namely bonds, swaps, caps and other interest rate products. On closer observation, however, one faces new phenomena. For example, the natural choice of a numeraire is not obvious in markets where interest rates change randomly over time.

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Appendix 1: Problems

The exercises correspond to the section with the same number.

14.1

Consider a consol bond, i.e. an asset paying 1€ at times T = 1, 2, 3, … Express its fair price π(t) at time t in terms of zero coupon bonds B(t, T), T = 1, 2, 3, … with T ≥ t. Moreover, compute π(0) in terms of the simple rate ℓ := L(0, T − 1, T) provided that the latter is the same for T = 1, 2, 3, …

14.2

Derive the forward rate dynamics (14.17) for the short-rate resp. factor models in Examples 14.5–14.10.

14.3

Reference [131] consider a HJM model of the form

$$\displaystyle \begin{aligned} df(t,T)&=-Y(t)e^{-\lambda(T-t)}{1-e^{-\lambda(T-t)}\over\lambda}dt+\sqrt{Y(t)}e^{-\lambda(T-t)}dW(t),{} \end{aligned} $$

(14.103)

$$\displaystyle \begin{aligned} dY(t)&=(m-\mu Y(t))dt+\beta\sqrt{Y(t)}d\widetilde W(t){} \end{aligned} $$

(14.104)

with parameters λ, m, μ, β > 0 and Q₀-Wiener processes $W,\widetilde W$ with correlation ϱ. This is inspired by the Hull–White model of Examples 14.7 and 14.19 where Y (t) is constant, cf. Problem 14.2. Their motivation is to come up with a setup such that the set of conceivable forward rate curves f(t, ⋅) for t > 0 is not rigidly linked to the initial curve f(0, ⋅).

Our goal is to derive semiexplicit formulas for call and put options in this model. To this end, verify that (14.103) is of the form (14.47) with deterministic σ(t, T) relative to the three-dimensional time-inhomogeneous affine process X = (r, ξ, Y ) given by

$$\displaystyle \begin{aligned}\xi(t):=\int_0^tY(s)e^{-2\lambda(t-s)}ds.\end{aligned}$$

Use this fact to express the function π in (14.69) in terms of the solution to some system of ODEs. As a side remark, note that (14.103, 14.104) correspond to an affine factor model in the sense of Sect. 14.2.1.

Hint: Use (14.47) in order to obtain

$$\displaystyle \begin{aligned}f(t,T)=f(0,T)+e^{-\lambda(T-t)}\big(r(t)-f(0,t)\big)+\xi(t){e^{-\lambda(T-t)}-e^{-2\lambda(T-t)}\over\lambda}\end{aligned}$$

and hence $\dot f(t,t)$ for (14.52).

14.4

Suppose that X is a univariate Wiener process and σ₁, …, σ_N are constant in the forward process dynamics (14.70). Determine call and put option prices as in Example 14.28 explicitly as well as Jamshidian’s representation of the swaption as a portfolio of bonds.

14.5

Determine the Itō process representation of the short rate in the lognormal rational model of Example 14.40. Does it correspond to an affine short-rate model in the sense of Sect. 14.2.1?

14.6

Determine π(t, z) in (14.101) in closed form if X in Example 14.44 is a Gamma-OU or an inverse-Gaussian OU process. Does this setup correspond to an affine short-rate model in the sense of Sect. 14.2.1?

Appendix 2: Notes and Comments

Titles devoted to or covering interest rate theory include [29, 42, 46, 105, 129, 223, 296]. For credit risk modelling, which is not treated in our introductory text, we refer also to [26, 62, 262].

Sections 14.1 and 14.2 owe much to the classical expositions of [29, 30, 223, 279]. The main difference is that we allow for general semimartingales and affine processes instead of Itō processes resp. continuous affine processes in the standard textbook literature. Jamshidian’s representation goes back to [156].

The Heath–Jarrow–Morton approach is due to [140]. It is extended to Lévy and other driving processes in [33, 34, 83, 84, 86, 89, 159, 274] and elsewhere. The completeness result in Sect. 14.3.3 is taken from [94]. For questions of existence and uniqueness of solutions to the HJM term structure equations we refer to [108] and the literature therein.

Section 14.4 is motivated by [85, 87, 193], where the driving semimartingale X is a time-inhomogeneous Lévy process. As is pointed out in Example 14.30, it differs in spirit from the LIBOR market model approach of [39, 157, 158, 221]. The affine LIBOR model of [190] is an instance of the affine forward process model in Sect. 14.4.4, where the integrands σ(⋅, n) are chosen in a specific way to ensure positive interest rates. For an overview of different approaches to LIBOR modelling we refer to [124, 229].

Section 14.5 and in particular Sect. 14.5.2 is an affine extension of the rational lognormal model in [223, Section 16.5], cf. also [113, 247, 255] in this context. Black–Scholes type formulae for the Gaussian case as in Example 14.38 can be found in [223, Propositions 16.5.2 and 16.5.3]. Note that the Lévy extension of the Flesaker–Hughston approach in Example 14.39 differs from the one in [44]. Linear-rational models are introduced in [110], cf. also [111, 224] for related approaches.

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Authors and Affiliations

Ernst Eberlein
- 1
Jan Kallsen
- 2

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

Interest Rate Models

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