The Mean-field Libor Market Model

Table of Links

1. Introduction
2. Mean-field Libor market model
3. Life insurance with profit participation
4. Numerical ALM modeling
5. Phenomenological assumptions and numerical evidence
6. Estimation of future discretionary benefits
7. Application to publicly available reporting data
8. Conclusions
9. Declarations
10. References

Mean-field Libor Market Model

To simulate market values of bonds, or other interest rate dependent assets, and discount factors, a stochastic interest rate model is needed. To this end, a popular choice in the insurance industry is the Libor market model (e.g., [5, 13]). A well-known downside of this model is, however, that it tends to suffer from blow-up ([16]).

This can have undesirable consequences in conjunction with a numerical ALM model (Section 3): when forward rates are too high over a prolonged period of time some quantities (e.g., the balance sheet item of declared bonuses, DB) may become so large that precision differences occur, and this can in turn trigger undesired and unrealistic management actions or even cause leakage: for example, when a rounding error between assets and liabilities is blown up such that the book value of assets does no longer suffice to cover the statutory value of liabilities management is required to inject additional capital; such a management action is then undesired because it causes leakage, and not realistic because it is triggered by a numerical artefact which does not have a counterpart in reality. With the aim of reducing the probability for exploding rates, while at the same time retaining the main features of the Libor market model, [7] introduced a mean-field Libor market model (MF-LMM). In this model the growth of the second moment is controlled via a mean-field interaction.

1.1. MF-LMM: model formulation. Let us fix a tenor structure 0 = t0 < t1 = δ < . . . < tN = Nδ where N ∈ N and δ > 0 are fixed. In the ALM model we consider yearly time steps such that δ = 1 and N corresponds to the projection horizon T of Section 2. For i = 0, . . . , N − 1 the i-th forward (Libor) rate valid on [t, ti+1] as seen at time t ≤ ti is F i t := 1 δ P(t, ti) − P(t, ti+1) P(t, ti+1) where P(t, ti) is the value at t of 1 currency unit paid at ti . In the following, we assume that each Brownian motion is defined on a complete filtered probability space which satisfies the usual assumptions (i.e. completeness of filtration and right-continuity).

The classical Libor market model (e.g., [5, 13]) is now specified as follows. For each index i = 0, . . . , N −1, there should be an R d -valued diffusion coefficient σi = σi(t), a forward measure Qi and a corresponding d-dimensional Brownian motion Wi (d ≥ 1), such that the dynamics of F i under Qi are given by dFi t = F i t σ ⊤ i dWi t (1.1) , t ≤ ti where the initial condition is given by the prevailing yield curve at t = 0. Let P(R k ) be the space of probability measures on R k and P2(R k ) = {µ ∈ P(R k ) : R Rk ⟨x, x⟩ µ(dx) < ∞} the subspace of probability measures with finite second moment. The idea of the mean-field extension is to allow σi to depend not only on time but also on the law of the process. Namely, we assume that σi : [0, ti ] × P2(R) → R d : σi = σi  t, µi t  = λi  t, µi t (h 1 i ), . . . , µi t (h l i )  (1.2) where: • µ i t = Law(F i t ) = (F i t )♯Qi = Qi ◦ (F i t ) −1 is the law of F i t under Qi , i.e., the push-forward of Qi with respect to F i t ; • λi : R+ × R l → R d is a vector valued function with components λ j i , j = 1, . . . , d; • hi : R → R l is a vector valued function with components h k i , k = 1, . . . , l; • µ i t (h k i ) = R R h k i (x) µ i t (dx) Conditions on λi and hi for the existence and uniqueness of solutions to the mean-field system (1.1)-(1.2) are given in Theorem 1.4 below.

The dimension l may be chosen arbitrarily following the modelling needs. We will be mostly concerned with the case l = 2. Indeed, a particularly interesting special case arises when h1(x) = x, h2(x) = x 2 , independently of i, and λi = λi(t, µi t (h1), µi t (h2)). Notice that µ i t (h2) − µ i t (h1) 2 = VarQi [F i t ] = Varµi t is the variance of F i t with respect to the forward measure Qi . By slight abuse of notation, we will write λi = λi  t, µi t (h2) − µ i t (h1) 2  in this case such that λi : R+ × R → R d . That is, we consider the model σi = σi  t, µi t  = λi  t, VarQi [F i t ]  (1.3) . The variance is a measure for the expected width of the scenario cone, therefore introducing such a dependency opens up the possibility to control the blow-up probability of the process without the necessity of including a state dependent mean reverting property.

1.2. Existence and uniqueness. The question of existence and uniqueness was answered only partially for one-dimensional stochastic drivers in [7]. Since we aim to simulate more general multi-dimensional processes in the MF-LMM framework in our numerical study, we derive below general conditions for existence and uniqueness. Mean-Field 24SDEs, also called McKean-Vlasov SDEs or nonlinear diffusions, were introduced and studied in the works of McKean [23] and Sznitman [27]. In the classical set-up it is assumed that the dependency on the measure variable is linear: in the context of equation (1.2), this would mean that σ should be of the form σ(t, µ) = R f(t, x) dµ(x) for a function f where we have suppressed the index. Since the variance is quadratic in the measure variable, the prescription (1.3) does not fit into this scheme. Conditions for existence and uniqueness for mean-field SDEs with nonlinear measure dependency were derived by [21], and the approach of this section is to use these results.

Thus we need to show that σ is Lipschitz in the measure variable. To do so, we use the L-derivative introduced by P.-L. Lions ([6]). Before turning to the question of existence and uniqueness, we introduce the notion of an interacting particle system (IPS) associated to a mean-field equation. The IPS associated to (1.1)-(1.2) is a system of SDEs that is defined as follows. Let n ∈ N and 1 ≤ p ≤ n. For a random variable X denote by δX the Dirac measure centered at X. The IPS associated to the mean-field Libor market model is dLi,p,n t = L i,p,n t σi(t, µi,n t ) ⊥ dWi,p (1.4) µ i,n t = 1 n Xn p=1 δL i,p,n

where Wi,p are mutually independent d-dimensional Brownian motions. Note that µ i,n t , which is called the empirical measure, is a stochastic measure. The initial conditions for L i,p,n t are the same as those for F i t , that is, given by the initial forward rate F i 0 . Firstly, the IPS (1.4) has structural relevance: Due to the empirical measure, two processes L i,p,n and L i,q,n will in general not be independent. However, in the limit as n goes to infinity one may hope that L i,p,n tends to a process L i,p which is a copy of the mean-field SDE (1.1)-(1.2). Thus the processes L i,p,n and L i,q,n would become independent in the limit n → ∞, and this leads to the notion of propagation of chaos (see [27] and [20] for a review).

Secondly, for our purposes, the approximating IPS is not only of structural importance but also crucial from a practical point of view since the numerical simulation of a mean-field SDE is realized as a Monte-Carlo simulation of the approximating IPS, and this is the basis for the numerical algorithm in Section 3. The corresponding result is provided in Theorem 1.4 below. Let P2(R) be equipped with the Wasserstein distance (1.6) W2(µ1, µ2) = infπ∈C(µ1,µ2)  Z R2 |x − y| 2 π(dx, dy) 1/2 where C(µ1, µ2) = {γ ∈ P(R × R) with marginals µ1 and µ2} denotes the set of all couplings. The space P2(R) is a Polish (i.e., separable and completely metrizable topological) space. Since P2(R) is a nonlinear space, the notion of differentiability requires attention.

Consider a continuous function f : P2(R) → R. Following [6, 3], the derivative of f at µ is defined by composing the function with a curve, µε, through µ and evaluating ∂f(µε)/∂ε|ε=0. Concretely, let id : x 7→ x be the identity map on R. Given µ ∈ P2(R) and ϕ ∈ L 2 (R, µ), we define a curve with values in P2(R) by ε 7→ µ ϕ ε := (id + εϕ)♯µ. Definition 1.1 ([3]). Let f : P2(R) → R be a continuous function. (1) f is called intrinsically differentiable at µ ∈ P2(R) with derivative DLf(µ) if L 2 (R, µ) → R, ϕ 7→ ∂ ∂ε    ε=0 f(µ ϕ ε ) = limε→0 f(µ ϕ ε ) − f(µ) ε =: DL ϕ f(µ) is a bounded linear functional. In this case, DLf(µ) is characterized via the natural L 2 (R, µ)-pairing through ⟨DLf(µ), ϕ⟩L2(R,µ)(R) = DL ϕ f(µ). (2) If additionally, lim ||ϕ||L2(R,µ)→0 |f((id + ϕ)♯µ) − f(µ) − DL ϕ f(µ)| ||ϕ||L2(R,µ) = 0 for all ϕ ∈ L 2 (R, µ), where ||ϕ||L2(R,µ) is the L 2 -norm of ϕ, then DLf(µ) is called the Lions or L-derivative of f at µ. Consider functions g and h such that • g : R → R is differentiable; • h : R → R is twice differentiable, and h ∈ L 1 (R, µ), h ′ ∈ L 2 (R, µ) and h ′′ ∈ L∞(R); Remark 1.2.

These assumptions ensure that the calculation below make sense. Incidentally, we are interested in the cases h(x) = x and h(x) = x 2 , cf. (1.3), whence these assumptions are not restrictive. On the other hand, it is relevant for our purposes that we do not need to assume more than the existence of the derivative of g. Indeed, this allows to consider regime switching functions depending on max(VarQi [F i t ] − v0, 0) where VarQi [F i t ] is the variance of the process and v0 is a pre-defined variance threshold. Such a dependency is used in the dampening construction in [7], and in the combined dampening-anti-correlation implementation in Section 3.2 Define f : P2(R) → R through (1.7) f(µ) = g(µ(h)) where µ(h) = Eµ[h] = R R h(x) µ(dx). In the following, the notation k(ε) = o(ε) means that an expression k(ε) satisfies limε→0 k(ε)/ε = 0. Observe that, for ϕ ∈ L 2 (R, µ) and ε > 0, Taylor’s expansion implies h(y + εϕ(y)) = h(y) + h ′ (y)εϕ(y) + h ′′(ξ) 2 ε 2ϕ(y) 2 = h(y) + h ′ (y)εϕ(y) + o(ε)ϕ(y) 2

for a ξ ∈ [y, y + εϕ(y)] and where o(ε) is independent of y since h ′′ ∈ L∞(R). Thus f  µ ϕ ε  = g  Z R h(x) ((id + εϕ)♯µ) (dx)  = g  Z R h(y + εϕ(y)) µ (dy)  = g  Z R  h(y) + εϕ(y)h ′ (y) + o(ε)ϕ(y) 2  µ (dy)  = g  µ(h) 

• g ′  µ(h) ε⟨ϕ, h′ ⟩L2(R,µ) + o(ε)||ϕ(y)||2 L2(R,µ) 

• o  ε⟨ϕ, h′ ⟩L2(R,µ) + o(ε)||ϕ(y)||2 L2(R,µ)  = f  µ 

• ε D g ′ (µ(h))h ′ , ϕE L2(R,µ)

• o  ε  Here we use that o(ε)g ′ (µ(h))||ϕ(y)||2 L2(R,µ) + o(ε⟨ϕ, h′ ⟩L2(R,µ) + o(ε)||ϕ(y)||2 L2(R,µ) ) = o(ε). Therefore, Definition 1.1 yields DL ϕ f(µ) = ⟨g ′ (µ(h))h ′ , ϕ⟩L2(R,µ) . The Cauchy-Schwarz inequality now implies that the intrinsic derivative is also a Lions derivative and is given by (1.8) DL f(µ)(x) = g ′ (µ(h))h ′ (x) for x ∈ R.

  Slightly more generally, we can consider vector valued functions f : P2(R) → R d of the form (1.9) f(µ) =  g j (µ(h1), . . . , µ(hl))d j=1 where g j , j = 1, . . . , d, and hk, k = 1, . . . , l, are functions satisfying the same conditions as g and h, respectively. Then we find that the Lions derivative DLf(µ) : R → R d at µ is given by (1.10) DL f(µ) = X l k=1 ∂kg j (µ(h1), . . . , µ(hl)) h ′ k d j=1 . The following lemma is provided for completeness’ sake. Lemma 1.3. Assume f : P2(R) → R d is continuous and that the Lions derivative DLf(µ) exists for all µ in P2(R). If ||DLf(µ)||L2(Rd,µ) ≤ K, for all µ in P2(R), then f is Lipschitz continuous with Lipschitz constant K. Proof. Let fe : L 2 = L 2 (Ω, F, P) → R d be an extension of f defined by fe(X) = f(Law(X)) where (Ω, F, P) is a probability space, see [6]. We use [6, Thm 6.2] which yields DLf(µ) = Dfe(X), where D is the Fr´echet derivative, and Dfe(X) does not depend on the extension but only on the law µ = Law(X).

  Here, the Fr´echet derivative is identified with the corresponding element in L 2 via the Hilbert space pairing EP [⟨X, Y ⟩] = ⟨X, Y ⟩L2 where the angle bracket without subscript denotes the Euclidean inner product. Consider elements µk = Law(Xk) with k = 0, 1 in P2(R). Using the Cauchy-Schwarz inequality   f(µ1) − f(µ0)    =    Z 1 0 ∂ ∂sfe  X0 + s(X1 − X0)  ds    =    Z 1 0 EP hDDfe  X0 + s(X1 − X0)  ,  X1 − X0 Ei ds    =    Z 1 0 D DL f  Law(X0 + s(X1 − X0)) ,  X1 − X0 E L2 ds    ≤ K       X1 − X0       L2 . Since this holds for all extensions and since the Wasserstein distance satisfies d2(µ0, µ1) ≤ ||X1 − X0||L2 when Law(X0) = µ0 and Law(X1) = µ1, the lemma follows.

  □ Theorem 1.4 (Existence, uniqueness and IPS approximation of MF-LMM). Consider the mean-field Libor market model be given by (1.1)-(1.2). Let λi = (λ j i ) d j=1 : [0, ti ] × R l → R d be Lipschitz continuous in t and L-differentiable in µ such that DLλi(t, µ) = X l k=1 ∂k+1 λ j i (t, µ(h1), . . . , µ(hl)) h ′ k d j=1 is bounded in L 2 (R d , µ), uniformly for all t ∈ R+ and all µ ∈ P2(R). If (1.2) is of the form (1.3), then this is the case if λi : [0, ti ] × R+ → R d , (t, v) 7→ λi(t, v) is Lipschitz continuous in t, differentiable in v, and satisfies the pointwise bound (1.11) v X d j=1 |∂2 λ j i (t, v)| 2 ≤ K with respect to a constant K < ∞ which is independent of (t, v) ∈ [0, ti ] × R+. (1) Then the system has a unique solution F i t such that E[sup0≤t≤ti |F i t | 2 ] < ∞. (2) Moreover, pathwise propagation of chaos holds: limn→∞ sup p≤n E h sup t≤ti    L i,p,n t − L i,p t    2i = 0 where L i,p,n is the approximating IPS (1.4) and L i,p is a version of (1.1)-(1.2) defined with respect to Wi,p .

  Proof. The result follows directly from [21, Prop. 2 and Thm. 3] if it can be shown that σi is Lipschitz continuous also in the measure variable. In view of Lemma 1.3, this, in turn, holds if the Lions derivative DLσi(t, µ) exists and is bounded in L 2 (R d , µ) with respect to some constant independent of t and µ. Given the system (1.1)-(1.2), the Lions derivative is of the form (1.10). Hence the result follows. In the special case where the system is given by (1.1)-(1.3) we have h1(x) = x and h2(x) = x 2 . For better readability we omit the index i and the time dependency from now on. Owing again to (1.10), σ(µ) = λ(µ(h2) − µ(h1) 2 ) has an L-derivative DLσ whose L 2 (R d , µ)-norm is       DLσ(µ)       2 L2(Rd,µ)

  X d j=1       − 2µ(h1)(λ j ) ′  µ(h2) − µ(h1) 2  h ′ 1 + (λ j ) ′  µ(h2) − µ(h1) 2  h ′ 2       2 L2(Rd,µ) = 4X d j=1   (λ j ) ′ (v)    2       h1 − µ(h1)       2 L2(Rd,µ) = 4v X d j=1   (λ j ) ′ (v)    2 where v = µ(h2) − µ(h1) 2 is the variance of µ. Hence boundedness of ||DLσ(µ)||2 L2(Rd,µ) is ensured by the pointwise assumption. □ Remark 1.5. The MF-LMM was introduced in [7] where it was shown in a numerical study that the mean-field interaction can lead to a reduction of blow-up probability, i.e. a reduction in the number of scenarios where the forwards exceed a pre-defined threshold.

  Such a reduction of blow-up probability is desirable for numerical reasons as mentioned in the introduction to this section. However, it is also a feature that is in agreement with the postulate to model a realistic forward dynamic: exploding interest rates are not observed in reality, and central banks act to stabilize rates whence such an explosion is not only not observed but also very implausible. It should be stressed that the mean-field interaction encoded in (1.3) is not interpreted as an interaction between different forward rate scenarios. Such an interpretation would not make sense since different unfoldings of reality cannot communicate with each other. Rather, the underlying idea is that of an ulterior agency (i.e., central bank policy) which controls yield curves to the effect that the variance of possible yield movements is restricted. It is this effect that is modelled by the mean-field dependency.

  1.3. Projection along tenor dates. Having shown existence and uniqueness, the mean-field Libor market model may be regarded as a generic Libor market model with time-dependent volatility structure. In particular, the transformation to the spot measure may be carried out. However, as shown in [7] the transformed system is in general not a mean-field system, unless one introduces an auxiliary process. For our envisaged purposes of performing long term simulations it is standard to consider yearly time steps along tenor dates. Along these dates the picture simplifies and the transformed system is, in fact, a mean-field SDE with respect to the spot measure.

  This is relevant since it implies that the approximating IPS can be used to define a Monte-Carlo routine based on the associated Euler-Maruyama scheme. Let (1.12) B(tj ) = (1 + δFj−1 tj−1 )B(tj−1), B(t0) = 1 , denote the implied money market account (i.e., the numeraire) and Q∗ the associated spot measure. If t = tj is a tenor date, then the auxiliary process, alluded to above, can be expressed as (1.13) Y m tj = B(tj ) −1 P(j, m) P(0, m) , where (1.14) P(j, m) = Πm−1 l=j (1 + δlF l tj ) −1 , is the time tj -value of one unit of currency paid at tm. With (1.15) Ψm j := EQ∗ " F m tj − EQ∗  F m tj B(tj ) −1 P(j, m) P(0, m) 2 B(tj ) −1 P(j, m) P(0, m)

  it follows that the evolution along the tenor dates of the mean-field Libor market model with respect to the spot measure is given by (1.16) dF m tj = F m tj   Xm k=j+1 δFk tj δFk tj + 1 λ k (tj , Ψ k j ) ⊤λ m(tj , Ψ m j ) dt + λ m(tj , Ψ m j ) ⊤dW∗ t   , since η(tj ) = j + 1. See [7]. Thus it remains to specify the functional form of λ i : [0, ti ] × R+ → R d which has to be Lipschitz continuous in t and should satisfy the pointwise condition (1.11). A numerical study concerning various such choices is provided in [7], where it is also shown numerically that, in the long run, the classical Libor market model can lead to exploding rates while the mean-field controlled models all significantly reduce blow-up probability.