Continuous-Time Risk Contribution and Budgeting for Terminal Variance
aa r X i v : . [ q -f i n . M F ] N ov Continuous-Time Risk Contribution and Budgeting forTerminal Variance
Mengjin ZHAO ∗ , Guangyan JIA † Abstract
Seeking robustness of risk among different assets, risk-budgeting portfolio selectionshave become popular in the last decade. Aiming at generalizing risk budgeting methodfrom single-period case to the continuous-time, we characterize the risk contributions andmarginal risk contributions on different assets as measurable processes, when terminal vari-ance of wealth is recognized as the risk measure. Meanwhile this specified risk contributionhas a aggregation property, namely that total risk can be represented as the aggregationof risk contributions among assets and ( t, ω ). Subsequently, risk budgeting problem thathow to obtain the policy with given risk budget in continuous-time case, is also exploredwhich actually is a stochastic convex optimization problem parametrized by given risk bud-get. Moreover single-period risk budgeting policy is related to the projected risk budget incontinuous-time case. Based on neural networks, numerical methods are given in order toget the policy with a specified budget process.
Portfolio construction techniques based on predicted risk, without expected returns, have be-come popular in the last decade. Mainly with diversifying risks, risk-control techniques throwone’s eyesight on allocating risk from that on money. Directing the applications on portfolioselections in the subsequent decades, mean-variance framework, foundation work of Markowitzfor his Nobel-Prize-winning result, can be considered the origin of risk-based investments [11,12].Together with Markowitz’s pioneering work, analytic solutions of efficient frontier obtained byMerton is considered the fundamental basis for portfolio construction in a single period [13]. Afterthis for years, literatures in multi-period portfolio selections were dominated by the maximizingexpected utility framework, and for this we recommend [5,14,15,19]. Different from expectation-utility framework, efficient frontier associated with variance of terminal wealth were obtained byembedding technique which led the problem into stochastic linear-quadratic framework [8, 21].The key role of the framework in single/multi-period case is the covariance matrix which char-acterizes the risk structure of assets. Sensitivity analysis for mean-variance portfolio selectionswhich substantially are quadratic programming were studied, implying the loss of robustnesswhen given non-stable inputs [1, 2]. Therefore people prefer robust policies instead of sensitiveones in practice with regularization or constraint techniques on their policies. ∗ Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, P.R. China. Email:[email protected] † Corresponding author. Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, P.R.China. Email: [email protected]
1n the view of regularization on policies, one may seek diversification rather than minimiza-tion of mean-variance goal since the solution meeting implementation usually behaves unattrac-tive. 1 /N or equally weighted portfolios can be considered a naive diversification policy withoutknowledge on risk and Markowitz global minimum variance portfolio with predicted covariancematrix is verified with concentrations on low-correlation assets. Between these two portfolioscome maximum diversification portfolios, equal risk contribution portfolios and inverse volatilityportfolios which pursue the diversification on risk [3, 16, 17]. Considering standard deviation asthe risk measure, risk parity problem is analytically discussed and solved by [10]. Just as [9]suggests, allocating risk, the second step after measuring risk, is much of importance. The faithin equal risk contribution or risk parity implies methodology in budgeting risk. A short sum-mary with theoretical and empirical results on risk-based investments in single-period case canbe found in [7]. Readers can also refer to the book [18] for classical theory and applications onrisk parity/budgeting. Continuous-time risk-parity policies, or more generally ones with given risk budgets, are of ourinterest. To make the story clearer, here we shall start with the methodology of risk parityin single-period case. The minimization of standard deviation or variance is actually a familiarmeans in portfolio construction. Working on the probability space (Ω , F , P ), we give an axiomaticdescription and derive the risk contribution in single-period case. Definition 1.1 (Deviation Risk Measure) . A deviation risk measure is a functional ρ D : L (Ω) → [0 , ∞ ] satisfying:D1. ρ D ( X + C ) = ρ D ( X ) for X ∈ L (Ω) and C ∈ R ;D2. ρ D ( hX ) = hρ D ( X ) for all h > ;D3. ρ D ( X + Y ) ≤ ρ D ( X ) + ρ D ( Y ) for X, Y ∈ L (Ω) ;D4. ρ D ( C ) = 0 for C an arbitrary constant, and ρ D ( X ) > for X any non-constant randomvariable.A deviation risk measure is call k -homogeneous if D2 is replaced byD2’. ρ D ( hX ) = h k ρ D ( X ) for all h > . Suppose that there are d assets in the market with R d -valued random vector r , namely thevalue of those assets, and R d -valued u the shares investors hold. Then we have random variable r p = u ⊤ r for the portfolio value. Definition 1.2 (Risk Contribution - Single Period) . Let ρ be a continuously differentiable riskmeasure(not necessary a deviation risk measure). • Marginal risk contribution of portfolio u in a vector style is defined by c := ∂ρ ( v ⊤ r ) ∂v (cid:12)(cid:12)(cid:12) v = u where the i -th element is the marginal risk contribution on i -th asset. Risk contribution of portfolio u in a vector style is χ := u ⊤ c = u ⊙ ∂ρ ( v ⊤ r ) ∂v (cid:12)(cid:12)(cid:12) v = u of which i -element is the risk contribution on i -th asset.Particularly, a portfolio is called Equally-Risk-Contribution(ERCP for short) or risk-parity if χ = u ⊙ c = λe d with notation e d = [1 , ..., ⊤ for some λ ∈ (0 , ∞ ) , namely, χ ( i ) = u ( i ) c ( i ) = u ( j ) c ( j ) = χ ( j ) = λ, for each i, j. (S-ERCP)Substantially based on Euler’s homogeneity theorem, [20] states the following aggregationproperty from an economic point of view d X i =1 u ( i ) maginal z}|{ c ( i ) | {z } i -th risk contribution = ρ D ( r p ) | {z } total risk which is also call Euler decomposition. One can always define the risk contributions of a riskmeasure from Definition 1.2 without homogeneous property, but can not get the aggregationof risk contribution. More generally, throwing light on positive k -homogeneous deviation riskmeasure ρ k , we can modify the marginal risk contribution of ρ k as1 k ∂ρ k ( v ⊤ r ) ∂v (cid:12)(cid:12)(cid:12) v = u to meet aggregation property above.With denotion Σ the covariance matrix of random vector r , it’s clear that variance σ ( u ⊤ r ) := u ⊤ Σ u is a 2-homogeneous deviation risk measure with marginal risk contribution 12 Σ u and thatstandard deviation σ ( u ⊤ r ) = √ u ⊤ Σ u is a 1-homogeneous deviation risk measure with marginalrisk contribution Σ uσ ( u ⊤ r ) . Problem 1.3 (Risk Budgeting - Single Period) . Let ρ be a continuously differentiable risk mea-sure and R d -valued risk budget β are exogenously assigned by the modeller. The modeller issupposed to find a suitable policy u ⋆ satisfying u ⋆ ∈ n u (cid:12)(cid:12)(cid:12) u ⊙ ∂ρ ( v ⊤ r ) ∂v (cid:12)(cid:12) v = u = β o . In other words, the risk contribution of ideal policy u ⋆ for each asset should be assigned theirpre-given risk budget β . For the sake of robustness, investors prefer risk-parity portfolios with variance as their riskmeasure. It has been shown that ERCP can be found by following optimization program [10]: u ⋆ = arg min u − λ d X i =1 log u ( i ) + 12 u ⊤ Σ u We denote a ( i ) as the i -th element of vector-valued a . With R d -valued a and b , vector a ⊙ b is defined by a ⊙ b := [ a (1) b (1) , . . . , a ( d ) b ( d ) ] ⊤ . u ∗ satisfies (S-ERCP). More generally, given R d -valued risk budget β >
0, we also have an associated optimization program u ⋆ = arg min u − d X i =1 β ( i ) log u ( i ) + 12 u ⊤ Σ u (S-Optimizition)whose solution u ∗ is the related policy with risk distribution β . To make the problem well-posed,covariance matrix Σ is assumed to be positive definite.Inspired by the work [21] where risk item is specified by terminal variance of wealth processand that of [10] above, we are interested in the risk contribution in continuous-time case. Asthe echo of single-period risk contribution/budgeting works, aspects listed below are answeredin this paper: • (Defining Risk Contribution) Without the covariance matrix, how to characterize therisk contribution of policies? Moreover, does the risk aggregation property still hold incontinuous-time case? • (Risk Budgeting) Guessing that risk contributions and budgets are stochastic processes,can we raise the inverse problem that how to get the policy coinciding with the pre-givenrisk budget? • (Connection) What is the connection between continuous-time solutions and single-periodones?The rest of our paper is organized as follows. In Section 2, terminal variance acting as a riskmeasure is represented as an unique integral of policy with respect to a signed measure induced bya Gateaux derivative and this signed measure then derive a continuous process which is actuallythe marginal risk contribution. In Section 3, we present a convex stochastic optimization problemsimilar to that in single-period case whose solution is the ideal policy corresponding to our pre-given risk budget process. Here, classic single-period solutions can be seemed as the solutionassociated to the projection of risk budget process onto the naive σ -algebra for measurableprocesses. Explanations of quadratic utility and embedding technique in [21] are presented bya risk contribution means. To give an illustration, SABR model will be discussed in Section 4for risk contributions, risk-parity budgeting and projections which are related to the parametersin model setting. Section 5 gives a numerical method using neural network and Monte Carlosampling to calculating the solution to the optimization in Section 3. Finally in Section 6,we conclude with a discussion of the enhancements from classic case, and we also raise severalpotential limitations as open problems. We will characterize the associated (marginal) risk contribution of the terminal variance actingas risk measure. Marginal risk contribution of a given policy will be firstly derived in a signedmeasure form whom the integral of the policy with respect to is the terminal variance, andthen the Radon-Nikodym derivative of the signed measure is proved to be a continuous processmeaning that total risk is continuously aggregated. Some preliminary settings are listed below.With the time horizon [0 , T ], we let (Ω , F , P ) be the random base equipped with a filtration F = {F t } t ∈ [0 ,T ] and { B t } t ∈ [0 ,T ] be a m -dim Brownian motion on this space. We also assume that F is the natural filtration generated by the Brownian motion B and completed by the collectionof P -null sets N , i.e., F t = σ ( B s , s ≤ t ); F t = F t ∨ N F = F T . And the processes we discussed in this paper are assumed to be F -adapted. Wealso denote Σ p the predictable σ -algebra for processes and H ◦ X the stochastic integral. Severalnecessary spaces are listed below: • With the well-known notation L p = L p (Ω , F ; P ) space on random variables for p ∈ [1 , ∞ ],we introduce a space S ∞ of processes where S ∞ = n X (cid:12)(cid:12)(cid:12)(cid:13)(cid:13) X (cid:13)(cid:13) S ∞ := (cid:13)(cid:13) sup s ≤ T | X s | (cid:13)(cid:13) L ∞ < ∞ o ; • For martingales and p ∈ [1 , ∞ ), we write H = n M (cid:12)(cid:12)(cid:12)(cid:13)(cid:13) M (cid:13)(cid:13) H := (cid:13)(cid:13) sup s ≤ T | M s | (cid:13)(cid:13) L < ∞ o ; • Integrable variation space is taken by A p = n A (cid:12)(cid:12)(cid:12)(cid:13)(cid:13) A (cid:13)(cid:13) A p := (cid:13)(cid:13) Z T | d A s | (cid:13)(cid:13) L p < ∞ with a.a. paths of finite variation o ; • H S , the space for continuous semi-martingales is denoted by H p S = ( X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) h M i / T (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) Z T | d A s | (cid:13)(cid:13)(cid:13) L < ∞ ) . with their canonical decomposition X = X + A + M . (Assets) The value processes of assets are characterised as a sequence of continuous specialsemi-martingales in H S . Without the loss of generality, the dynamics of assets value are assumedto be in the following form ( d S ( i ) t = b ( i ) t d t + σ ( i ) t d B t S ( i )0 = s ( i )0 , for i = 1 , . . . , d where the instantaneous diffusion is a vector-valued process σ ( i ) t , the i -th row of σ t = [ σ ( i,j ) t ] d,mi,j =1 .We shall denote the R d -valued process S = [ S (1) , S (2) , . . . , S ( d ) ] ⊤ as the assets for short. (Policy/Control) The policy u is assumed to be an R d -valued predictable process with u ∈ S ∞ . u t is the shares we buy at time t and the space S ∞ implies that we cannot afford to holdinfinitely many shares in any case. (Investment Process) The Ito-type integral of one policy u with respect to assets S is definedas the value of investment X u where X ut = ( u ⊤ ◦ S ) t = Z t u τ d S τ , for t ∈ [0 , T ] (2.1)is the value of investment X u at time t . Moreover, an investment is called a self-financingportfolio else if X ut = u ⊤ t S t = d X i =1 u ( i ) t S ( i ) t (2.2)for each t . With u ∈ S ∞ and S ∈ H S , it’s easy to check X u is in H S and we can say that X u isa special asset. 5 Terminal Variance as the Risk Measure)
Here we consider the terminal variance Var(( u ◦ S ) T ) of portfolio X u as the risk measure whereVar(( u ◦ S ) T ) = E [( u ◦ S ) T ] − ( E ( u ◦ S ) T ) . It’s easy to check that u Var(( u ◦ S ) T ) is a convex positive 2-homogeneous mapping. Moreover,space H S ensures that the portfolio X u has a second-order moments implying the variance isfinite. Definition 2.1 (Non-Degeneration) . The market is non-degenerate if for arbitrary policy u = 0 we always have Var (( u ◦ S ) T ) > . With the slope-mapping θ θ [Var( X u + θvT ) − Var( X uT )] = 2Cov( X uT , X vT ) + θ Var( X vT ) atpoint u in direction v , non-degeneration condition above ensures that u Var( X uT ) is strictlyconvex. In this subsection, we put d = 1, namely there is only one risky asset in our sight. It is muchimportant through which we can investigate the structure of its terminal variance so that we canget the generalised multi-dimensional result on the base of it.By Itˆo’s formula, we can rewrite the terminal variance of our naive portfolio u ◦ S and divideit into three parts,Var (( u ◦ S ) T )= E (cid:20) ( u ◦ S ) T − ( u ◦ S ) T Z Ω ( u ◦ S ) T ( ω ) P (d ω ) (cid:21) = E " Z T u ◦ S ) t u t d S t | {z } I + Z T u t d h S i t | {z } II − Z T u t d S t Z Ω ( u ◦ S ) T ( ω ′ ) P (d ω ′ ) | {z } III . For the (I) part, we denote the operator of which the image is a random variableΦ I : u Z T u ◦ S ) t u t d S t Similarly we can define the operator Φ , Φ for the (II) and (III) parts:Φ II : u Z T u t d h S i t Φ III : u Z T u t d S t Z Ω ( u ◦ S ) T ( ω ′ ) P (d ω ′ )The images of these three operators are equipped with the L -norm.6 emma 2.2 (Gˆateaux differential) . Under L -norm, the Gˆateaux differentials of Φ I , Φ II and Φ III are dΦ I ( u ; v ) = Z T u ◦ S ) t v t + 2( v ◦ S ) t u t d S t dΦ II ( u ; v ) = Z T u t v t d h S i t dΦ III ( u ; v ) =( v ◦ S ) T Z Ω ( u ◦ S ) T ( ω ′ ) P (d ω ′ ) + ( u ◦ S ) T Z Ω ( v ◦ S ) T ( ω ′ ) P (d ω ′ ) Proof.
With given θ ∈ [0 ,
1] and v ∈ S ∞ , we take the variation of Φ x ( x = I,II,III). For (I) part,we have Φ I ( u + θv ) − Φ I u = Z T h (cid:0) ( u + θv ) ◦ S (cid:1) t ( u + θv ) t − u ◦ S ) t u t i d S t = Z T h ( θv ◦ S ) t u t + 2( u ◦ S ) t θv t + 2( θv ◦ S ) t θv t i d S t = Z T h θ (cid:2) ( v ◦ S ) t u t + ( u ◦ S ) t v t (cid:3) + 2 θ ( v ◦ S ) t v t i d S t by the linearity of stochastic integral. Let L I v := Z T v ◦ S ) t u t + ( u ◦ S ) t v t ]d S t . Obviously, ∀ v ∈ S ∞ the linear operator L I satisfieslim θ → (cid:13)(cid:13)(cid:13)(cid:13) Φ I ( u + θv ) − Φ I uθ − L I v (cid:13)(cid:13)(cid:13)(cid:13) L = lim θ → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T θ ( v ◦ S ) t v t d S t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ≤ lim θ → k ( v ◦ S ) k H S = 0 . Hence the linear operator L I is the Gˆateaux differential of Φ I . Considering that L I v is associatedwith u , we denote it by dΦ I ( u ; v ) in the fashion of directional derivative.Similarly, for the (II)part we have the convex variation of Φ II Φ II ( u + θv ) − Φ II u = Z T θu t v t + θ v t d h S i t and the Gˆateaux differential of Φ II dΦ II ( u, v ) = Z T u t v t d h S i t . As for part (III), we can getΦ
III ( u + θv ) − Φ III u =( θv ◦ S ) T Z Ω ( u ◦ S ) T ( ω ′ ) P (d w ′ ) + ( u ◦ S ) T Z Ω ( θv ◦ S ) T ( ω ′ ) P (d w ′ )+ (( θv ◦ S ) T Z Ω ( θv ◦ S ) T ( ω ′ ) P (d w ′ ) . L III v := ( v ◦ S ) T Z Ω ( u ◦ S ) T ( ω ′ ) P (d w ′ ) + ( u ◦ S ) T Z Ω ( v ◦ S ) T ( ω ′ ) P (d w ′ ) . Then we get lim θ → (cid:13)(cid:13)(cid:13)(cid:13) Φ ( u + θv ) − Φ uθ − A v (cid:13)(cid:13)(cid:13)(cid:13) L = 0since θ (cid:13)(cid:13)(cid:13)(cid:13) ( v ◦ S ) T Z Ω ( v ◦ S ) T ( ω ′ ) P (d w ′ ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ θ E [ | ( v ◦ S ) T | ] (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( v ◦ S ) T ( ω ′ ) P (d w ′ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ (cid:0) E [ | ( v ◦ S ) T | ] (cid:1) < ∞ . Finally the Gˆateaux differential of Φ
III at u reads dΦ III ( u ; v ) = L III v . Definition 2.3 (Dol´eans measure, [4]) . If A ∈ A is non-decreasing, then a non-negative measure µ A can be defined on ([0 , T ] × Ω , B [0 ,T ] ⊗ F ) by µ A ( C ) = E [(1 C ◦ A ) T ] = E "Z T C d A s for each set C ∈ B ⊗ F . We call µ A the Dol´eans measure associated with A . We shall introduce a signed measure to distribute the terminal risk over the time interval[0 , T ]. Theorem 2.4 (Marginal Risk Measure - 1-dim) . Given d = 1 , the terminal variance of aportfolio can be represented as an integralVar (( u ◦ S ) T ) = 12 Z [0 ,T ] × Ω u ( t, ω ) µ (d t, d ω ) (2.3) where µ ( E ) = E [dΦ I ( u, E )+ dΦ II ( u, E )+ dΦ III ( u, E )] , E ∈ Σ p . µ ( E ) is called the risk gradientmeasure of Var (( u ◦ S ) T ) at u .Moreover within non-degenerate market, the signed measure µ introduced by the mapping u µ u above is unique on the support supp( u ) = { u = 0 } .Proof. When we take the direction policy v in the form of 1 E where E is a Σ p measurable set,the Gˆateaux differentials in the Lemma 2.2 give us three set functions with the common domainΣ p µ I ( E ) := E [dΦ I ( u, E )] µ II ( E ) := E [dΦ II ( u, E )] µ III ( E ) := E [dΦ III ( u, E )] . We want to show that µ x ( x = I,II,III) are signed measures so that we can get the result onrepresentation by integrating our policy u with respect to the summation of these µ x . For µ I , µ II and µ III , it is easy to check the following properties Measure µ is actually induced by u , therefore we use the notation µ u . Without causing ambiguity, we prefer µ . µ x ( ∅ ) = 0 • µ x ([0 , T ] × Ω) < ∞ • (Finite Additivity). For arbitrary disjoint E , E ∈ Σ p , we have µ x ( E + E ) = µ x ( E ) + µ x ( E ) . And it is left us to seek the countably additivity property of µ x .For µ I part, we should notice that µ I ( E ) = E [dΦ I ( u, E )] = E Z T (cid:16) u ◦ S ) t E + 2(1 E ◦ S ) t u t (cid:17) d F t where F is the part of finite variation in the canonical decomposition of S . With the decompo-sition of u = u + − u − , F = F + − F − and S = ( F + + M + ) − ( F − + M − ) = S + − S − , we canwrite the first item ν ( E ) = E Z T u ◦ S ) t E d F t = E Z T (cid:0) ( u + − u − ) ◦ ( S + − S − ) (cid:1) t E d( F + t − F − t )= E Z T (cid:0) ( u ◦ S ) + t − ( u ◦ S ) − t (cid:1) E d( F + t − F − t )=2 E Z T E (cid:16) ( u ◦ S ) + t d F + t + ( u ◦ S ) − t d F − t (cid:17) − E (cid:16) ( u ◦ S ) − t d F + t + ( u ◦ S ) + t d F − t (cid:17) . By Definition2.3, obviously ν is a combination of Dol´eans measures induced by four increasingprocess, hence a signed measure. And the decomposition of the second item(denoted by ν ) reads ν ( E ) = E Z T E ◦ S ) t u t d F t =2 E Z T (cid:16) E ◦ ( S + − S − ) (cid:17) t ( u + t − u − t )d( F + t − F − t )=2 E Z T (cid:16) (1 E ◦ S + ) t u + t dF + t + (1 E ◦ S + ) t u − t dF − t + (1 E ◦ S − ) t u + t dF − t + (1 E ◦ S − ) t u − t dF + t (cid:17) + (cid:16) (1 E ◦ S + ) t u + t dF − t + (1 E ◦ S + ) t u − t dF + t − (1 E ◦ S − ) t u + t dF + t + (1 E ◦ S − ) t u − t dF − t (cid:17) . Noticing that (cid:8) n X i =1 E i (cid:9) n is a non-decreasing sequence for disjoint set sequence { E i } i , we take f + , + n = (cid:16) n X i =1 E i ◦ S + (cid:17) u + , then n f + , + n o n is also a non-decreasing sequence and respectivelynon-decreasing n − f + , − n o n , n − f − , + n o n , n f − , − n o n . Applying monotone convergence theorem,we can get the countably additivity of ν . Consequently µ is a signed measure on Σ p .As for µ II , it can be considered the difference of two Dol´eans measures induced by tworespected processes u + ◦ h S i and u − ◦ h S i since µ II ( E ) = 2 E Z T E u t d h S i t = 2 E Z T E ( u + t − u − t )d h S i t = 2 µ u + ◦h S i ( E ) − µ u − ◦h S i ( E ) . µ II is a signed measure.Within the decomposition technique we treat ν with, we can see that µ III ( E ) = 2 E [(1 E ◦ S ) T ] E [( u ◦ S ) T ] = 2 E (cid:2)(cid:0) E ◦ ( F + − F − ) (cid:1) T (cid:3) E [( u ◦ S ) T ] . Then µ is a signed measure on Σ p .Finally we take µ := µ I + µ II + µ III , and the integral of u with respect to µ shows us Z [0 ,T ] × Ω u ( t, ω ) µ (d t, d ω ) = 2Var( X uT )If there is a signed measure µ ′ also representing the integral above, then we have Z [0 ,T ] × Ω u ( t, ω )d( µ − µ ′ ) = 0for arbitrary u . This singular result together with the non-degenerate market ensures the unique-ness of mapping u µ u . Remark 2.5.
For any C -continuous positive homogeneous function f ( x ) of order k , we havethe Euler’s homogeneity theorem to characterise it kf ( x ) = (cid:28) ∂f∂x (cid:12)(cid:12)(cid:12) x = x , x (cid:29) where ∂f∂x (cid:12)(cid:12)(cid:12) x = x = ∇ f ( x ) is the gradient of f at x and also the marginal risk contribution inDefinition1.2. Compared with the property above, Theorem2.4 inherits that in the case k = 2 .That is the reason why we call µ marginal risk measure. The risk of a portfolio, terminal risk in our model, is supposed to be accumulated over the timeinterval [0 , T ] and cases Ω where the investment value process behaves differently. In the languageof integral, the terminal variance of our naive portfolio should be in the following suspicious formVar( X uT ) ? = E Z T u t c t d t where the measurable process c ( t, ω ) can be considered the instantaneous marginal risk contri-bution to the whole risk. Noticing the terminal variance is actually an integral by Theorem 2.4,the guess above substantially is the representation of marginal risk measure. Theorem 2.6 (Flow Representation - 1-dim) . Given non-degenerate market, the signed measure µ on Σ p derived in Theorem2.4 can be uniquely represented as µ ( E ) = E Z T E ( t, ω )d C ( t, ω ) = E Z T E ( t, ω ) c ( t, ω )d t (2.4) where the measurable set E is taken in Σ p and the process C is continuous in A . As theinstantaneous marginal risk contribution, the process c is the Radon-Nikodym derivative of C with respect to t . roof. Noticing that the predictable σ -algebra Σ p is generated by the simple left-continuousprocesses in the form φ ( s,t ] where s, t ∈ [0 , T ] and φ ∈ F s , we can just take E = ( s, t ] × H, H ∈ F s the simple form into consideration. The measure of E in the form above can be split as µ (( s, t ] × H ) = µ (( s, T ] × H ) − µ (( t, T ] × H ) . We can define a sequence of set functions on F t parametrised by t as m t ( H ) := µ (( t, T ] × H )where m t is defined on the σ -algebra F t .We claim that m t is a signed measure on F t absolutely continuous with respect to P (cid:12)(cid:12) F t sinceit is induced by signed µ . It’s left to show the absolute continuity of m t for every fixed index t . F is taken P -null sets with F ∈ F t and we take E = ( t, T ] × F . Three parts of µ in Theorem2.4satisfies the following properties. • For µ I part, we have µ I ( E ) = E Z T (cid:16) u ◦ S ) τ E ( τ, ω ) + 2(1 E ◦ S ) τ u τ d S τ (cid:17) = E Z T (cid:16) u ◦ S ) τ ( t,T ] × F ( τ, ω ) + 2(1 ( t,T ] × F ◦ S ) τ u τ d S τ (cid:17) = E (cid:16) Z T u ◦ S ) τ ( t,T ] ( τ )1 F ( ω )d S τ (cid:17) + 2 E (cid:16) Z T (cid:0) F ◦ (1 ( t,T ] ◦ S ) (cid:1) τ u τ d S τ (cid:17) = E (cid:16) F Z Tt u ◦ S ) τ d S τ (cid:17) + E (cid:16) F Z Tt ( t,T ] ◦ S ) τ u τ d S τ (cid:17) =0since F is F t -measurable. • In a same manner, µ II part reads µ II ( E ) = E Z T E ( τ, ω ) u τ d h S i τ = E (cid:16) F Z Tt u τ d h S i τ (cid:17) = 0 . • As for µ III part, only to notice (1 E ◦ S ) T = 1 F ( S T − S t ), we have µ III ( E ) − E (cid:2) (1 E ◦ S ) T (cid:3) E (cid:2) ( u ◦ S ) T (cid:3) = 2 E (cid:2) F ( S T − S t ) (cid:3) E (cid:2) ( u ◦ S ) T (cid:3) = 0 . Consequently, for every t and P -null set F ∈ F t we have m t ( F ) = µ (( t, T ] × F ) = µ (( t, T ] × F ) + µ (( t, T ] × F ) + µ (( t, T ] × F ) = 0hence m t ≪ P (cid:12)(cid:12) F t .For every index t , we write A t as the Radon-Nikodym derivative of m t with respect to P (cid:12)(cid:12) F t .We see that for every F T -measurable set F , we have E [1 F A + T ] < ∞ E [1 F A − T ] < ∞ µ and the Hahn-Jordan decomposition of A , and hencesup (cid:16) X i (cid:12)(cid:12) A t i +1 ( ω ) − A t i ( ω ) (cid:12)(cid:12) (cid:17) < ∞ for a.a. ω where the supremum is taken over the possible partitions of [0 , T ]. And then the process A is anintegrable process of finite variation. To see that A is right-continuous, we should notice (cid:26) inf r>t A + r ∈ [ c, ∞ ) (cid:27) = \ r>t (cid:8) A + r ∈ [ c, ∞ ) (cid:9) ∈ F t + = F t . Let { t n } n ↓ t with H ∈ F t (hence ( t n , T ] × H ↑ ( t, T ] × H ), and also we have E (cid:2) H A + t (cid:3) = µ + (( t, T ] × H ) = lim t n ↓ t m + t n ( H ) = lim t n ↓ t E (cid:2) H A + t n (cid:3) = E h lim n (cid:0) H A + t n (cid:1)i = E (cid:2) H A + t + (cid:3) . Then the process A has a right-continuous version since sup r>t A − r is handled in a same manner.Living in the natural filtration generalized by Brownian motion B , for an arbitrary set H ∈ F t we have a representation H = { ω | ( B t ( ω ) , B t ( ω ) , . . . , B t n ( ω ) , . . . ) ∈ G } where { t i } i is a sequence of countable points with t i ≤ t and G ∈ B ( R ∞ ) with B ( R ∞ ) given by ⊗ i ∈ N B ( R ). Here we let H n be the projection of H in head-most coordinates ( t , t , . . . , t n ), andwe have ( t ′ n , T ] × H n ↓ ( t, T ] × H with t ′ n := sup i ≤ n t i and H n ∈ F t ′ n . Process A is left-continuoussince with ∗ = + , − E [1 H A ∗ t ] = µ ∗ (( t, T ] × H ) = lim n m ∗ t ′ n ( H n ) = lim n E h H n A ∗ t ′ n i = E h lim n (cid:16) H n A ∗ t ′ n (cid:17)i = E (cid:2) H A ∗ t − (cid:3) by the continuity of µ from above. Consequently, we can say that A is an integrable continuousmeasurable process of finite variation.Finally we can see that µ (( s, t ] × H ) = E [1 H A s − H A t ] = E (cid:20) − Z ts H ( ω )d A τ ( ω ) (cid:21) = E " − Z T E ( τ, ω )d A τ ( ω ) . Conversely, if there is a process A ′ satisfies equation (2.4), A ′ is distinguishable from A sincethe measure introduced by A is unique up to distinguishability.With C := − A and c ( t, ω ) the Radon-Nikodym derivative of { C ( t, ω ) } t with respect toLebesgue measure, we complete the proof. When it comes to a generalized multi-asset situation, we should stress the benefits why we takeinvestment value processes in equation2.1 into consideration instead of self-financing portfoliousually characterized by a linear stochastic differential equation. Here are the reasons: • With d = 1, a self-financing portfolio acts only in a form X ut = u S t where u is aconstant determined by investor’s initial wealth, instead of the generalized form X u =( u ◦ S ) t . Through the method in Theorem2.4 and 2.6 it’s hard to derive the (marginal)risk contribution in this constant policy case.12 With d >
1, people prefer a controlled linear stochastic differential equation to characteringthe value process of the self-financing portfolio where control is usually a ( d − d -dimprocess instead of ( d − • Substantially in mathematics, value process of our investment should be linear with respectto our policy u , and stochastic integral u ◦ S keeps this property of importance.Comparing to classical single-period case, risk contribution is certainly related to the structureof covariance matrix. Result following can be viewed as a generalization of Theorem2.4 andfurther gives a characterization of the covariance-like structure. Theorem 2.7 (Risk Gradient Measure - Generalized Version) . For a generalized portfolio in thecase where dimension d > , its terminal variance can be uniquely expressed asVar ( X uT ) = d X i =1 Z [0 ,T ] × Ω u ( i ) ( t, ω ) µ i (d t, d ω ) + d X i =1 d X j =1 ,j = i Z [0 ,T ] × Ω u ( i ) ( t, ω ) η i,j (d t, d ω ) (2.5) where µ i is the marginal risk measure of i -th asset S ( i ) mentioned in Theorem2.4 and η i,j is anew signed measure related to mutual effect between i -th and j -th assets for each i, j , or brieflyVar ( X uT ) = Z [0 ,T ] × Ω h u ( t, ω ) , µ (d t, d ω ) i where µ is the product measure defined by µ := d Y i =1 (cid:0) µ i + X j = i η i,j (cid:1) . roof. Noticing u ⊤ ◦ S = d X i =1 u ( i ) ◦ S ( i ) , we can writeVar(( u ⊤ ◦ S ) T )= E " d X i =1 Z T (( u ( i ) ◦ S ( i ) ) t u ( i ) t d S ( i ) t | {z } I-( i,i ) + X i = j Z T ( u ( i ) ◦ S ( i ) ) t u ( j ) t d S ( j ) t | {z } I-( i,j ) + d X i =1 Z T u ( i )2 t d h S ( i ) i t | {z } II-( i,i ) + X i = j Z T u ( i ) t u ( j ) t d h S ( i ) , S ( j ) i t | {z } II-( i,j ) − d X i =1 Z T u ( i ) t d S ( i ) t Z Ω ( u ( i ) ◦ S ( i ) ) T ( ω ′ ) P (d ω ′ ) | {z } III-( i,i ) − X i = j Z T u ( i ) t d S ( i ) t Z Ω ( u ( j ) ◦ S ( j ) ) T ( ω ′ ) P (d ω ′ ) | {z } III-( i,j ) = d X i =1 Z [0 ,T ] × Ω u ( t, ω ) µ i (d t, d ω ) + E " X i = j Z T ( u ( i ) ◦ S ( i ) ) t u ( j ) t d S ( j ) t | {z } I-( i,j ) + X i = j Z T u ( i ) t u ( j ) t d h S ( i ) , S ( j ) i t | {z } II-( i,j ) − X i = j Z T u ( i ) t d S ( i ) t Z Ω ( u ( j ) ◦ S ( j ) ) T ( ω ′ ) P (d ω ′ ) | {z } III-( i,j ) by Itˆo’s formula and Theorem2.4. Firstly for convenience we give some notations of those partsby taking operators Ψ i,j I u := Z T ( u ( i ) ◦ S ( i ) ) t u ( j ) t d S ( j ) t , Ψ i,j II u := Z T u ( i ) t u ( j ) t d h S ( i ) , S ( j ) i t , Ψ i,j III u := Z T u ( i ) t d S ( i ) t Z Ω ( u ( j ) ◦ S ( j ) ) T ( ω ′ ) P (d ω ′ ) . We use the notation 1 E ( t, ω ) = [1 E (1) ( t, ω ) , . . . , E ( d ) ( t, ω )] ⊤ where E = × di =1 E ( i ) to treat multi-dim case.To deal with part I-( i, j ), we can see that L i,j I v := Z T (cid:16) ( v ( i ) ◦ S ( i ) ) t d( u ( j ) ◦ S ( j ) ) t + ( u ( i ) ◦ S ( i ) ) t d( v ( j ) ◦ S ( j ) ) t (cid:17) is actually the Gˆateaux differential of Ψ i,j I under L -norm and we denote it by dΨ i,j I ( u ; v ).Similarly to Theorem 2.7, the signed measure η i,j on Σ p induced by dΨ i,j I ( u ; v ) reads η i,j I ( E ( i ) ) := E [dΨ i,j I ( u ; 1 E )]= E h Z T (cid:16) E ( i ) ◦ S ( i ) (cid:17) t u ( j ) t dS ( j ) t + (cid:16) u ( i ) ◦ S ( j ) (cid:17) t E ( j ) dS ( j ) t i . i, j ) and III-( i, j ), can also be treated in a same approach and derivethe associated signed measures η i,j II ( E ( i ) ) and η i,j III ( E ( i ) ) via finding their Gˆateaux differentialsdΨ i,j II ( u ; v ) and dΨ i,j III ( u ; v ). Aggregating the measures η i,jx ( x = I, II, III) with fixed pair ( i, j ),we can get the associated signed measure η i,j on Σ ( i ) p . Then for every pair ( i, j ) we have Z [0 ,T ] × Ω u ( i ) ( t, ω ) η i,j (d t, d ω )= Z [0 ,T ] × Ω u ( i ) ( t, ω ) η i,j I (d t, d ω ) + Z [0 ,T ] × Ω u ( i ) ( t, ω ) η i,j II (d t, d ω ) − Z [0 ,T ] × Ω u ( i ) ( t, ω ) η i,j III (d t, d ω )= E h Z T (2( u ( i ) ◦ S ( i ) ) t u ( j ) t d S ( j ) t | {z } I-( i,j ) + Z T u ( i ) t u ( j ) t d h S ( i ) , S ( j ) i t | {z } II-( i,j ) + Z T u ( i ) t d S ( i ) t Z Ω ( u ( j ) ◦ S ( j ) ) T ( ω ′ ) P (d ω ′ ) | {z } III-( i,j ) i . Finally, with the aggregation of measures µ i and η i,j we get the Equation 2.5.Uniqueness of the expression in Equation 2.5 is ensured by taking indicator functions 1 E overthe σ -algebra Σ p in the manner from Theorem2.7.Comparing result above to single-period case with variance as the risk measure in Definition1.2where ∂ρ ( v ⊤ r ) ∂v (cid:12)(cid:12)(cid:12) v = u single = 2Σ single u single gives structure of marginal risk contribution, we can say that product measure µ represent theproperties of Σ single u single somehow and respectively σ single i,j u single j associated to η i,j . Actuallywhenever S is a vector-valued martingale, product measure µ ( E ) can be calculated as µ ( E ) = E h Z T E ⊙ ( σ t σ ⊤ t u t )d t i = E h Z T E ⊙ (cid:10) u t , d h S i t (cid:11)i for some predictable set E . Corollary 2.8 (Covariance) . Terminal covariance of two portfolio X u and X v can be calculated Cov( X uT , X vT ) = Z [0 ,T ] × Ω v ( t, ω ) µ u (d t, d ω ) = Z [0 ,T ] × Ω u ( t, ω ) µ v (d t, d ω ) . Proof.
In the manners same to Theorem2.4 and Theorem2.7 by deriving the Gateaux derivativetogether with the symmetry of bilinear mapping Cov( · , · ), we can easily get the result. Corollary 2.9 (Linearity of Marginal Contribution) . Risk gradient measure µ u induced by policy u in Theorem2.7 is linear in u .Proof. Only noticing dΦ( u ; v ) and dΨ( u ; v ) are both linear to u for arbitrary directions v , wecan get the assertion above. 15 orollary 2.10. With the notation E ( t, ω ) = [1 E (1) ( t, ω ) , . . . , E ( d ) ( t, ω )] ⊤ , the marginal riskmeasure µ in Theorem2.7 for an investment X u can be uniquely represented as the vector inte-gration µ ( E ) = E Z T E ( t, ω ) ⊙ d C ( t, ω ) = E Z T E ( t, ω ) ⊙ c ( t, ω )d t by some continuous adapted process C with c as the Radon-Nikodym derivative of C with respectto Lebesgue measure.Proof. Taking E (1) = ( t, T ] × H the section set of E , we can check that m t ( H ) := ˜ µ (cid:16)(cid:0) ( t, T ] × H (cid:1) × ∅ × · · · × ∅ | {z } d − (cid:17) is a signed measure over F t absolutely continuous with respect to P (cid:12)(cid:12) F t . Hence we can write C (1) t the Radon-Nikodym derivative of m t . Varying t over time interval [0 , T ], we can get a process C (1) by what we have done in Theorem2.6. Similarly C ( i ) for i = 1 , . . . , d are calculated in asame manner. Finally we put them together C := [ C (1) , · · · , C ( d ) ] ⊤ and get our result.The significance of expression above is that it can help us seek the structure of terminalvariance and how the instantaneous risk contributions be aggregated. Finally, we give the formaldefinition of continuous-time risk contribution also the centre concept in this paper. Definition 2.11 ((Marginal) Risk Contribution - Continuous-time) . Given a R d -valued policy u within c defined in Corollary2.10, we can define c ( i ) ( t, ω ) the marginal risk contribution of i -thasset at time t in the case ω and respectively u ( i ) ( t, ω ) c ( i ) ( t, ω ) the risk contribution.Particularly, policy u is said to be purely risk-parity if the associated risk distribution satisfies u ( i ) ( t, ω ) c ( i ) ( t, ω ) = λ for a.a. ( t, ω ) (C-ERCP) namely u ⊙ c is a constant vector valued process λe d with some constant λ > . As a result, the risk of an investment X u can be represented asVar( X uT ) = E Z T u ⊤ t c t d t (2.6)implying that the total risk is accumulated continuously among the support of u . Corollary 2.12 (Continuity of Marginal Risk Contribution on Policy) . Suppose we have twopolicies u and u ′ with notation δu = u − u ′ . We have an estimation on marginal risk contribution E " Z T ⊤ E c δut d t ≤ K (cid:13)(cid:13) δu (cid:13)(cid:13) S ∞ for arbitrary predictable sets E . roof. For convenience here we provide the proof for case d = 1, and the left part for multi-dimcase can be easily get in a same manner. Starting with three Gateaux differential originallyappearing in Lemma 2.2, we have E [dΦ I ( u ; v )] ≤ E " sup s ≤ T (cid:12)(cid:12) u s (cid:12)(cid:12) Z T (cid:16) S t v t + ( v ◦ S ) t (cid:17) d S t ≤ (cid:13)(cid:13) u (cid:13)(cid:13) S ∞ (cid:13)(cid:13) v (cid:13)(cid:13) S ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z T S t d S t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L , E [dΦ II ( u ; v )] ≤ E h sup s ≤ T (cid:12)(cid:12) u s (cid:12)(cid:12) sup s ≤ T (cid:12)(cid:12) v s (cid:12)(cid:12) h S i T i = 2 (cid:13)(cid:13) u (cid:13)(cid:13) S ∞ (cid:13)(cid:13) v (cid:13)(cid:13) S ∞ (cid:13)(cid:13)(cid:13)(cid:10) S (cid:11) / T (cid:13)(cid:13)(cid:13) L , E [dΦ III ( u ; v )] ≤ (cid:13)(cid:13) u (cid:13)(cid:13) S ∞ (cid:13)(cid:13) v (cid:13)(cid:13) S ∞ E (cid:2)(cid:12)(cid:12) S T (cid:12)(cid:12)(cid:3) . Finally replacing u by δu and v by 1 E , with representation in Theorem 2.7 we get the 1-dimresult E h Z T E c δut d t i ≤ K k δu k S ∞ , ∀ E ∈ Σ p . Compared to classical result, except distinction among assets, (marginal) risk contribution candiffer among time [0 , T ] and cases in Ω. These two newly-presented phenomena are of our interestin allocating risk among not only assets but also different ( t, ω ). Having depicted the structure ofterminal variance, we are facing the risk budgeting problem. Within fixed assets, we write D thecollection of risk distributions induced by all possible policies in S ∞ . That means the elementsin D can be reached by applying some policies. Problem 3.1 (Risk Budgeting - Continuous-time) . With terminal variance as the risk measureof investments and a continuous R d -valued process β ∈ D , the modellers seek to find suitablepolicies u ⋆ satisfying u ⋆t ⊙ c u ⋆ t = β t for each t . Put in the single-period case, getting a policy with given risk budget is equal to solving the opti-mization problem(C-ERCP). Moreover, it’s left us to envisage the connection between continuous-time policy with risk budget β and the optimization problemMinimize J ( u ) = E h Z T − d X i =1 β ( i ) t log u ( i ) t d t i + Var[( u ◦ S ) T ] (C-Optimization)over possible policies u ∈ S ∞ . 17 heorem 3.2 (Continuous-time Risk Budgeting with Positive Budget) . Given positive R d -valuedprocess β ∈ D , optimization problem (C-Optimization) has at least one solution u ∗ in S ∞ . And u ∗ is just the solution of Problem 3.1 with pre-given risk budget β .The solution is unique else if the condition in Definition2.1 is satisfied.Proof. Noticing J ( · ) is a convex mapping when given a strictly positive coefficient β , we onlyneed to seek the first order condition of the functional J ( · ) which reads lim θ ↓ J ( u ⋆ + θv ) − J ( u ⋆ ) θ = E h Z [0 ,T ] × Ω − β ( t, ω ) ⊙ v ( t, ω ) ./u ⋆ ( t, ω ) + v ( t, ω ) ⊙ c u ⋆ ( t, ω )d t i for arbitrary policies v . Hence we have − β ( i ) ( t, ω ) + u ⋆ ( i ) ( t, ω ) c u ⋆ ( i ) ( t, ω ) = 0 , for a.a. ( t, ω ) in Σ p and i = 1 , . . . , d.β can be reached by some u ⋆ (may not unique) since β ∈ D .As for the uniqueness, non-degeneration in Definition2.1 implying the strict convexity of J ( · )which ensures the uniqueness of solution u ⋆ . Remark 3.3 (Connection to Diversifying Assets Shares) . Optimization problem above can alsothought as a constrained convex minimization of terminal variance, i.e.,minimize Var ( X uT ) s.t. u ∈ C, where the convex constraint area is given by − log u ( i ) ( t, ω ) ≤ α ( i ) ( t, ω ) for a.a. ( t, ω ) in Σ p and i = 1 , . . . , d. This convex constraint area C somehow can be recognized as a regulation preventing the concen-tration of shares u around zero and for each ( t, ω ) Karush-Kuhn-Tucker multiplier β ( t, ω ) canbe considered as the log-tolerance around α ( t, ω ) when varying u . Therefore the optimizationproblem is equivalent to seeking a trade-off between maximizing diversification and minimizingtotal risk. Though continuous-time risk-parity case is covered by Theorem 3.2, risk budget β with neg-ativeness on negative support C − ⊂ ([0 , T ] × Ω) d is still interesting. In the view of regulation inRemark 3.3, we can consider the constraint area with two part ( − log u ( j ) ( t, ω ) ≤ α ( j ) ( t, ω ) for some ( t, ω ) ∈ C + and j = 1 , . . . , d with positive budget; − log − u ( k ) ( t ′ , ω ′ ) ≤ α ( k ) ( t ′ , ω ′ ) for some ( t ′ , ω ′ ) ∈ C − and k = 1 , . . . , d with negative budget.and hence we can deduce a generalized optimizationMinimize J ( u ) = E h Z T − d X i =1 β ( i ) t log( δ ( i ) t u ( i ) t )d t i + Var[( u ◦ S ) T ]where risk budget β can be negative on C − and { , − } -valued sign process δ is also negativeon C − . This optimization is still convex and we can get from the solution that negative risk For two non-zero R d vectors a and b , a./b = [ a (1) b (1) , . . . , a ( d ) b ( d ) ] ⊤ . C − . We can also see that u andits opposite policy − u have the same risk distribution.However, we should notice that the sign of ˙ u ( i ) ( t, ω ) ˙ c ( i ) ( t, ω ) and ˙ u ( i ) ( t, ω ) may be oppositefor some special ˙ u . These policies cannot be reached by convex optimization above, but we knowthat ˙ u live in the saddle points of J ( u ) = E h Z T − d X i =1 ˙ β ( i ) t log | u ( i ) t | d t i + Var[( u ◦ S ) T ]with ˙ β ∈ D . Corollary 3.4 (Projection) . With integrable positive risk budget β not necessarily in D andadmissible policy set U = S ∞ ∩ L (Σ ′ p ) where Σ ′ p is a sub σ -algebra of Σ p , the solution ofoptimization min u ∈U J ( u ) = E h Z T − d X i =1 β ( i ) t log u ( i ) t d t i + Var [( u ◦ S ) T ] gives a policy u ⋆ whose associated risk distribution is the projection of β onto Σ ′ p .Proof. The assertion can be easily get from the first order condition E h Z [0 ,T ] × Ω − β ( t, ω ) ⊙ E ( t, ω ) ./u ⋆ ( t, ω ) + 1 E ( t, ω ) ⊙ c u ⋆ ( t, ω )d t i = 0and − ¯ β ( i ) ( t, ω ) + u ⋆ ( i ) ( t, ω ) c u ⋆ ( i ) ( t, ω ) = 0 , for a.a. ( t, ω ) in Σ ′ p and i = 1 , . . . , d where ¯ β is the projection within measure P × Lebesgue.Sometimes it’s not easy to find a policy whose risk contribution happens to be the givenbudget β particularly when we are handling less information. The projection corollary aboveillustrates that both single-period risk contributions and budgeting problems are degenerate casesof continuous-time ones.Different from the usual Expectation-Utility problems, optimization(C-Optimization) is nota pure stochastic control problem. Though embedding technique is used to solve a mean-variancefrontier which is converted to a stochastic control LQ problem, here we also adopt this idea inorder to stress its connection to risk contribution. Lemma 3.5 (Embedding, [21]) . Denote C ( γ ) the solutions of following parametrized auxiliaryproblem minimize ˆ J ( u ; γ ) = E h Z T − d X i =1 β ( i ) t log u ( i ) t d t + γ ( u ◦ S ) T + ( u ◦ S ) T i with same condition in Theorem 3.2. We have u ⋆ ∈ [ γ ∈ R C ( γ ) . And moreover u ⋆ is optimal for C ( γ ⋆ ) where γ ⋆ = − E [( u ⋆ ◦ S ) T ] . roof. We define a function f ( x, y, z ) = x − y + z which is a concave function with f (cid:16) E (cid:2) ( u ◦ S ) T (cid:3) , E (cid:2) ( u ◦ S ) T (cid:3) , E h Z T − d X i =1 log u ( i ) t d t i(cid:17) = J ( u ) . If u ⋆ is not optimal for C ( γ ⋆ ), there will be a u ′ optimal for ˆ J ( · ; γ ⋆ ) satisfying E h Z T − d X i =1 β ( i ) t log u ⋆ ( i ) t d t + γ ( u ⋆ ◦ S ) T +( u ⋆ ◦ S ) T i > E h Z T − d X i =1 β ( i ) t log u ′ ( i ) t d t + γ ( u ′ ◦ S ) T +( u ′ ◦ S ) T i . On the one hand, the convexity of f implying f (cid:16) E (cid:2) ( u ⋆ ◦ S ) T (cid:3) , E (cid:2) ( u ⋆ ◦ S ) T (cid:3) , E h Z T − d X i =1 log u ⋆t ( i ) d t i(cid:17) + 2 E (cid:2) ( u ⋆ ◦ S ) T (cid:3)(cid:16) E (cid:2) ( u ′ ◦ S ) T (cid:3) − E (cid:2) ( u ⋆ ◦ S ) T (cid:3)(cid:17) + (cid:16) E (cid:2) ( u ′ ◦ S ) T (cid:3) − E (cid:2) ( u ⋆ ◦ S ) T (cid:3)(cid:17) + (cid:16) E h Z T − d X i =1 log u ′ ( i ) t d t i − E h Z T − d X i =1 log u ⋆t ( i ) d t i(cid:17) ≥ f (cid:16) E (cid:2) ( u ′ ◦ S ) T (cid:3) , E (cid:2) ( u ′ ◦ S ) T (cid:3) , E h Z T − d X i =1 log u ′ ( i ) t d t i(cid:17) which induces a contradiction to original problem and hence we get the result. J ( u ⋆ ) ≤ J ( u ′ ). Remark 3.6.
With Lemma3.5 the portfolios aiming at allocating risk can be reached by solvingsuch a stochastic optimal control problem which is equivalent to the original problem(C-Optimization).Moreover, thinking through the item E [ γ ( u ◦ S ) T ] and comparing it to µ III = E [dΦ III ( u ; 1 E )] de-fined in Theorem2.4, we can define a new signed measure ˆ µ III ( E ) = E [ γ Z T E d S t ] where γ can be seemed as a modification of E [( u ⋆ ◦ S ) T ] . And when we get the right guess γ ⋆ , therelated ˆ µ III meets true µ III . Actually in the view of risk contribution, this embedding techniquegives a modification of risk contribution.
Here this section we give a risk-budgeted forward interest rate case, the dynamic of which isannounced by SABR model, to illustrate the connection between classical risk contributions inDefinition 1.2 and continuous-time ones specified in Definition 2.11.Known as a stochastic volatility model which is widely used in the financial industry espe-cially in the interest rate derivative markets, SABR model was originally presented in [6] tocapture the volatility smile for improvement of Markovian local volatility models. Just as thename ’stochastic, alpha, beta, rho’ implies, we consider a single forward interest rate with thestochastic-volatile underlying dynamic ( d F t = σ t F βt d W ,t F = f , ( d σ t = ασ t d W ,t σ = s h W , W i t = ρ d t . Controlling the implied at-the-money volatilities of European options,parameter α ≥ F . Power β ∈ [0 ,
1] characterizesthe curvature of implied at-the-money volatilities. Remained constant ρ ∈ ( − ,
1) reflects thecorrelation of two risk sources W and W . For convenience equivalent Brownian motion ( B , B )is defined by d W ,t = d B ,t , d W ,t = ρ d B ,t + p − ρ d B ,t . We now explore the risk contribution of some given policy u . Terminal variance is given byVar(( u ◦ F ) T ) = E [( u ◦ F ) T ] = E [ Z T u t σ t F βt d t ] (4.1)from which we get the marginal risk contribution ( c t ) t in an exponential martingale fashion c t = u t σ t F βt = u t s E ( αρB ) t E ( α p − ρ B ) t F βt . (4.2)Essentially the parameters setting in SABR model are given for calibration of volatility smile,however they can also influence our risk contribution in several aspects: • Prospective level of stochastic volatility s results in a positive relationship to marginal riskcontribution; • Stochastic exponential gives a log-normal distribution of volatility varying through ( t, ω )with scale factor α ; • Correlation ρ reconciles volatility generated from rewritten risk resource B and B ; • The price of underlying F also has the right to say that high price gives a high contribution; • Controlling the curvature of implied skewness when facing the calibration of options, pa-rameter β also distribute the influence of F in different price levels.Risk-budgeted policies are of our interest particularly in the cases mentioned in Corollary 3.4where we are restricted to some small σ -algebras. Suppose that we are handling policies with afixed risk level λ >
0, namely that with Var( F uT ) = λ. Assume that ρ = 0 and a small filtration H := {H t } t generated by B . The following four casesare projected one-by-one.1. (Purely Risk-Parity) In the view of Theorem 3.2, we can indicate the optimal policy asso-ciated to risk-parity budget λT for almost all ( t, ω ) in F by u t · (cid:0) u t s E ( αB ) t F βt (cid:1) = λT . Then purely risk-parity policy with total risk λ is given by u ⋆t = r λT s E ( αB ) t F βt .
21. ( H -Projection) Considering that stochastic volatility of underlying F is not easy to perceive,we have a second-best solution restricted on H by Corollary 3.4 u ′ t = r λT s ˆ F βt where ˆ F is the associated projection of underlying. Comparing this solution to u ⋆ , wecannot distinguish the policy among different E ( αB ) t ( ω ) namely the volatility rate σ t ( ω )and ( σ t ) t is substituted by its expectation s in the formula above. This projected policy issubstantially equivalent to that of Black-Scholes model where the volatility rate is assumedconstant.3. (Non-stochastic Time-varying) Furthermore another shrunk σ -algebra F ⊗ B [0 ,T ] gives anon-stochastic time-varying policy u ′′ by u ′′ ( t ) E [ s E ( αB ) t F βt ] = λT for each t .4. (Naive Single-Period) Finally, a buy-and-hold policy ¯ u can give the degenerated single-period solution by the equation¯ u E [ Z T s E ( αB ) t F βt d t ] = ¯ u Var( F T ) = λ. These four policies demonstrate the distinctions from single-period case to the continuous-time case. Similar to extended mean value theorem of integrals, single-period policies act likelow-resolution version of continuous-time ones. It means that continuous-time risk budgetingcan give a more exquisite policy with robust risk distribution through changing the position ofunderlying.
We should notice that it’s not easy to get the explicit form of optimal policy since the solutiondoes not have a feedback formula or meet some universal equations. Having established theform of risk contributions and its related inverse budgeting problem, we will give a numericalresult through neural networks because the solutions to budgeting problems are substantiallymeasurable mappings living in a canonical space without typical structures with respect toBrownian motions.Procedure for calculating budgeting problem is stated below:1. Divide [0 , T ] equally into N parts ( t = 0 , . . . , t N − , t N = T ). Increments of m -dimBrownian motion are generated by independent random sampling with normal distribu-tion , r TN ! .2. As a result, policy u = { u t } t N − t = t is a collection of N random variables with structures givenby u t ( ω ) = u t (∆ B t , ∆ B t , . . . , ∆ B t − N ) for t = t i . u t is identified in a network form, with Corollary 2.12 it is reasonable to assign aparametrized structure u t ( ω ) = N N t (∆ B t , ∆ B t , . . . , ∆ B t − N | θ t )where N N t is a network with its parameter θ t . For each t i , graph of u t i (∆ B t , ∆ B t , . . . , ∆ B t − N | θ t i )is given by Figure 1. Figure 1: Structure of u t i
3. Substitute population moments appearing in budgeting problem by sample moments. Seek-ing an optimal policy u ⋆ of J ( u ) becomes finding the best parameter Θ := { θ t } t N − t = t of J ( u | Θ). Θ is specified by stochastic gradient descent with a quantity of independent sam-pling paths of m -dim Brownian motion paths. Example 5.1 (Numerical solution: SABR in Section 4) . We are handling the numerical solutionwith different restrictions mentioned in Section 4. Parameters in SABR dynamics are assumedto be s = 0 . , α = 0 . , β = 0 . , ρ = 0 . . With a purely risk-parity budget, the risk budgetingproblem becomes J ( u ) = E (cid:20)Z − u t d t (cid:21) + Var ( X uT ) where time horizon is set by T = 1 and total risk is then calculated Var ( X u ⋆ T ) = 10 × .Time axis [0 , T ] is then divided into parts equivalently denoted by partition π [0 , T ] andpolicy structure is assigned a parametrized network keeping predictable property u t i (cid:0) ∆ B t , ∆ B t , . . . , ∆ B t n − (cid:1) = N N t i (cid:0) ∆ B t , ∆ B t , . . . , ∆ B t n − (cid:12)(cid:12) θ t i (cid:1) or each t i ∈ π [0 , T ] with ReLU activation. Our risk-parity goal then becomes minimizing J ( u Θ ) over possible parameters Θ through stochastic gradient descent method with iterations .Loss of iterations is presented in Figure 2. In a view to accumulation of risk contribution, (a) Purely Risk-Parity (b) H -Projection(c) Non-stochastic Time-Varying (d) Naive Single-Period Figure 2: SGD Loss namely the manner how total risk is accumulated in the following formulaVar ( X u ⋆ T ) = E [ Z u ⊤ t c ⋆t d t ] , To meet a good estimation of E ( X uT ), we generate 20000 paths of 2-dim Brownian motion for each iterationand learning rate is specified as a constant 1E-5. e are also interested in numerical result of ( E [ u ⊤ t c t ]) t ∈ [0 , shown in Figure 3. It’s easy to Figure 3: Comparison between given budget β and expected risk contribution of u ⋆ conclude that single-period policy has a non-constant risk contribution among time interval [0 , T ] and that other policies behave inversely. Several key aspects are concerned in this paper: • Marginal risk contribution together with risk contribution is properly defined in continuous-time case when terminal variance of the investment is recognized as the risk measure. • Inverse problem of the point above, process-valued risk budget problem, is then explored.Solution to this continuous-time risk budgeting optimization is then connected to pre-givenbudget process. • Of importance, the bridge connecting single-period and continuous-time cases is the pro-jection of risk budget process.Based on the results above, some useful applications make them sparkle. With fixed policies, therisk distribution of them can be specified and moreover we can judge whether the concentrationof risk distributions to reach the robustness. On the other hand, when given some proper riskbudgets which may depending on some special observed factors, the associated policies can alsobe obtained by solving their related stochastic optimizations.However, it also remains that results for self-financing portfolios have not been settled whichof our interest are usually linked to stochastic differential equations characterizing the value ofportfolios. With market driven by general processes rather than merely Brownian motion, it’squite a problem if risk budgets of portfolios can be obtained since the risk contribution may notbe continuously aggregated. 25
Acknowledgements
The authors would like to thank the referee for the helpful comments.
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