Continuous-time limits of multi-period cost-of-capital margins

Definition 1.

For Y ∈ L 1 ⁢ ( ℱ τ k + δ k ) , α δ k ∈ ( 0 , 1 ) and a nonnegative η τ k ∈ L 0 ⁢ ( ℱ τ k ) , the one-step valuation mapping is defined as

Remark 1.

Notice that the backward recursion for the discrete-time cost-of-capital-margin process { V t } t = 0 T in Section 1.1 may be expressed as V t = φ t 1 ⁢ ( L t + 1 - L t + V t + 1 ) , V T = 0 , and corresponds to partitioning [ 0 , T ] into subintervals of lengths one.

Theorem 1.

Consider a partition τ of [ 0 , T ] , 0 = τ 0 < … < τ m = T . For each k, φ τ k δ k is a mapping from L 1 ⁢ ( F τ k + δ k ) to L 1 ⁢ ( F τ k ) satisfying

1. if λ ∈ L 1 ⁢ ( ℱ τ k ) and Y ∈ L 1 ⁢ ( ℱ τ k + 1 ) , then φ τ k δ k ⁢ ( Y + λ ) = φ τ k δ k ⁢ ( Y ) + λ ,

2. if Y , Y ~ ∈ L 1 ⁢ ( ℱ τ k + 1 ) and Y ≤ Y ~ , then φ τ k δ k ⁢ ( Y ) ≤ φ τ k δ k ⁢ ( Y ~ ) ,

3. φ τ k δ k ⁢ ( 0 ) = 0 .

Definition 2.

Given L ∈ L 1 ⁢ ( 𝔽 ) and a partition τ of [ 0 , T ] , 0 = τ 0 < … < τ m = T , the cost-of-capital-margin (CoCM) process { V t τ ⁢ ( L ) } t ∈ τ of L with respect to the partition τ and filtration 𝔽 is defined in terms of a sequence of one-step valuation mappings defined in Definition 1 as follows:

where Δ ⁢ L τ k + 1 :- L τ k + 1 - L τ k .

Remark 2.

From Theorem 1, it follows that

where ∘ denotes composition of mappings. In particular, if L , L ~ ∈ L 1 ⁢ ( 𝔽 ) satisfy L T ≤ L ~ T , then V 0 τ ⁢ ( L ) ≤ V 0 τ ⁢ ( L ~ ) . Moreover, since each φ τ k δ k inherits positive homogeneity, i.e. φ τ k δ k ⁢ ( c ⁢ Y ) = c ⁢ φ τ k δ k ⁢ ( Y ) for c ≥ 0 , from VaR τ k , 1 - α δ k , V 0 τ ⁢ ( c ⁢ L ) = c ⁢ V 0 τ ⁢ ( L ) for c ≥ 0 .

Definition 3.

Given L ∈ L 1 ⁢ ( 𝔽 ) and a partition τ of [ 0 , T ] , 0 = τ 0 < … < τ m = T , the lower and upper CoCM processes { V ˇ t τ , β ⁢ ( L ) } t ∈ τ and { V ^ t τ , β ⁢ ( L ) } t ∈ τ of L with respect to the partition τ and filtration 𝔽 are given by

Assumption 1.

η τ k ≡ η δ k is nonrandom and depends only on the length δ k of the subinterval [ τ k , τ k + 1 ) .

Remark 3.

In our setup, as well as in [16, 22], discrete-time multi-period risk-adjusted values, given a partition τ = { τ k } k = 0 m of [ 0 , T ] , may be expressed as

where the mapping φ τ k , τ k + 1 is a mapping from a subspace of L 0 ⁢ ( ℱ τ k + 1 ) to a subspace of L 0 ⁢ ( ℱ τ k ) . In our setting,

where δ k = τ k + 1 - τ k and

with 1 - α δ ∼ - δ ⁢ log ⁡ α and η δ ∼ δ ⁢ log ⁡ ( 1 + η ) as δ → 0 . Notice from the last expression above for φ τ k δ k ⁢ ( Y ) that, for very small values η δ k , φ τ k δ k ⁢ ( Y ) < 𝔼 τ k ⁡ [ Y ] . This inequality is a consequence of the limited liability for the capital provider. Notice also that φ τ k δ k ⁢ ( Y ) > 𝔼 τ k ⁡ [ Y ] is equivalent to

which, for 1 - α δ k ∈ ( 0 , 1 2 ) small and reasonable models for Y, holds for moderately large values η δ k .

In [16], actuarial valuation rules are extended from discrete to continuous time. Modified to our setting where financial values are expressed in the numéraire, the valuation rule most similar to the one considered here is

where η , α ∈ ( 0 , 1 ) are fixed constants. Notice that φ τ k , τ k + 1 ⁢ ( Y ) > 𝔼 τ k ⁡ [ Y ] if VaR τ k , 1 - α ⁡ ( - Y ) > 𝔼 τ k ⁡ [ Y ] . Although the expressions for φ τ k , τ k + 1 may appear similar, they are fundamentally different. The mapping φ τ k , τ k + 1 in [16] is a priori given by an actuarial valuation rule whereas, in our case, it is the result of the capital providers’ acceptability condition for financing the repeated capital requirements, taking the capital providers’ limited liability into account.

In [22], a negative liability value corresponds to a positive value in our setting, and vice versa. The mappings φ τ k , τ k + 1 in [22], modified to our sign convention, are of the form

and F τ k may be chosen as

which gives

which coincides with (2.3).

Notice that, in [16, 22], the quantities 𝔼 τ k ⁡ [ Y ] and VaR τ k , 1 - α ⁡ ( - Y ) appearing in the mapping φ τ k , τ k + 1 ⁢ ( Y ) are scaled appropriately in order to obtain convergence of discrete-time value processes to continuous-time value processes, and α and η are constants that do not depend on the partition of the time interval [ 0 , T ] . In our setting, the sequences { α δ k } and { η δ k } are chosen so that, regardless of the partition of [ 0 , T ] , there is a reasonable nontrivial probability of successful financing of the capital requirements through the entire time period and a reasonable expected excess return to the capital providers for providing capital.

Remark 4.

There are natural alternatives to the capital provider’s acceptability criterion (1.1) considered in this paper. Consider a sequence { U t } t = 0 T - 1 of so-called conditional monetary utility functions, where, for each t, U t is a mapping from L 1 ⁢ ( ℱ t + 1 ) to L 1 ⁢ ( ℱ t ) satisfying

1. if λ ∈ L 1 ⁢ ( ℱ t ) and Y ∈ L 1 ⁢ ( ℱ t + 1 ) , then U t ⁢ ( Y + λ ) = U t ⁢ ( Y ) + λ ,

2. if Y , Y ~ ∈ L 1 ⁢ ( ℱ τ k + 1 ) and Y ≤ Y ~ , then U t ⁢ ( Y ) ≤ U t ⁢ ( Y ~ ) ,

3. U t ⁢ ( 0 ) = 0 .

Assume that, at time t, the capital provider will accept providing capital if the utility of the excess capital at time t + 1 ,

is not smaller than the utility of the capital asked for at time t,

Equating these expressions and solving for V u , t gives the backward recursion

where

It is shown in [7, Proposition 1] that φ u , t satisfies φ u , t ⁢ ( 0 ) = 0 , the monotonicity property φ u , t ⁢ ( Y ) ≤ φ u , t ⁢ ( Y ~ ) if Y ≤ Y ~ , and the translation-invariance property: if λ ∈ L 1 ⁢ ( ℱ t ) and Y ∈ L 1 ⁢ ( ℱ t + 1 ) , then φ u , t ⁢ ( Y + λ ) = φ u , t ⁢ ( Y ) + λ , i.e. the analogue of Theorem 1 holds.

One example of a conditional monetary utility function is the certainty equivalent Y ↦ u t - 1 ⁢ ( 𝔼 t ⁡ [ u t ⁢ ( Y ) ] ) when u t is an exponential utility function u t ⁢ ( y ) = - γ t ⁢ exp ⁡ { - y γ t } . In general, certainty equivalents obtained from (strictly increasing and concave) utility functions u t do not satisfy the translation invariance property in Remark 4  i of conditional monetary utility functions; see e.g. [9, Proposition 2.46].

Definition 4.

Let { τ m } m = 1 ∞ be a sequence of partitions of [ 0 , T ] , 0 = τ m , 0 < … < τ m , m = T , such that

Let L ∈ L 0 ⁢ ( 𝔽 ) . If there exists { V t ⁢ ( L ) } t ∈ [ 0 , T ] ∈ L 0 ⁢ ( 𝔽 ) that is càdlàg such that

then { V t ⁢ ( L ) } t ∈ [ 0 , T ] is the continuous-time limit of the sequence of discrete-time CoCM processes { V t τ m ⁢ ( L ) } t ∈ τ m .

Remark 5.

Consider a sequence of partitions { τ m } m = 1 ∞ and two corresponding càdlàg continuous-time limits { V t ⁢ ( L ) } t ∈ [ 0 , T ] and { V ~ t ⁢ ( L ) } t ∈ [ 0 , T ] . Clearly, V T ⁢ ( L ) = V ~ T ⁢ ( L ) = 0 . Take any t ∈ [ 0 , T ) and a sequence { k m } m = 1 ∞ of natural numbers such that τ m , k m ↓ t as m → ∞ . From Definition 4,

and by the càdlàg property,

For all m,

Letting m → ∞ establishes uniqueness of the continuous-time limit.

Remark 6.

If L ∈ L 1 ⁢ ( 𝔽 ) , then { 𝔼 t ⁡ [ L T ] } t ∈ [ 0 , T ] is a martingale with a unique càdlàg modification [17, p. 8]. In particular, if L ∈ L 1 ⁢ ( 𝔽 ) is càdlàg, then { 𝔼 t ⁡ [ L T - L t ] } t ∈ [ 0 , T ] is a possible candidate for the continuous-time limit of discrete-time CoCM processes. This particular limit will appear naturally for a wide class of processes L; see Theorem 2 below.