Profit and loss decomposition in continuous time and approximations

Junike, Gero; Stier, Hauke; Christiansen, Marcus Article — Published Version Profit and loss decomposition in continuous time and approximations Finance and Stochastics Provided in Cooperation with: Springer Nature Suggested Citation: Junike, Gero; Stier, Hauke; Christiansen, Marcus (2025) : Profit and loss decomposition in continuous time and approximations, Finance and Stochastics, ISSN 1432-1122, Springer, Berlin, Heidelberg, Vol. 29, Iss. 4, pp. 1075-1107, https://doi.org/10.1007/s00780-025-00571-7 This Version is available at: https://hdl.handle.net/10419/330649 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Finance and Stochastics (2025) 29:1075–1107 https://doi.org/10.1007/s00780-025-00571-7 Profit and loss decomposition in continuous time and approximations Gero Junike1·Hauke Stier1·Marcus Christiansen1 Received: 4 July 2024 / Accepted: 19 December 2024 / Published online: 18 July 2025 © The Author(s) 2025 Abstract Financial institutions and insurance companies that analyse the evolution and sources of profits and losses often look at risk factors only at discrete reporting dates, ignoring the detailed paths. Continuous-time decompositions avoid this weakness and also make decompositions consistent across different reporting grids. We construct a large class of continuous-time decompositions from a rearranged version of Itô’s formula, and uniquely identify a preferred decomposition from the axioms of exactness, symmetry and normalisation. This unique decomposition turns out to be a stochastic limit of recursive Shapley values, but it suffers from a curse of dimensionality as the number of risk factors increases. We develop an approximation that breaks this curse when the risk factors almost surely have no simultaneous jumps. Keywords Profit and loss attribution ·Sequential decompositions ·Change analysis ·Risk decomposition ·Itô’s formula Mathematics Subject Classification 60H05 ·60H30 ·91G10 ·91G60 ·91G30 · 91G40 JEL Classification C02 ·C30 ·C63 ·G10 ·G12 1Introduction Profit and loss (P&L) attribution, also known as change analysis, has a long history in risk management. P&L attribution is the process of analysing the change between ✉G. Junike [email protected] H. Stier [email protected] M. Christiansen [email protected] 1Carl von Ossietzky Universität, Institut für Mathematik, 26129 Oldenburg, Germany 1076 G. Junike et al. two valuation dates and explaining the evolution of the P&L by the movement of the sources (risk factors) between the two dates; see Candland and Lotz [4]. In other words, the change in the P&L over time is decomposed in terms of the different risk factors to explain how each factor contributes to the P&L. In the literature, there are many ways to obtain a P&L attribution. For example, consider a portfolio in EUR consisting of a long position in the S&P 500, Yfor short. The P&L of such a portfolio is driven by two risk factors, namely Yand the USD/EUR exchange rate, Xfor short. To decompose the P&L over one year, we look for two random variables DXand DY such that X1Y1−X0Y0=DX+DY. The numbers DXand DYare interpreted as the contributions of Xand Yto the P&L. In the literature, we can find many desirable properties that a decomposition should possess; see Shubik [28], Friedman and Moulin [12] and Shorrocks [26] among many others. The authors argue that a decomposition should be symmetric, i.e., the contributions of the risk factors should be independent of the way in which the risk factors are labelled or ordered. These authors also require that the sum of all contributions equals the P&L; such decompositions are called exact. Further, Christiansen [6]argues that a decomposition should be normalised, i.e., if a risk factor remains constant, its contribution to the P&L should be zero. It is also desirable for a decomposition to consider the full path of each risk factor, i.e., to use all available information; see Mai [18] and Flaig and Junike [9]. A common method for creating decompositions is to sequentially update the risk factors one by one while “freezing” all other risk factors. This idea dates back at least to Oaxaca [20] and Blinder [3], who developed a sequential updating (SU) decomposition technique in a one-period setting. The SU decomposition is given by DX=X1Y0−X0Y0,D Y=X1Y1−X1Y0, when we update the risk factor Xfirst. Alternatively, one may update Yfirst to obtain DX=X1Y1−X0Y1,D Y=X0Y1−X0Y0. Each SU decomposition is exact, but if there are drisk factors, there are d!different updating orders and therefore d!different SU decompositions. Candland and Lotz [4] call the one-period SU decomposition waterfall and apply it to P&L attribution. See Fortin et al. [10] for an overview on how the SU decomposition is used in various fields of economics. The SU decomposition can also be defined in a multi-period setting by dividing the time horizon into subintervals and applying the SU decomposition recursively on each subinterval. Jetses and Christiansen [16] and Christiansen [6] analysed the limit of the SU decomposition when the mesh size of the time grid converges to zero. In the limit, the decomposition takes the whole path into account, and the limiting SU decomposition is called the infinitesimal sequential updating (ISU) decomposition. The ISU decomposition is independent of any time grid, which is helpful “to prevent inconsistencies when using conflicting subintervals for different purposes”; see Flaig and Junike [9, Sect. 1]. P&L decomposition in continuous time and approximations 1077 The averaged sequential updating (ASU) decomposition, also known as the Shapley value, is simply the arithmetic average of the d!possible SU decompositions. It has many desirable properties; in particular, it is exact and symmetric. Shapley [27] introduces the ASU decomposition for cooperative games. Shubik [28] defines the ASU decomposition for cost functions. Sprumont [29] and Friedman and Moulin [12] provide an axiomatisation of the ASU decomposition for cost functions. Jetses and Christiansen [16] define the infinitesimal averaged sequential updating (IASU) decomposition as the average of the d!possible ISU decompositions. We now summarise our main contributions. In this paper, we start directly in a time-continuous setting. If the portfolio is a C2-function of the risk factors and the latter have continuous paths, Itô’s formula provides a natural additive decomposition of the portfolio. Our main contributions are as follows. In order to treat risk factors with jumps, we provide a rearranged version of Itô’s formula and use it to define a large class of reasonable decompositions, which we call Itô decompositions and which include all d!ISU and the IASU decompositions as special cases. We prove that there is a unique Itô decomposition (up to indistinguishability) that satisfies the three axioms of exactness, symmetry and normalisation. We show that it is indistinguishable from the IASU decomposition. We further show that the IASU decomposition can be interpreted as the limiting case of the ASU decomposition. Compared to Jetses and Christiansen [16], who assume that the covariations between the risk factors are zero, we use much weaker assumptions to prove the convergence of the SU/ASU decompositions to the ISU/IASU decompositions. In summary, we propose to use the IASU decomposition to obtain a P&L attribution because it considers the whole paths of the risk factors and satisfies the axioms of exactness, symmetry and normalisation. However, in practical applications, the IASU decomposition has two drawbacks: a) similarly to the ASU decomposition, it suffers from the curse of dimensionality; b) the IASU decomposition is defined by stochastic integrals, which somehow must be approximated in practice. Naively approximating these integrals can lead to decompositions that are no longer exact. As another important contribution of this paper, we show that the IASU decomposition does not suffer from the curse of dimensionality if the risk factors do not have simultaneous jumps. In this case, the IASU decomposition is indistinguishable from the average of two (suitably selected) ISU decompositions. To avoid point b), we suggest approximating ISU/IASU by SU/ASU. Up to now, most practitioners have applied an arbitrary SU decomposition in a one-period setting to obtain an annual P&L attribution; see Candland and Lotz [4]. Working with real market data, Flaig and Junike [9] empirically show that the SU decomposition depends significantly on the order or labelling of the risk factors, and that some SU decompositions may even ignore relevant risk factors, which may “lead to wrong trading and hedging decisions”; see Flaig and Junike [9, Sect. 1]. Our theoretical analysis suggests using the average of only two SU decompositions with a sufficiently fine time grid to obtain a P&L attribution, since such a decomposition is arbitrarily close to the IASU decomposition when the risk factors do not have simultaneous jumps. To obtain these two SU decompositions, define one SU decomposition in any order, e.g. alphabetically ascending, and another SU decomposition by the reverse order, e.g. alphabetically descending; see Theorem 3.10 for details. 1078 G. Junike et al. Thus our analysis is highly relevant for practitioners: we recommend computing two SU decompositions instead of one and using a finer grid than just annual data to obtain a decomposition that is much closer to the IASU decomposition than a single SU decomposition. While the choice of the decomposition (the average of two SU decompositions) is theoretically justified, we have only numerical experiments available to estimate the time grid, and we recommend using monthly or weekly data. Is there any other way to break the curse of dimensionality? Christiansen [6] proves that the ISU decomposition is symmetric if it is stable with respect to small perturbations in the empirical observation of the risk factors. In Appendix A.3,we show that the ISU decomposition of a simple product of two correlated Brownian motions is not stable. This shows that stability is a rather strong assumption. There are other decomposition principles as well. There is the so-called one-ata-time (OAT) decomposition, which is also known as bump and reset; see Candland and Lotz [4]. The OAT decomposition is closely related to the SU decomposition. It is symmetric, but in general not exact. Frei [11] analyses the limit of the OAT decomposition when the mesh size of the time grid converges to zero. There are also completely different approaches. Fischer [8] uses a conditional expectations approach. Rosen and Saunders [24] use the Hoeffding method for a decomposition of credit risk portfolios. Frei [11] and Bielecki et al. [1] use the Euler principle for risk attribution. Ramlau-Hansen [23] and Norberg [19] decompose surplus in life insurance by heuristic integral representations, where the integrators are interpreted as the driving forces of change and determine the attribution. A similar idea is used in Schilling et al. [25] based on the martingale representation theorem. This article is structured as follows. In Sect. 2, we establish some notation. In Sect. 3, we develop a rearranged version of Itô’s formula and introduce the family of Itô decompositions. We show that the IASU decomposition is the only exact and symmetric Itô decomposition, and we break the curse of dimensionality of the IASU decomposition in Theorem 3.10. In Sect. 4, we prove that the IASU decomposition can be approximated by the ASU decomposition. In Sect. 5, we provide some numerical applications. Section 6concludes. 2Notation Let (Ω, ℱ,𝔽=(ℱt)t≥0,P)be a filtered probability space satisfying the usual conditions, i.e., ℱ0contains all nullsets and 𝔽is right-continuous. Let 𝒳be the set of all real-valued 𝔽-semimartingales. A so-called risk basis or information basis is a d-dimensional semimartingale X∈𝒳d, and its dcomponents are called risk factors or sources of risk. We denote the stopped semimartingale by Xσ=(X1,σ ,...,Xd,σ) for a stopping time σ. Equality of random variables is understood in the almost sure sense, and equality of stochastic processes is understood up to indistinguishability. Let C2be the set of twice continuously differentiable functions from ℝdto ℝ.For f∈C2and i, j =1,...,d, we write fiand fij for the partial derivatives ∂ifand ∂i∂jf.Byx∧y, we denote the minimum of two real numbers xand y. We call a map F:𝒳d→𝒳non-anticipative if for any stopping time σ, it holds that for all P&L decomposition in continuous time and approximations 1079 X∈𝒳d, Ft(Xσ)=Ft∧σ(X), t ≥0.(2.1) Such a non-anticipative mapping depends only on the information up to time t, i.e., on Xt.Byℳ, we denote some subspace of all non-anticipative mappings. By ℳ(C2), we denote the space of functionals F:𝒳d→𝒳such that F(X) =(f (Xt))t≥0for X∈𝒳dand some f∈C2, which are clearly nonanticipative. By σd, we denote the set of all d!permutations of {1,...,d}.Letid∈σd be the identity. In a slight abuse of notation, we define for π∈σd, π(x) =(xπ(1),...,xπ(d)), x ∈ℝd, π(X) =(Xπ(1),...,Xπ(d)), X ∈𝒳d. For two one-dimensional semimartingales Zand Yand a càglàd process H, we denote by ∫︁t 0HsdZs:= ∫︁(0,t]HsdZsthe stochastic integral. In particular, ∫︁0 0HsdZs=0 by convention. We further set Z0−=0, Zt−=lim ε↘0Zt−ε,t>0, ΔZt=Zt−Zt−,t≥0, [Z,Y]=ZY −Z0Y0−∫︂· 0Zu−dYu−∫︂· 0Yu−dZu, [Z,Y]c=[Z,Y]− ∑︂ 0<s≤· ΔZsΔYs. We write p →for the convergence in probability of a sequence of random variables. For A⊆{1,...,d}, we define the projection pA:ℝd→ℝd,x↦→ (︁x11A(1), . . . , xd1A(d))︁, where the function 1A(h) is 1 if h∈Aand 0 otherwise. 3 Family of Itô decompositions Similarly to Shorrocks [26], Christiansen [6], we define a decomposition as follows. Definition 3.1 Amap δ:ℳ×𝒳d→𝒳d,(F,X)↦→ (︁δ1(F,X),...,δd(F, X))︁ is called a decomposition. 1080 G. Junike et al. We interpret δi t(F, X) as the contributionofXito the profitand loss Ft(X)−F0(X) in [0,t]. We recall the following three axioms from the literature: i) A decomposition is called exact if for all F∈ℳand X∈𝒳d, it holds that F(X)−F0(X) =δ1(F, X) +···+δd(F, X). ii) A decomposition is called symmetric if for all π∈σd,F∈ℳand X∈𝒳d,it holds that F(X) =F(︁π(X))︁=⇒ δi(F, X) =δπ−1(i)(︁F,π(X))︁. iii) A decomposition is called normalised if for all 0 ≤r<s<∞,i=1,...,d, F∈ℳand X∈𝒳d, it holds that Xiis indistinguishable from a constant process on (r, s] =⇒ δi(F, X) is indistinguishable from a constant process on (r, s]. Axiom i) ensures that a decomposition is able to fully explain the P&L; see Shorrocks [26] and Christiansen [6]. Axiom ii) considers symmetric maps Fand states that if Fdoes not depend on the order or labelling of the risk factors, then neither does the decomposition. The symmetry axiom is motivated by the fact that δi(F, X) represents the contribution of Xiand that the term δπ−1(i)(F, π(X)) also describes the contribution of (︁π(X))︁π−1(i) =(︁Xπ(1),...,Xπ(d))︁π−1(i) =Xi. The symmetry axiom has already been mentioned in similar form in Friedman and Moulin [12] and Shorrocks [26]. Finally, for axiom iii), if the risk factor Xiremains constant during (r, s], it does not contribute to Fs(X)−Fr(X), and so the contribution of Xiin (r, s]should also be zero. This is exactly reflected by the normalisation axiom, taken from Christiansen [6]. Next, we indicate how Itô’s formula helps to define decomposition principles. Let f:ℝd→ℝbe in C2.Fori, j =1,...,d,let Ii:= ∫︂· 0fi(Xs−)dXi s,I ij := ∫︂· 0fij (Xs−)d[Xi,Xj]c s,(3.1) S:= ∑︂ 0<s≤·(︃f(X s)−f(X s−)− d ∑︂ i=1 fi(Xs−)ΔXi s)︃.(3.2) Itô’s formula states that for t≥0, we have for any semimartingale X∈𝒳dthat f(X t)−f(X 0)= d ∑︂ i=1 Ii t+1 2 d ∑︂ i=1 Iii t+1 2 d ∑︂ i,j=1 i=j Iij t+St.(3.3) P&L decomposition in continuous time and approximations 1081 If we assume that Xhas continuous paths without interaction effects, i.e., Iij =0, i=j, and S=0, then (3.3) provides a natural way to additively decompose the P&L f(X t)−f(X 0). Indeed, by the normalisation axiom, Iiand Iii should be assigned to δi, which is interpreted as the contribution of Xi. To see this, assume that some δj depends on Iior Iii for i= j. Assume that Xjis constant everywhere. According to the normalisation axiom, we should then have δj=0. So δjmust not depend on Ii or on Iii. However, how to handle the interaction effects Iij ,i= j, and the jump part S is not so obvious. Therefore, we provide in Proposition 3.3 a rearranged version of Itô’s formula. Based on that result, we define the large family of Itô decompositions in Definition 3.4 and show in Sect. 3that this family contains many well-known decomposition principles as special cases. Within the family of Itô decompositions, we identify a single decomposition that satisfies the axioms of exactness, symmetry and normalisation. For A⊆{1,...,d},i∈{1,...,d}and s>0, define Yi,A s:=f(︁Xs−+pA(ΔXs))︁−f(︁Xs−+pA\{i}(ΔXs))︁−fi(Xs−)ΔXi s and Si,A(X) := ∑︂ 0<s≤· Yi,A s. For π∈σd, define Si,π (X) := Si,{j:π(j)≤π(i)}(X). (3.4) To obtain Si,π (X), all time points swhere Xijumps are considered. All risk factors except Xiare fixed at sor s−, depending on the choice of π, and only Xivaries between s−and s. Lemma 3.2 Fix i∈{1,...,d},X∈𝒳dand A⊆{1,...,d}.If i∈A,then Si,A(X) is a semimartingale with a.s.paths of finite variation on compacts. Proof Fix X∈𝒳d.LetNbe a nullset such that u↦→ |Xi u(ω)|,i=1,...,d,is càdlàg for ω∈Ω\Nand d ∑︂ h,j=1∑︂ 0<s≤t|ΔXh s(ω)ΔXj s(ω)|<∞,ω∈Ω\N,t ≥0.(3.5) Such an Nexists as Xis a semimartingale. Let ω∈Ω\Nand t≥0. Let Mω⊆ℝd be the closure of the set {Xu(ω) :u∈[0,t]}, which is compact. The function fand its derivatives are continuous and reach a maximum and minimum on the convex hull of Mω, which is compact by Carathéodory’s theorem; see Grünbaum [14, Sect. 2.3]. Hence fand its derivatives are bounded on the convex hull of Mω.Fors∈(0,t],we develop faround Xs−(ω) using a Taylor expansion. We have that f(︂Xs−(ω) +pA(︁ΔXs(ω))︁)︂=f(︁Xs−(ω))︁+∑︂ h∈A fh(︁Xs−(ω))︁ΔXh s(ω) +R(ω), 1082 G. Junike et al. where R(ω) is the remainder term of the Taylor expansion, i.e., for some θ(ω) ∈[0,1], it holds that R(ω) =1 2∑︂ h,j∈A fhj (︂Xs−(ω) +θ(ω)pA(︁ΔXs(ω))︁)︂ΔXh s(ω)ΔXj s(ω). The term f(X s−(ω)+pA\{i}(ΔXs(ω))) can be treated similarly. Since i∈A, it holds for some C(ω) > 0, which does not depend on sor θ(ω), that Yi,A s≤C(ω) ∑︂ h,j∈A|ΔXh s(ω)ΔXj s(ω)|. It follows by (3.5) that ∑︂ 0<s≤t|Yi,A s(ω)|<∞,ω∈Ω\N. (3.6) Since twas arbitrary, (3.6) implies that u↦→ Si,A u(X)(ω),ω∈Ω\N, is càdlàg and of finite variation on compacts. Therefore Si,A(X) is a semimartingale. □ Proposition 3.3 Let π∈σd,f∈C2and X∈𝒳d.For all t≥0, it holds that f(X t)−f(X 0)= d ∑︂ i=1(︃Ii t+1 2Iii t+1 2 d ∑︂ j=1 j=i Iij t+Si,π t)︃, where Iiand Iij are defined in (3.1)and Si,π is defined in (3.4). Proof Since the series telescopes, we have that f(X s)−f(X s−) = d ∑︂ i=1 f(︁Xs−+p{j:π(j)≤π(i)}(ΔXs))︁−f(︁Xs−+p{j:π(j)<π(i)}(ΔXs))︁. By (3.6), it holds for any t≥0 that d ∑︂ i=1 Si,π t(X) =∑︂ 0<s≤t d ∑︂ i=1 Yi,{j:π(j)≤π(i)} s=St,(3.7) where Sis defined in (3.2). The claim then follows by Itô’s formula. □ Definition 3.4 Let λij ∈[0,1]for i, j =1,...,d.Letμπ∈[0,1]for π∈σd.The decomposition δItô :ℳ(C2)×𝒳d→𝒳d,(F,X)↦→ (︁δItô,1(F,X),...,δItô,d (F, X))︁, P&L decomposition in continuous time and approximations 1089 semimartingales and hence a semimartingale. By Protter [21, Sect. II.2] and since σn→∞a.s. for n→∞, the process δSU,i,π,γ (F, X) is a semimartingale and the decomposition δSU,π,γ is therefore well defined. The fact that σn→∞a.s. implies that the sum in (4.1) evaluated at tis a.s. finite. We show exactness. Let x∈ℝd. Since p{j:π(j)≤π(π−1(d))}(x) =xand p{j:π(j)<π(π−1(1))}(x) =0, we have for any t∈[0,∞)and n∈ℕby (4.3) that d ∑︂ i=1 δSU,i,π,γ t∧σn(F, X) = d ∑︂ i=1 i=π−1(d) n−1 ∑︂ ℓ=0 Ft∧σn(︁Xσℓ+p{j:π(j)≤π(i)}(Xσℓ+1−Xσℓ))︁+ n−1 ∑︂ ℓ=0 Ft∧σn(Xσℓ+1) − d ∑︂ i=1 i=π−1(1) n−1 ∑︂ ℓ=0 Ft∧σn(︁Xσℓ+p{j:π(j)<π(i)}(Xσℓ+1−Xσℓ))︁− n−1 ∑︂ ℓ=0 Ft∧σn(Xσℓ). For each i∈{1,...,d}\{π−1(d)}, there is exactly one k∈{1,...,d}\{π−1(1)} such that p{j:π(j)≤π(i)}(x) =p{j:π(j)<π(k)}(x), since π(k) =π(i) +1 if and only if k=π−1(π(i) +1). Thus we get d ∑︂ i=1 δSU,i,π,γ t∧σn(F, X) = n−1 ∑︂ ℓ=0 Ft∧σn(Xσℓ+1)− n−1 ∑︂ ℓ=0 Ft∧σn(Xσℓ) =Ft∧σn(Xσn)−Ft∧σn(Xσ0) =Ft∧σn(X) −F0(X). Since tand nwere arbitrary and σn→∞a.s., the decomposition δSU,π,γ is exact. To see the last point, note that two processes with càdlàg paths are indistinguishable if they are modifications. □ Example 4.4 Assume d=2. The SU decomposition with respect to γdefines d!=2 decompositions, namely δSU,id,γ (F, X) and δSU,ϱ,γ (F, X) with ϱ(1)=2 and ϱ(2)=1, by δSU,1,id,γ (F, X) =∞ ∑︂ ℓ=0(︁F(X1,σℓ+1,X2,σℓ)−F(X1,σℓ,X2,σℓ))︁, δSU,2,id,γ (F, X) =∞ ∑︂ ℓ=0(︁F(X1,σℓ+1,X2,σℓ+1)−F(X1,σℓ+1,X2,σℓ))︁ 1090 G. Junike et al. and δSU,1,ϱ,γ (F, X) =∞ ∑︂ ℓ=0(︁F(X1,σℓ+1,X2,σℓ+1)−F(X1,σℓ,X2,σℓ+1))︁, δSU,2,ϱ,γ (F, X) =∞ ∑︂ ℓ=0(︁F(X1,σℓ,X2,σℓ+1)−F(X1,σℓ,X2,σℓ))︁. Definition 4.5 Let γ={0=σ0<σ 1<···}be an unbounded random partition. The ASU (averaged sequential updating) decomposition δASU,γ :ℳ×𝒳d→𝒳dis defined by δASU,i,γ (F, X) =1 d!∑︂ π∈σd δSU,i,π,γ (F, X), i =1,...,d. Remark 4.6 As in Shorrocks [26], we observe that δASU,i,γ (F, X) =1 d!∑︂ π∈σd δSU,i,π,γ (F, X) =∑︂ A⊆{1,...,d} i∈A δSU,i,A,γ (F, X)ξi,A for ξi,A defined in (3.8) and δSU,i,A,γ (F, X) := ∞ ∑︂ ℓ=0(︂F(︁Xσℓ+pA(Xσℓ+1−Xσℓ))︁−F(︁Xσℓ+pA\{i}(Xσℓ+1−Xσℓ))︁)︂. Thereby, the computational cost to obtain δASU,i,γ can be reduced from 𝒪(d!) to 𝒪(2d−1). Theorem 4.7 Fix π∈σdand let (γn)n∈ℕbe a sequence of unbounded random partitions tending to the identity.Let F∈ℳ(C2),X∈𝒳d,t≥0and i∈{1,...,d}. Then it holds for n→∞that δSU,i,π,γn t(F, X) p −→ δISU,i,π t(F, X), δASU,i,γn t(F, X) p −→ δIASU,i t(F, X). Proof See Appendix A.2.□ The next example shows that the assumption F∈ℳ(C2)in Theorem 4.7 is important to ensure convergence. Example 4.8 Let Zbe a stochastic process with independent increments and Z0=0. Suppose the jumps of Zonly occur at fixed times J={2−ℓ−1:ℓ∈ℕ}, and for P&L decomposition in continuous time and approximations 1091 each ℓ∈ℕ, the process jumps by ±ℓ−1with equal probability. The process Zstays constant between jumps. Then Zis a semimartingale; see ˇ Cerný and Ruf [5]. Let f(x1,x2)=|x1−x2| so that f/∈C2.Let(γn={0=σn 0<σ n 1<···})n∈ℕbe a deterministic sequence of unbounded partitions tending to the identity such that γncontains the first nsmallest elements of J, but the intersection with (2−n−1,2]is empty. Assume that X=(Z, Z). Then for t≥2, it follows that ∞ ∑︂ ℓ=0(︁f(X1,σn ℓ+1 t,X2,σn ℓ t)−f(X1,σn ℓ t,X2,σn ℓ t))︁= n ∑︂ ℓ=1 ℓ−1, which is divergent for n→∞; so the SU decomposition does not converge for the map Ft(X) := f(X t),t≥0. How can the IASU decomposition be computed efficiently in practice? If we naively approximate the integrals in Definition 3.6 numerically, we may lose exactness of the decomposition, which is undesirable in many applications. Theorem 4.7 suggests using the ASU decomposition as an approximation of the IASU decomposition. However, this becomes computationally infeasible for moderately large d since the computational cost to obtain δASU,i,γ scales like 𝒪(2d−1). The next result provides an elegant solution when there are no simultaneous jumps. Definition4.9 Let γ={0=σ0<σ 1<···}be an unbounded random partition. The 2SU (average of two sequential updating) decomposition δ2SU,π,γ :ℳ×𝒳d→𝒳d with updating order π∈σdis defined by δ2SU,i,π,γ (F, X) =1 2(︁δSU,i,π,γ (F, X) +δSU,i,π′,γ (F, X))︁,i=1,...,d, where π′=d+1−π. Corollary 4.10 Fix π∈σdand let (γn)n∈ℕbe a sequence of unbounded random partitions tending to the identity.Let F∈ℳ(C2),X∈𝒳d,i∈{1,...,d}and t≥0. i) If ΔXhΔXj=0for all h, j ∈{1,...,d}with h= j,then δ2SU,i,π,γn t(F, X) p −→ δIASU,i t(F, X), n →∞. ii) If [Xh,Xj]=0for all h, j ∈{1,...,d}with h= j,then δSU,i,π,γn t(F, X) p −→ δIASU,i t(F, X), n →∞. Proof If ΔXhΔXj=0, h=j, Theorem 3.10 implies that δIASU,i(F, X) =1 2(︁δISU,i,π (F, X) +δISU,i,π′(F, X))︁, 1092 G. Junike et al. Fig. 1 Overview of discrete approximations of the IASU decomposition which is the limit of δ2SU,i,π,γn(F, X) by Theorem 4.7.If[Xh,Xj]=0, h= j, apply Corollary 3.12 and Theorem 4.7.□ In particular, the 2SU decomposition with arbitrary updating order πis exact and approximates the IASU decomposition when the risk factors do not have simultaneous jumps. In this case, the computationally expensive averaging to obtain the ASU decomposition can be omitted and the computational complexity to approximate δIASU,i decreases from 𝒪(2d−1)to 𝒪(1). Theorem 4.7 and Corollary 4.10 are also illustrated in Fig. 1. Finally, we define the OAT decomposition. To obtain the contribution of Xi,all risk factors are fixed at the origin and only Xiis allowed to change from the beginning of a subinterval to the end of that subinterval. Definition 4.11 Let γ={0=σ0<σ 1<···}be an unbounded random partition. The OAT (one-at-a-time) decomposition δOAT,γ :ℳ×𝒳d→𝒳dis defined by δOAT,i,γ (F, X) =∞ ∑︂ ℓ=0(︁F(X1,σℓ,...,Xi−1,σℓ,Xi,σℓ+1,Xi+1,σℓ,...,Xd,σℓ)−F(Xσℓ))︁. Remark 4.12 The OAT decomposition is symmetric, but in general not exact. Let (γn)n∈ℕbe a sequence of unbounded random partitions tending to the identity. For each i∈{1,...,d}, choose a permutation πi∈σdsuch that πi(i) =1. Then δOAT,i,γnis indistinguishable from δSU,i,πi,γn.IfF∈ℳ(C2), Theorem 4.7 gives for t≥0 that δOAT,i,γn t(F, X) p −→ δISU,i,πi t(F, X), i =1,...,d, for n→∞. Thus by Corollary 3.12, the three decompositions principles OAT, SU (with arbitrary order π∈σd) and ASU are asymptotically indistinguishable if there are no interaction effects. 5 Applications Investment portfolios of financial institutions or insurance companies may include instruments such as stocks, plain vanilla or callable bonds, convertible bonds, inflation- P&L decomposition in continuous time and approximations 1093 linked bonds, contingent convertible bonds (CoCos), basket options, foreign exchange options and structured products. These instruments often depend on multiple risk factors such as different foreign exchange rates, interest rates for different maturities, credit spreads, inflation rate, some trigger activations for CoCos, multiple equities and time decay. Candland and Lotz [4] also considered defaults and rating changes as risk factors. In order to obtain a P&L attribution of such instruments, we propose the IASU decomposition because it is exact, symmetric and normalised, and it takes into account the whole paths of the risk factors, i.e., uses all available information. The last point also avoids inconsistencies when reporting a P&L attribution for different time grids, e.g. on an annual, quarterly, monthly and weekly basis. The IASU decomposition involves a stochastic integral. To approximate the IASU decomposition, we propose the ASU or 2SU decomposition with a sufficiently fine time grid, as such an approximation is always an exact decomposition. The use of the 2SU decomposition is theoretically justified when the risk factors do not have simultaneous jumps. In Sect. 5.1, we provide an exemplary decomposition of a plain vanilla call option with stochastic interest rates on a foreign stock. A change in the P&L of this option can be explained by movements in the stock, the yield curve, the foreign exchange rate and time decay. Thus there are d=4 risk factors. We analyse the unexplained P&L of the OAT decomposition, the range of the SU and 2SU decompositions over all possible updating orders π∈σdfor different time grids, and the convergence of the ASU decomposition to the IASU decomposition. Computing the ASU decomposition to approximate the IASU decomposition becomes infeasible when the number of risk factors dis moderately large; for example, a plain vanilla bond paying coupons may depend on dyield curves. A basket option may depend on dstocks. In practice, d=30 is a common case for basket options; see Grzelak et al. [15]. In Sect. 5.2, we decompose a digital cash-or-nothing basket put option. We illustrate that it is impossible to obtain the ASU decomposition in reasonable time when d=30, and we show how the 2SU decomposition is able to break the curse of dimensionality. 5.1 Decomposing a call option with stochastic interest rates In this section, we allocate the P&L of the price of a plain vanilla European call option with strike Kand maturity T=10 with stochastic interest rates and foreign exchange exposure. The stock price Sis given by a Black–Scholes model with constant volatility σS>0 and with stochastic interest rates r. The dynamics under the risk-neutral measure are given by dSt=rtStdt +σSStdBS t, drt=κ(η −rt)dt +σrdBr t with constant volatility σr>0, long-term mean η∈ℝand speed κ>0 of meanreversion. Under the physical measure, the stock has drift μS∈ℝ, and the foreign exchange rate Yis assumed to follow a geometric Brownian motion with drift μY∈ℝ 1094 G. Junike et al. and volatility σY>0 driven by the Brownian motion BY. The Brownian motions are assumed to have correlations d⟨BS,Br⟩t=ρSrdt, d⟨BS,BY⟩t=ρSY dt, d⟨BY,Br⟩t=ρYrdt. The time to maturity is denoted by τ(t) =T−t. The price pcall(t) at time tof the plain vanilla call option is given by a C2-function f:ℝd→ℝ, see Rabinovitch [22], i.e., pcall(t) =f(︁St,r t,Y t,τ(t) )︁=:Ft(S,r,Y,τ), t >0, with f(s,r,y,τ)=ysΦ(︁d+(s,r,τ) )︁−yKP(r,τ)Φ(︁d−(s,r,τ) )︁, where Φdenotes the distribution function of a standard normal distribution and d±(s,r,τ)=1 √v(τ)(︃log s KP(r,τ) ±1 2v(τ))︃, v(τ) =σ2 Sτ+σ2 r τ−2gκ(τ) +g2κ(τ) κ2−2ρSrσSσr τ−gκ(τ) κ, gκ(τ) =1−e−κτ κ. The bond price P(r,τ)is given by P(r,τ)=A(τ)e−gκ(τ )r , where A(τ) =exp (︃(︂η+σ2 rλ κ−σ2 r 2κ2)︂(︁gκ(τ) −τ)︁−1 κ(︂σrgκ(τ) 2)︂2)︃ and λdenotes the market price of risk. For simplicity, we set the market price of risk to zero and hence assume that the dynamics of runder the physical and the risk-neutral measure are identical. Björk [2, Sect. 24.2] describes how to estimate the parameters for rfrom market data. We simulate 1000 paths of the stock, interest rate and foreign exchange rate under the physical measure over one year. For each path, we decompose the price of the call option at time t=1 with respect to the d=4 risk factors X:= (S,r,Y,τ). We use the following parameters: K=S0=100, μS=0.05, σS=0.4, Y0=1.1, μY=0, σY=0.05, r0=0.08, κ=0.1, η=0.05, σr=0.01 and ρSr =−0.7, ρSY =−0.4, ρYr =0.7. Figure 2shows the relative unexplained P&L of the OAT decomposition, i.e., |(F1(X) −F0(X)) −∑︁d i=1δOAT,i,γ 1(F, X)| |F1(X) −F0(X)|. P&L decomposition in continuous time and approximations 1095 Fig. 2 Relative unexplained P&L for the OAT decomposition of a plain vanilla call option in a foreign currency at time t=1 for different time grids Fig. 3 Relative range of all SU and 2SU decompositions for the risk factor S Weuseastimegridsγannual, quarterly, monthly, weekly and daily time steps. As observed in Flaig and Junike [9], we also see that the unexplained P&L of the OAT decomposition is significant for all time grids. Figure 3shows the relative range of the d!SU decompositions for the risk factor S, i.e., max π∈σd δSU,1,π,γ 1(F, X) δIASU,1 1(F, X) −min π∈σd δSU,1,π,γ 1(F, X) δIASU,1 1(F, X) , and the relative range of the d! 22SU decompositions for the risk factor S. The limiting IASU decomposition is approximated by an ASU decomposition with 10’000 time steps per year. We observe that the range is significant for the SU decompositions and insignificant for the 2SU decompositions. The speed of convergence of the ASU to the IASU decomposition is illustrated in Fig. 4for the risk factor S, i.e., we show the convergence δASU,1,γ 1(F, X) δIASU,1 1(F, X) −→ 1asγtends to the identity. Figures 3and 4look similar for other risk factors. 1096 G. Junike et al. Fig. 4 Convergence of the ASU decomposition to the IASU decomposition for the risk factor S In further numerical experiments, we calculate the relative difference between the ASU decomposition and the 2SU decompositions,  δ2SU,i,π,γ 1(F, X) −δASU,i,γ 1(F, X) δIASU,i,γ 1(F, X) , over all risk factors i∈{1,...,d}, time grids γand updating orders π∈σd, and observe values of less than 0.6% in 95% of the simulations. In conclusion, we find that the ASU decomposition and the 2SU decompositions are strongly dependent on the time grid, but using monthly or weekly instead of annual time steps significantly reduces the deviation of the ASU and 2SU decompositions from the IASU decomposition. 5.2 Decomposing a basket option In this section, we compare the computational cost of obtaining a one-year P&L attribution of a basket option using a naive SU decomposition with annual time grid to the computational cost of obtaining an ASU and a 2SU decomposition based on a monthly time grid, respectively. We consider drisk factors, namely time decay and d−1 different stocks. A digital cash-or-nothing basket put option pays $1 at maturity Tif S1 T≤K,...,Sd−1 T≤Kand zero otherwise. The stock prices are given by a Black–Scholes model. We set the interest rate rto zero. We set Si 0=K=100, i=1,...,d −1 and T=2. The price of the option at time t∈[0,T) is equal to Φ(log K,...,log K), where Φis the distribution function of a (d −1)-dimensional normal distribution with location (︃logS1 t−(︂r−1 2σ2)︂(T −t),...,logSd−1 t−(︂r−1 2σ2)︂(T −t))︃∈ℝd−1 and covariance matrix Σ(T −t), where we set σ=0.2, ρ=0.5 and Σij ={︄σ2,i=j, ρσ2,i= j. P&L decomposition in continuous time and approximations 1097 Table 1 CPU time to compute the dcontributions of the SU, ASU and 2SU decompositions of a basket option over one year using different time grids. The CPU time of Φis obtained from a Monte Carlo simulation. The CPU times in brackets are estimated using the CPU time of Φand the known complexities of the three decompositions Number of evaluations of Φ d=4d=15 d=30 Evaluation of Φ1 0.018 s 0.15 s 0.54 s SU with annual grid d+1 0.09 s 2.4 s 16.7 s 2SU with monthly grid (12d+1)2 1.76 s 54.3 s 390 s ASU with monthly grid (12d+1)2(d−1)7.06 s 123.6 hr 3318.7 yrs Basket options are often priced by using Monte Carlo techniques; see Glasserman [13, Sect. 3.2.3]. For moderate dimensions, many basket options can also be priced by using faster Fourier techniques; see Eberlein et al. [7] and Junike and Stier [17]. We compute Φby using a simple Monte Carlo simulation implemented in C++ with 100’000 simulations. The experiments are performed on a laptop with Intel i7-11850H processor and 32 GB RAM. Table 1shows the CPU time needed to obtain Φfor d∈{4,15,30}. We measure CPU times by averaging over 100 runs. Since in some cases, the arguments of Φto obtain an SU decomposition with a certain update order πare the same for different contributions, we need to evaluate Φonly dL +1 times, where Lis the number of subintervals of [0,T], to obtain the dindividual contributions. For example, (12d+1)2 and (12d+1)2d−1evaluations of Φare required for the 2SU and ASU decompositions with a monthly time grid. Table 1also shows the CPU time to compute the SU, ASU and 2SU decompositions. A naive SU decomposition based on an annual time grid is at most 24 times faster than a 2SU decomposition with a monthly time grid. The computational cost of the 2SU decomposition for each contribution is dimension-independent, except for the longer time required to evaluate Φ. Compared to the ASU decomposition, the 2SU decomposition is 2d−2times faster. The ASU decomposition cannot be computed in reasonable time for d≥30. Remark 5.1 To reduce the computational time, it is possible to compute the dcontributions for the SU, 2SU and ASU decompositions in parallel, which would reduce the numerical effort by a factor of d. Furthermore, the sums for the SU, 2SU and ASU decompositions can also be parallelised. For example, for the 2SU decomposition, we need to perform 2(dL+1)function evaluations to obtain all dcontributions. If a function evaluation takes 0.54 s in d=30 dimensions as in Table 1, the computation time for the 2SU decomposition with monthly time grid could be reduced from 390 s to about 0.54 s using 722 cores for parallelisation. 6Conclusions We showed that the IASU decomposition is the only (up to indistinguishability) exact and symmetric decomposition in the family of Itô decompositions, which is a large 1098 G. Junike et al. class of normalised decompositions based on a rearranged version of Itô’s formula. This axiomatic result, together with the fact that the IASU decomposition is gridindependent and considers the full paths of the risk basis, makes it a decomposition of choice from a theoretical perspective. In practice, the calculation of the IASU decomposition comes with two challenges: it involves stochastic integrals that must be approximated, and the computational effort explodes as the number of risk factors increases. We have shown that the IASU decomposition can be approximated by the ASU decomposition (which is always exact and symmetric) if we use a sufficiently fine time grid, but the ASU decomposition also suffers from the curse of dimensionality as the number of risk factors increases. For applications where different risk factors may have interactions, but almost surely do not have simultaneous jumps, we have shown that the IASU decomposition is indistinguishable from the average of two ISU decompositions, thus breaking the curse of dimensionality. Therefore, from a theoretical point of view, the 2SU decomposition with a sufficiently fine time grid is an appropriate approximation of the IASU decomposition. Based on our own numerical experiments and the empirical analysis of Flaig and Junike [9], we recommend using monthly or even weekly instead of annual time steps. The additional computational cost of our two recommendations is moderate, but the theoretical properties of the decomposition are dramatically improved. Appendix A.1 Auxiliary results Lemma A.1 Let i, j ∈{1,...,d}.Let π,η ∈σdand x∈ℝd.Then it holds that η−1(︂p{j:π(j)≤π(η−1(i))}(︁η(x))︁)︂=p{j:π(η−1(j))≤π(η−1(i))}(x). (A.1) Proof Let k∈{j:π(j) ≤π(η−1(i))}, which is equivalent to η(k) ∈{︁j:π(︁η−1(j))︁≤π(︁η−1(i))︁}︁. Since (η−1(x))η(k) =xkand (η(x))k=xη(k), we obtain that (︃η−1(︂p{j:π(j)≤π(η−1(i))}(︁η(x))︁)︂)︃η(k) =(︂p{j:π(j)≤π(η−1(i))}(︁η(x))︁)︂k =(︁p{j:π(η−1(j))≤π(η−1(i))}(x))︁η(k), which leads to (A.1). □ P&L decomposition in continuous time and approximations 1105 let 𝒯also contain (˜τn)n∈ℕ=((τn 2,τn 1))n∈ℕ. Since τn 2(an ℓ,1)=τn 2(bn ℓ,1)=τn 1(bn ℓ,1) and by the multidimensional Taylor theorem, δISU,1 t(X ⋄τn)=∑︂ ℓ(︂ϱ(︁(X ⋄τn)bn ℓ,1∧t)︁−ϱ(︁(X ⋄τn)an ℓ,1∧t)︁)︂ =∑︂ ℓ ϱ1(︁(X ⋄τn)an ℓ,1∧t)︁(︁X1 τn 1(bn ℓ,1∧t) −X1 τn 1(an ℓ,1∧t))︁. By the definitions of X1,X2and ρ, δISU,1 t(X ⋄τn)=∑︂ ℓ Bτn 2(an ℓ,1∧t)(Bτn 1(bn ℓ,1∧t) −Bτn 1(an ℓ,1∧t)) =∑︂ ℓ Bτn 1(bn ℓ,1∧t)(Bτn 1(bn ℓ,1∧t) −Bτn 1(an ℓ,1∧t)) =∑︂ ℓ Btℓ(Btℓ∧t−Btℓ−1∧t) =2∑︂ ℓ (Btℓ+Btℓ−1) 2(Btℓ∧t−Btℓ−1∧t)−∑︂ ℓ Btℓ−1(Btℓ∧t−Btℓ−1∧t) for tn ℓ:= τn 1(bn ℓ,1)=τn 1(an ℓ+1,1)=τn 2(bn ℓ−1,2)=τn 2(an ℓ,2).Let∫︁t 0Bs◦dBsdenote the Stratonovich integral and ∫︁t 0BsdBsthe Itô integral. It holds that δISU,1 t(X ⋄τn)p −→ 2∫︂t 0Bs◦dBs−∫︂t 0BsdBs=1 2B2 t+1 2t for n→∞. By the same arguments, δISU,1 t(X ⋄˜τn)=∑︂ ℓ(︂ϱ(︁(X ⋄˜τn)bn ℓ,2∧t)︁−ϱ(︁(X ⋄˜τn)an ℓ,2∧t)︁)︂ =∑︂ ℓ ϱ1(︁(X ⋄˜τn)an ℓ,2∧t)︁(X2 τn 2(bn ℓ,2∧t) −X2 τn 2(an ℓ,2∧t)) =∑︂ ℓ Bτn 1(an ℓ,2∧t)(Bτn 2(bn ℓ,2∧t) −Bτn 2(an ℓ,2∧t)) =∑︂ ℓ Bτn 2(an ℓ,2∧t)(Bτn 2(bn ℓ,2∧t) −Bτn 2(an ℓ,2∧t)) =∑︂ ℓ Btℓ(Btℓ+1∧t−Btℓ∧t) p −→ ∫︂t 0BsdBs =1 2B2 t−1 2t 1106 G. Junike et al. for n→∞. Therefore, p-limn→∞δISU,1 t(X ⋄τn)= p-limn→∞δISU,i t(X ⋄˜τn), i =1,2, for t>0, and hence the ISU decomposition of ϱ(X) cannot be stable at X.□ Acknowledgements We thank Bernd Buchwald and Andreas Märkert from Hannover Re, Solveig Flaig from Deutsche Rück, Julien Hambuckers from University of Liège, Jan-Frederik Mai from XAIA Investment GmbH, Thorsten Schmidt from Universität Freiburg and an anonymous referee for very helpful comments that improved this paper. Funding information Open Access funding enabled and organized by Projekt DEAL. Declarations Competing interests The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/ 4.0/. References 1. Bielecki, T.R., Cialenco, I., Pitera, M., Schmidt, T.: Fair estimation of capital risk allocation. Stat. Risk. 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