Non-Parametric Robust Model Risk Measurement with Path-Dependent Loss Functions
aa r X i v : . [ q -f i n . M F ] M a r NON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITHPATH DEPENDENT LOSS FUNCTIONS
YU FENG
Abstract.
Understanding and measuring model risk is important to financial practi tioners. However, there lacks a non parametric approach to model risk quantification ina dynamic setting and with path dependent losses. We propose a complete theory gen eralizing the relative entropic approach by Glasserman and Xu (2014) to the dynamiccase under any f divergence. It provides an unified treatment for measuring both theworst case risk and the f divergence budget that originate from the model uncertaintyof an underlying state process. Introduction
As a working definition, model risk refers to the quantification of unanticipatedlosses resulting from the use of inappropriate models to value and manage financialsecurities, including widely traded securities like stocks and bonds, for which marketprices are readily available, and less traded derivatives written on such securities.Unlike other financial risks, which are concerned with the impact of randomness withinthe paradigm of a chosen model, model risk is concerned with the possibility thatthe wrong modelling paradigm was chosen in the first place. This makes it a muchmore challenging proposition, both conceptually and in terms of implementation. It isthus unsurprising that model risk continues to languish behind its more traditionalcounterparts, such as price risk, interest rate risk and credit risk, both in terms ofidentifying an appropriate theoretical methodology and in the development of specificmetrics.A simple approach of accounting for model uncertainty is to assign weights to alter native models and then calculate the average market risk (Branger and Schlag 2004).Perhaps a better way is to separate the model risk component from the market riskcomponent. In addition, from the risk management point of view, one may be moreinterested in the worst case scenario instead of the average scenario. Kerkhof et al.(2002) proposed a risk differencing measure that separates the market risk under theworst case model from the nominal market risk. Following the worst case approach,Cont (2006) formulated a quantitative framework for measuring the model risk in deriv ative pricing. This approach applies to a parametric set of alternative measures whichprice some benchmark instruments within their respective bid ask spreads. FollowingCont’s work, Gupta et al. (2010) proposed the definition of the spread of a contingentclaim to be the set of the prices given by all legitimate models. Bann¨or and Scherer(2013) proposed a parametric risk framework that unifies the proposals of Cont (2006),Gupta et al. (2010) and Lindstr¨om (2010). This approach incorporates a distributionof parameter values to capture the risk of parameter uncertainty, resulting in bid askspreads in instruments that face parameter risk. Detering and Packham (2016) ap proach the problem of model risk measurement based on the residual profit and loss
Date : March 6, 2019. from hedging in the reference model. Kerkhof et al. (2010) propose a procedure totake model risk into account when computing capital reserves. Instead of formulatingmodel risk in terms of a collection of probability measures, they consider the realitythat practitioners may evaluate risk based on models of different natures. From apractical point of view, Boucher et al. (2014) proposed an approach that incorporatesmodel risk into the usual market risk measures.The approaches described above are parametric in the sense that they consideralternative models parametrised by a finite set of parameters. To go beyond that,Glasserman and Xu (2014) proposed a non parametric approach. Under this frame work, a worst case model is found among alternative models in a neighborhood of areference model. Glasserman and Xu adopted the relative entropy (or the Kullback Leibler divergence) to measure the distance between the probability measure given bythe reference model and an (equivalent) alternative measure. By imposing a constrainton the relative entropy budget, the set of legitimate alternative models is defined in anon parametric fashion, and the worst case scenario can then be solved analyticallywithin a finite distance to the reference model. This approach is formulated w.r.t thedistribution of a state variable, thus less applicable when the state variable evolves dy namically. In this paper, we apply it conceptually to the problem of measuring modelrisk w.r.t a state process. We solve the problem in a dual formulation and handle itspath dependency with the help of the functional Ito calculus (Cont 2016). The con straint that defines the legitimate alternative models is w.r.t the f divergence, a moregeneral choice than the Kullback Leibler divergence.2. Problem Formulation
Fix T ∈ (0 , ∞ ) and d ∈ N , and let Ω := D ([0 , T ] , R d ) denote the set of c `adl `ag paths ω : [0 , T ] → R d . Let [0 , T ] ∋ t X ( t ) be the canonical process on Ω , which meansto say that X ( t )( ω ) := ω ( t ) , for all ( t, ω ) ∈ [0 , T ] × Ω . Let F = ( F t ) t ∈ [0 ,T ] denote thefiltration on Ω generated by X , which is to say that F t := _ s ∈ [0 ,t ] ¶ X ( s ) − ( U ) (cid:12)(cid:12)(cid:12) U ∈ B ( R d ) © = _ s ∈ [0 ,t ] [ U ∈ B ( R d ) { ω ∈ Ω | ω ( s ) ∈ U } , for all t ∈ [0 , T ] . In particular, F := ¶ X (0) − ( U ) (cid:12)(cid:12)(cid:12) U ∈ B ( R d ) © = [ U ∈ B ( R d ) { ω ∈ Ω | ω (0) ∈ U } . Fix a reference probability measure P on (Ω , F T ) , subject to the condition P Ä X (0) − ( U ) ä = P ( { ω ∈ Ω | ω (0) ∈ U } ) = if ∈ U ;0 if / ∈ U , for all U ∈ B ( R d ) , which is to say that almost all paths start at zero under P . Note thatthis condition ensures that P ( A ) = 0 or P ( A ) = 1 , for all A ∈ F .To be consistent with the notation in Cont (2016), we shall write ω t := ω ( t ∧ · ) ∈ Ω to denote the path ω ∈ Ω stopped at time t ∈ [0 , T ] . We impose an equivalence relation ∼ on [0 , T ] × Ω , by specifying that ( t, ω ) ∼ ( t ′ , ω ′ ) if and only if t = t ′ and ω t = ω ′ t ′ , for all ( t, ω ) , ( t ′ , ω ′ ) ∈ [0 , T ] × Ω . That is to say, two pairs, each consisting of a time anda path, are equivalent if the times are equal and the corresponding stopped paths are ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 3 the same. The quotient set Λ dT := [0 , T ] × Ω / ∼ forms a complete metric space, whenendowed with the metric d ∞ : (Λ dT ) → R + , defined by d ∞ Ä ( t, ω ) , ( t ′ , ω ′ ) ä := sup s ∈ [0 ,T ] | ω ( s ∧ t ) − ω ′ ( s ∧ t ′ ) | + | t − t ′ | = k ω t − ω ′ t ′ k ∞ + | t − t ′ | , for all ( t, ω ) , ( t ′ , ω ′ ) ∈ Λ dT . We refer to (Λ dT , d ∞ ) as the space of stopped paths .A measurable function F : Λ dT → R is called a non anticipative functional , where Λ dT is endowed with the Borel sigma algebra generated by d ∞ and R is endowed with theBorel sigma algebra generated by the usual Euclidean metric. Since ( t, ω ) ∼ ( t, ω t ) ,for all ( t, ω ) ∈ [0 , T ] × Ω , we may regard a non anticipative functional F : Λ dT → R asan appropriately measurable function F : [0 , T ] × Ω → R that satisfies the condition F ( t, ω ) = F ( t, ω t ) . That is to say, the value of a non anticipative functional, whenapplied to a particular time and path, depends only on the behaviour of the path upto the time. Note that ( F ( t, · )) t ∈ [0 ,T ] is a progressively measurable process, adapted tothe filtration F .Let M denote the family of (right continuous versions of) martingales on the filteredprobability space (Ω , F T , F , P ) , over the compact time interval [0 , T ] , and let M + (1) := { Z ∈ M | Z > and Z (0) = 1 } denote the sub family of non negative martingales starting at one. Each Z ∈ M + (1) defines a probability measure Q Z on (Ω , F T ) satisfying Q Z ≪ P (i.e. Q Z is absolutelycontinuous w.r.t P ), according to the recipe Q Z ( A ) := E Ä A Z ( T ) ä , for all A ∈ F T .Conversely, each probability measure Q on (Ω , F T ) satisfying Q ≪ P can be written as Q = Q Z , where Z ∈ M + (1) is determined by Z ( t ) := E Ç d Q d P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t å , for all t ∈ [0 , T ] .Consider a twice differentiable strictly convex function f : R + → R satisfying f (1) =0 . For any probability measure Q on (Ω , F T ) satisfying Q ≪ P , the f divergence of Q with respect to P is defined by D f ( Q k P ) := E Ç f Ç d Q d P åå (2.1)(see Basseville 2013, Section 2). Intuitively, f divergence provides a measure of thedistance between two probability measures. Hence, the set Z η := { Z ∈ M + (1) | D f ( Q Z k P ) η } , where η > , corresponds to the family of absolutely continuous probability measuresthat are close to the reference probability measure P .Finally, fix a non anticipative functional ℓ : Λ dT → R satisfying ℓ (0 ,
0) = 0 . We shallinterpret ℓ ( t, ω ) as the cumulative realized loss up to time t , incurred by a portfolioof financial securities. The state of the portfolio is completely determined by the path ω ∈ Ω . The condition of the reference probability measure guarantees P Ä X (0) − { } ä = P ( { ω ∈ Ω | ω (0) = 0 } ) = 1 , It follows that ℓ (0 , · ) = 0 P a.s. That is to say, the initial realized loss incurred bythe portfolio is zero under the reference probability measure. If we interpret P as theprobability measure associated with a nominal model for the dynamics of the portfolio, YU FENG then E Ä ℓ ( T, · ) ä gives the expected total loss under the nominal model. In financialapplications, we usually set the terminal time T as the point when the entire portfoliogets liquidated, thus realizing the cumulative loss.Suppose, now, that there is some uncertainty about which model best describesthe portfolio. In particular, suppose that each probability measure determined by amember of Z η , for some η > , corresponds to a plausible model for the dynamics of theportfolio. In that case, a risk manager would be interested in the following quantities: sup Z ∈Z η E Q Z Ä ℓ ( T, · ) ä and sup Z ∈Z η E Q Z Ä ℓ ( T, · ) ä − E Ä ℓ ( T, · ) ä . (2.2)The former expression may be regarded as the worst case expected loss suffered by theportfolio under all plausible models, while the latter expression quantifies the differencebetween the worst case expected loss and the expected loss under the default model.As such, it serves as a measure of model risk.Problem defined in (2.2) may be formulated in a dual form (Glasserman and Xu2014). We first define the Lagrangian L : M + (1) × (0 , ∞ ) × (0 , ∞ ) → R by L ( Z, ϑ, η ) := E Q Z Ä ℓ ( T, · ) ä − D f ( Q Z k P ) − ηϑ = E Q Z Ç ℓ ( T, · ) − f Ä Z ( T ) ä ϑZ ( T ) å + ηϑ , The Lagrangian leads to a dual function defined by d ( ϑ, η ) := sup Z ∈ M + (1) L ( Z, ϑ, η ) = sup Z ∈ M + (1) E Q Z Ä b ℓ ( T, Z ) ä + ηϑ Given t ∈ [0 , T ] and Z ∈ M + (1) , b ℓ ϑ ( t, Z ) := ℓ ( t, · ) − f Ä Z ( t ) ä ϑZ ( t ) (2.3)defines a F t measurable function b ℓ ϑ ( t, Z ) : Ω → R . As with ℓ : [0 , T ] × Ω → R , b ℓ ϑ ( · , Z ) may be regarded as a non anticipative functional.If the primal problem is convex and the constraint satisfies Slater’s condition (Slater2014), then strong duality holds, giving sup Z ∈Z η E Q Z ( ℓ ( T, · )) = inf ϑ ∈ (0 , ∞ ) d ( ϑ, η ) (2.4)This is proved in the following lemma. Lemma 2.1.
The following statements are true:(1) The set Z η is convex.(2) The function Z η ∋ Z E Q z Ä ℓ ( T, · ) ä is convex.(3) Strong duality Eq. 2.4 holds.(4) Given ϑ ∗ ∈ (0 , ∞ ) , and suppose that Z ∗ ∈ M + (1) satisfies Z ∗ = arg max Z ∈ M + (1) E Q Z Ä b ℓ ϑ ∗ ( T, Z ) ä then Z ∗ = arg max Z ∈Z η E Q Z Ä ℓ ( T, · ) ä The idea here is that all absolutely continuous probability measures close enough to the referencemeasure (in the sense of f divergence) correspond with models that are plausibly close to the referencemodel. ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 5 with η := E ( f ( Z ∗ ( T ))) .Proof. (1) Given Z , Z ∈ Z η , observe that D f ( Q λZ +(1 − λ ) Z k P ) = D f Ä λ Q Z + (1 − λ ) Q Z (cid:13)(cid:13)(cid:13) P ä = E Ä f Ä λZ ( T ) + (1 − λ ) Z ( T ) ää E Ä λf Ä Z ( T ) ä + (1 − λ ) f Ä Z ( T ) ää = λ E Ä f Ä Z ( T ) ää + (1 − λ ) E Ä f Ä Z ( T ) ää = λD f ( Q Z k P ) + (1 − λ ) D f ( Q Z k P ) , for all λ ∈ [0 , , by virtue of the convexity of f and Jensen’s inequality. Since D f ( Q Z k P ) η and D f ( Q Z k P ) η , the inequality above leads to D f ( Q λZ +(1 − λ ) Z k P ) η . This implies that λZ + (1 − λ ) Z ∈ Z η , by virtue of the fact that λZ + (1 − λ ) Z ∈ M + (1) .(2) Given Z , Z ∈ Z η , observe that E Q λZ − λ ) Z Ä ℓ ( T, · ) ä = E ÄÄ λZ ( T ) + (1 − λ ) Z ( T ) ä ℓ ( T, · ) ä = λ E Ä Z ( T ) ℓ ( T, · ) ä + (1 − λ ) E Ä Z ( T ) ℓ ( T, · ) ä = λ E Q Z Ä ℓ ( T, · ) ä + (1 − λ ) E Q Z Ä ℓ ( T, · ) ä , for all λ ∈ [0 , . Hence, the function Z η ∋ Z E Q Z Ä ℓ ( T, · ) ä is linear and thereforealso convex.(3) For a given η ∈ (0 , ∞ ) , the constant process Z = 1 satisfies D f ( Q Z || P ) = D f ( P || P ) =0 < η . It is also an interior point of the subset Z η ⊆ M + (1) . According to Slater’scondition (Slater 2014), the strong duality holds.(4) Let η := E ( f ( Z ∗ ( T ))) , and observe that inf ϑ ∈ (0 , ∞ ) d ( ϑ, η ) d ( ϑ ∗ , η )= sup Z ∈ M + (1) E Q Z Ä b ℓ ϑ ∗ ( T, Z ) ä + ηϑ ∗ = E Q Z ∗ Ä b ℓ ϑ ∗ ( T, Z ∗ ) ä + ηϑ ∗ = E Q Z ∗ Ä ℓ ( T, · ) ä − ϑ ∗ E Q Z ∗ Ç f ( Z ∗ ( T )) Z ∗ ( T ) å + ηϑ ∗ = E Q Z ∗ Ä ℓ ( T, · ) ä − ϑ ∗ E ( f ( Z ∗ ( T ))) + ηϑ ∗ = E Q Z ∗ Ä ℓ ( T, · ) ä sup Z ∈Z η E Q Z Ä ℓ ( T, · ) ä Lemma 2.1(3) then ensures that inf ϑ ∈ (0 , ∞ ) d ( ϑ, η ) = E Q Z ∗ Ä ℓ ( T, · ) ä = sup Z ∈Z η E Q Z Ä ℓ ( T, · ) ä To see this point, consider the continuous function H : M + (1) → R defined by H ( Z ) = E ( f ( Z ( T ))) (we endow M + (1) with the topology induced by the metric d ( Z , Z ) = E ( | f ( Z ( T )) − f ( Z ( T )) | ) . Thecontinuity ensures that S := H − (( − η, η )) is an open subset of M + (1) . Furthermore, S ⊆ { Z ∈ M + (1) | D f ( Q Z || P ) < η } ⊆ Z η suggesting that S ⊆ int ( Z η ) . As an element in S , the constant process Z = 1 is an interior point of Z η . YU FENG and the result follows. (cid:3)
For the primal problem formulated in Eq. 2.2, Lemma. 2.1(4) implies the existenceof a solution Z ∗ that lies on the boundary of Z η given η > (i.e. E ( f ( Z ∗ ( T ))) = η ), aslong as Z ∗ solves max Z ∈ M + (1) E Q Z Ä b ℓ ϑ ( T, Z ) ä (2.5)for some ϑ ∈ (0 , ∞ ) . In the following context, we will consider the dual problemformulated in Eq. 2.5 instead of the primal problem. For simplicity, we will regard θ > as given and express b ℓ ϑ by b ℓ .3. Characterising the Worst Case Expected Loss
This section provides implicit characterisation of the solution to the worst case ex pected loss problem formulated in (2.2).Given t ∈ [0 , T ] and ¯ Z ∈ M + (1) , define the family of ¯ Z consistent martingale densi ties up to time t by Z ( t, ¯ Z ) := { Z ∈ M + (1) | Z ( t ) = ¯ Z ( t ) } . Note that Z (0 , ¯ Z ) = M + (1) , since Z (0) = 1 = ¯ Z (0) for all Z ∈ M + (1) . Note that themartingale property of the members of Z ( t, ¯ Z ) ensures that Z ( s ) = E Ä Z ( t ) | F s ä = E Ä ¯ Z ( t ) | F s ä = ¯ Z ( s ) , for all Z ∈ Z ( t, ¯ Z ) and all s ∈ [0 , t ] . In other words, Z ( t, ¯ Z ) is the set of processes in M + (1) that are consistent with ¯ Z over the interval [0 , t ] . Moreover, we observe that Q Z ( A ) = E Ä A Z ( T ) ä = E Ä E Ä A Z ( T ) | F t ää = E Ä A Z ( t ) ä = E Ä A ¯ Z ( t ) ä = E Ä E Ä A ¯ Z ( T ) | F t ää = E Ä A ¯ Z ( T ) ä = Q ¯ Z ( A ) , for all Z ∈ Z ( t, ¯ Z ) and all A ∈ F t . That is to say, the probability measures associatedwith members of Z ( t, ¯ Z ) agree with each other on all F t measurable events. This isthe set of feasible alternative measures by looking forward (from time t ).Given ¯ Z ∈ M + (1) , we now define the F adapted process ( “ L ( t, ¯ Z )) t ∈ [0 ,T ] by “ L ( t, ¯ Z ) := max Z ∈Z ( t, ¯ Z ) E Q Z Ä b ℓ ( T, Z ) − b ℓ ( t, Z ) (cid:12)(cid:12)(cid:12) F t ä (3.1)for all t ∈ [0 , T ] , assuming the maximum always exists. Since b ℓ ( · , Z ) is a non anticipative functional satisfying b ℓ (0 , Z ) = 0 P a.s. and Z (0) = 1 implies that Q Z | F = P | F , it follows that b ℓ (0 , Z ) = 0 Q Z a.s. as well. Consequently, “ L (0 , ¯ Z ) = max Z ∈ M + (1) E Q Z Ä b ℓ ( T, Z ) (cid:12)(cid:12)(cid:12) F ä = max Z ∈ M + (1) E Q Z Ä b ℓ ( T, Z ) ä , (3.2)where the second equality follows from the fact that F and F T are independentsigma algebras, with respect to Q Z . This is simply the problem given in Eq. 2.5. First observe that Z (0) = 1 implies that Q Z ( A ) = P ( A ) = 0 or Q Z ( A ) = P ( A ) = 1 , for all A ∈ F .Consequently, given A ∈ F and B ∈ F T , we obtain Q Z ( A ∩ B ) Q Z ( A ) = 0 = Q Z ( A ) Q Z ( B ) , ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 7
Definition 3.1.
A worst case density process is some Z ∗ ∈ M + (1) that solves themaximisation problem (3.1) w.r.t the family of Z ∗ consistent martingale densities: E Q Z ∗ Ä b ℓ ( T, Z ∗ ) − b ℓ ( t, Z ∗ ) (cid:12)(cid:12)(cid:12) F t ä = “ L ( t, Z ∗ ) (3.3)for each t ∈ [0 , T ] .Suppose Z ∗ ∈ M + (1) is a worst case martingale density according to the definitionabove, then Z ∗ solves the problem formulated in Eq. 2.5. This is confirmed by sub stituting Eq. 3.2 into Eq. 3.3 which leads to E Q Z ∗ Ä b ℓ ( T, Z ∗ ) ä = max Z ∈ M + (1) E Q Z Ä b ℓ ( T, Z ) ä .In the proposition below, we characterizes such worst case density by its martingaleproperty. Proposition 3.2.
Fix ¯ Z ∈ M + (1) and suppose the maximum in (3.1) exists for each t ∈ [0 , T ] . Then the process [0 , T ] ∋ t “ L ( t, ¯ Z ) + b ℓ ( t, ¯ Z ) is a Q ¯ Z supermartingale. It isa Q ¯ Z martingale iff ¯ Z is a worst case density process.Proof. Given an arbitrary t ∈ [0 , T ] , we suppose Z ′ ∈ Z ( t, ¯ Z ) solves the maximisationproblem (Eq. 3.1). Applying the law of iterated expectation, we have E Q Z ′ (cid:16) b ℓ ( T, Z ′ ) − b ℓ ( t, Z ′ ) (cid:12)(cid:12)(cid:12) F s (cid:17) = E Q Z ′ Ç E Q Z ′ (cid:16) b ℓ ( T, Z ′ ) − b ℓ ( t, Z ′ ) (cid:12)(cid:12)(cid:12) F t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F s å = E Q Z ′ (cid:16) “ L ( t, ¯ Z ) (cid:12)(cid:12)(cid:12) F s (cid:17) (3.4)for all s ∈ [0 , t ] . By virtue of Z ′ ( s ) = ¯ Z ( s ) , b ℓ ( s, Z ′ ) = b ℓ ( s, ¯ Z ) for all s ∈ [0 , t ] . The samecondition also leads to Z ′ ∈ Z ( s, ¯ Z ) . According to the definition of “ L (Eq. 3.1), we havethe following inequality “ L ( s, ¯ Z ) > E Q Z ′ (cid:16) b ℓ ( T, Z ′ ) − b ℓ ( s, Z ′ ) (cid:12)(cid:12)(cid:12) F s (cid:17) = E Q Z ′ (cid:16) b ℓ ( t, Z ′ ) − b ℓ ( s, Z ′ ) (cid:12)(cid:12)(cid:12) F s (cid:17) + E Q Z ′ (cid:16) b ℓ ( T, Z ′ ) − b ℓ ( t, Z ′ ) (cid:12)(cid:12)(cid:12) F s (cid:17) = E Q ¯ Z (cid:16) b ℓ ( t, ¯ Z ) − b ℓ ( s, ¯ Z ) + “ L ( t, ¯ Z ) (cid:12)(cid:12)(cid:12) F s (cid:17) (3.5)for all s ∈ [0 , t ] . In the last equality, we replace Q Z ′ by Q ¯ Z because b ℓ ( t, ¯ Z ) , b ℓ ( s, ¯ Z ) and “ L ( t, ¯ Z ) are all F t measurable. Since t ∈ [0 , T ] is chosen arbitrarily, Eq. 3.5 holds forany s and t that satisfies s t T . in the case when Q Z ( A ) = 0 , while Q Z ( A ) Q Z ( B ) = Q Z ( B ) > Q Z ( A ∩ B ) = Q Z (cid:0) ( A c ∪ B c ) c (cid:1) = 1 − Q Z ( A c ∪ B c ) > − (cid:0) Q Z ( A c ) + Q Z ( B c ) (cid:1) = 1 − Q Z ( B c )= Q Z ( B )= Q Z ( A ) Q Z ( B ) , in the case when Q Z ( A ) = 1 . The conditional expectation of a F t measurable function X : Ω → R w.r.t a sub σ algebra F s ⊆ F t is E Q Z ′ (cid:0) X | F s (cid:1) = E Å Z ′ ( T ) Z ′ ( s ) X (cid:12)(cid:12)(cid:12)(cid:12) F s ã = E Å Z ′ ( t ) Z ′ ( s ) E Å Z ′ ( T ) Z ′ ( t ) X (cid:12)(cid:12)(cid:12)(cid:12) F t ã (cid:12)(cid:12)(cid:12)(cid:12) F s ã = E Å Z ′ ( t ) Z ′ ( s ) E Q Z ′ (cid:0) X | F t (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) F s ã = E Å ¯ Z ( t )¯ Z ( s ) X (cid:12)(cid:12)(cid:12)(cid:12) F s ã = E Q ¯ Z (cid:0) X | F s (cid:1) YU FENG
By re arranging Eq. 3.5, we obtain the supermartingale property of the F adaptedprocess [0 , T ] ∋ t “ L ( t, ¯ Z ) + b ℓ ( t, ¯ Z ) : “ L ( s, ¯ Z ) + b ℓ ( s, ¯ Z ) > E Q ¯ Z (cid:16) “ L ( t, ¯ Z ) + b ℓ ( t, ¯ Z ) (cid:12)(cid:12)(cid:12) F s (cid:17) (3.6)The process is a Q ¯ Z martingale iff the equality holds for all s t T . If ¯ Z isa worst case density process, then according to Definition 3.1 ¯ Z solves Eq. 3.1 for all t ∈ [0 , T ] . We may set Z ′ = ¯ Z in Eq. 3.5 so that the first line takes the equal sign forall s ∈ [0 , t ] . Conversely, if the equality holds for all s t T , then it holds for all s t = T . By taking the equal sign in Eq. 3.6 and replacing t by T , we get “ L ( s, ¯ Z ) = E Q ¯ Z (cid:16) b ℓ ( T, ¯ Z ) − b ℓ ( s, ¯ Z ) (cid:12)(cid:12)(cid:12) F s (cid:17) for all s ∈ [0 , T ] , confirming that ¯ Z is a worst case density process by Definition 3.1. (cid:3) Proposition. 3.2 can be regarded as generalization of the dynamic programmingequation. In fact, given an optimal martingale density Z ∗ ∈ M + (1) , we take an ar bitrary ¯ Z ∈ Z ( s, Z ∗ ) and substitute it into Eq. 3.6. By observing that ¯ Z ∈ Z ( s, Z ∗ ) matches Z ∗ up to time s , we transform Eq. 3.6 into “ L ( s, Z ∗ ) + b ℓ ( s, Z ∗ ) > E Q ¯ Z (cid:16) “ L ( t, ¯ Z ) + b ℓ ( t, ¯ Z ) (cid:12)(cid:12)(cid:12) F s (cid:17) The inequality holds for all ¯ Z ∈ Z ( s, Z ∗ ) . It takes the equal sign when ¯ Z = Z ∗ .This leads to the following dynamic programming equation with respect to the densityprocess, “ L ( s, Z ∗ ) + b ℓ ( s, Z ∗ ) = max Z ∈Z ( s,Z ∗ ) E Q Z ∗ (cid:16) “ L ( t, Z ) + b ℓ ( t, Z ) (cid:12)(cid:12)(cid:12) F s (cid:17) for all s and t that satisfies s t T .4. General Result of Model Risk Measurement
We have shown in Proposition. 3.2 that the F adapted process [0 , T ] ∋ t “ L ( t, Z ∗ )+ b ℓ ( t, Z ∗ ) is a Q Z ∗ martingale iff Z ∗ is a worst case density process. In this section, wewill show that such Z ∗ indeed exists under certain conditions and is characterized byan equation. This leads to a complete solution to the problem formulated in Eq. 2.2.First we prove a lemma. Lemma 4.1.
Fix a martingale density ¯ Z ∈ M + (1) . A measurable process C : [0 , T ] × Ω → R , satisfying E Q Z Ä C ( t, · ) | F t ä E Q ¯ Z Ä C ( t, · ) | F t ä (4.1) for all t ∈ [0 , T ] and all Z ∈ Z ( t, ¯ Z ) , admits a progressively measurable modification,i.e. there exists a progressively measurable process ˜ C : [0 , T ] × Ω → R , regarded asa non anticipative functional, satisfying Q ¯ Z Ä { ω ∈ Ω | C ( t, ω ) = ˜ C ( t, ω ) } ä = 1 for every t ∈ [0 , T ] .Proof. The F t measurable function u ( t, · ) := E Q ¯ Z Ä C ( t, · ) | F t ä forms a F adapted pro cess ( u ( t, · )) t ∈ [0 ,T ] . It admits a progressively measurable modification Ä ˜ C ( t, · ) ä t ∈ [0 ,T ] (Karatzas and Shreve 1991). We would like to show that Q ¯ Z Ä { ω ∈ Ω | C ( t, ω ) =˜ C ( t, ω ) } ä = 1 for every t ∈ [0 , T ] . ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 9
We prove this lemma by contradiction. Suppose there exists a t ∈ [0 , T ] such that Q ¯ Z Ä { ω ∈ Ω | C ( t, ω ) = ˜ C ( t, ω ) } ä < , then Q ¯ Z Ä { ω ∈ Ω | C ( t, ω ) = u ( t, ω ) } ä < . Thisimplies that either Q ¯ Z Ä { ω ∈ Ω | C ( t, ω ) < u ( t, ω ) } ä > or Q ¯ Z Ä { ω ∈ Ω | C ( t, ω ) >u ( t, ω ) } ä > . Without losing generality, we assume Q ¯ Z Ä { ω ∈ Ω | C ( t, ω ) < u ( t, ω ) } ä > . For notational simplicity, in the rest of the proof we use C to denote the randomvariable C ( t, · ) and u to denote the F t measurable function u ( t, · ) . We construct analternative martingale density Z ′ ∈ Z ( t, ¯ Z ) by Z ′ ( s ) = ¯ Z ( s ) s ∈ [0 , t ] E Q ¯ Z (cid:16) e C C , Z ′ ( t ) = ¯ Z ( t ) and Z ′ is a P martingale. The first three conditions are obvious from the definition.The martingale property of ( Z ′ ( s )) s ∈ [0 ,t ] is clear. The martingale property of ( Z ′ ( s )) s ∈ [ t,T ] is confirmed by E Ä Z ′ ( r ) | F s ä = E Ö E (cid:16) ¯ Z ( T ) Ä e C C , there exists a ω ∈ Ω such that E Q ¯ Z ( C . We define w l := E Q ¯ Z Ä C − Q ¯ Z (cid:0) { ω ∈ Ω | C ( t, ω ) = u ( t, ω ) } ∪ { ω ∈ Ω | u ( t, ω ) = ˜ C ( t, ω ) } (cid:1) > − Q ¯ Z (cid:0) { ω ∈ Ω | C ( t, ω ) = u ( t, ω ) } (cid:1) − Q ¯ Z (cid:0) { ω ∈ Ω | u ( t, ω ) = ˜ C ( t, ω ) } (cid:1) = 1 then the LHS of Eq. 4.1 (with Z replaced by Z ′ ) satisfies E Q Z ′ Ä C | F t ä ( ω ) = E Q ¯ Z (cid:16) Ce C Cw l c l + w u c u (4.3)Note that the inequality is given by the Chebyshev’s sum inequality, which states that w , w > and w + w = 1 , one have ( w a + w a )( w b + w b ) < w a b + w a b if a < a and b < b . This inequality can be easily proved by expanding the left handside. In Eq. 4.3, we have w l > , w u > and c l = E Q ¯ Z (cid:16) Ce C C E ( x ) ln E ( x ) > E ( x ) E (ln x ) (4.4)Following the inequality above, we take expectation w.r.t F t and under the alternativemeasure generated by the Radon Nikodym derivative ¯ Z ( T )¯ Z ( t ) C E (cid:16) ¯ Z ( T ) e C C E Q ¯ Z ( C C E Q Z ′ Ä C C
The progressively measurable process ˜ C is adapted to the filtration F . Therefore E Q ¯ Z Ä C ( t, · ) | F t ä = ˜ C ( t, · ) = E Q Z Ä C ( t, · ) | F t ä for all t ∈ [0 , T ] and Z ∈ Z ( t, ¯ Z ) . We use Lemma 4.1 to prove the following proposition. Proposition 4.2. Z ∗ ∈ M + (1) is a worst case martingale density iff the random variable Ω ∋ ω ℓ ( T, ω ) − f ′ ( Z ∗ ( T )( ω )) ϑ equals constant Q Z ∗ a.s., and is dominated by the same constant P a.s.Proof. Suppose Z ∗ ∈ M + (1) is a worst case martingale density. According to Defini tion. 3.1, Z ∗ = arg max Z ∈Z ( t,Z ∗ ) E Q Z Ä b ℓ ( T, Z ) − b ℓ ( t, Z ) (cid:12)(cid:12)(cid:12) F t ä (4.5)for all t ∈ [0 , T ] . Given any t ∈ [0 , T ] and any Z ∈ Z ( t, Z ∗ ) , we construct a newmartingale density that lies between Z ∗ and Z by Z λ = (1 − λ ) Z ∗ + λZ where λ ∈ [0 , . Z λ ∈ Z ( t, Z ∗ ) for all λ ∈ [0 , due to the convexity of Z ( t, Z ∗ ) . Since Z ∗ solves Eq. 4.5, the maximum value of K ( λ ) := E Q Zλ Ä b ℓ ( T, Z λ ) − b ℓ ( t, Z λ ) (cid:12)(cid:12)(cid:12) F t ä (4.6)is reached when λ = 0 . Taking the first and second derivatives with respect to λ , weget K ′ ( λ ) = E Ç Z ( T ) − Z ∗ ( T ) Z ∗ ( t ) Ç ℓ ( T, Z ∗ ) − ℓ ( t, · ) − f ′ ( Z λ ( T )) ϑ å (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t å (4.7) K ′′ ( λ ) = − E Ñ ( Z ( T ) − Z ∗ ( T )) Z ∗ ( t ) f ′′ ( Z λ ( T )) ϑ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t é (4.8)Notice that the twice differentiable function f : R + → R is convex as required by thenon negativity of the f divergence (Ali and Silvey 1966). This implies that f ′′ ( z ) > forall z ∈ R + . Combined with Eq. 4.8, this condition leads to K ′′ ( λ ) < for all λ ∈ [0 , .For K (0) = max λ ∈ [0 , K ( λ ) to hold, the first derivative at λ = 0 must satisfy K ′ (0) ⇔ E Q Z Ä C Z ∗ ( t, · ) | F t ä E Q Z ∗ Ä C Z ∗ ( t, · ) | F t ä (4.9)where the process C Z ∗ : [0 , T ] × Ω → R is defined by C Z ∗ ( t, · ) := ℓ ( T, · ) − ℓ ( t, · ) − f ′ ( Z ∗ ( T )) ϑ The inequality above holds for all t ∈ [0 , T ] and all Z ∈ Z ( t, Z ∗ ) . According toLemma. 4.1, C Z ∗ admits a progressively measurable modification, say ˜ C Z ∗ . In par ticular, at t = 0 C Z ∗ (0 , · ) = ℓ ( T, · ) − f ′ ( Z ∗ ( T )) ϑ takes a constant value c := ˜ C Z ∗ (0 , , Q Z ∗ a.s. In fact, ˜ C Z ∗ is regarded as a non anticipative functional so that ˜ C Z ∗ (0 , ω ) = ˜ C Z ∗ (0 ,
0) = c for all ω ∈ Ω satisfying (0 , ω ) ∼ (0 , . As a result, Q Z ∗ Ä C Z ∗ (0 , · ) = c ä > Q Z ∗ Ä ˜ C Z ∗ (0 , · ) = c ä − Q Z ∗ Ä C Z ∗ (0 , · ) = ˜ C Z ∗ (0 , · ) ä = Q Z ∗ Ä ˜ C Z ∗ (0 , · ) = c ä − > Q Z ∗ Ä (0 , · ) ∼ (0 , ä = Q Z ∗ ( { ω ∈ Ω | ω (0) = 0 } ) = 1 (4.10)Next we prove P Ä C Z ∗ (0 , · ) c ä = 1 by contradiction. Suppose on the contrary that P Ä C Z ∗ (0 , · ) > c ä > . We construct a martingale density Z ′ ∈ Z (0 , Z ∗ ) = M + (1) bysetting Z ′ ( t ) = E (cid:16) C Z ∗ (0 , · ) >c (cid:12)(cid:12)(cid:12) F t (cid:17) P Ä C Z ∗ (0 , · ) > c ä for all t ∈ [0 , T ] . This leads to E Q Z ′ Ä C Z ∗ (0 , · ) | F ä = E ( Z ′ ( T ) C Z ∗ (0 , · )) = E Ä C Z ∗ (0 , · ) >c C Z ∗ (0 , · ) ä P Ä C Z ∗ (0 , · ) > c ä > c E Ä C Z ∗ (0 , · ) >c ä P Ä C Z ∗ (0 , · ) > c ä = c Because we have already shown that C Z ∗ (0 , · ) = c , Q Z ∗ a.s. (Eq. 4.10), E Q Z ∗ Ä C Z ∗ (0 , · ) | F ä = c < E Q Z ′ Ä C Z ∗ (0 , · ) | F ä According to Eq. 4.9, K ′ (0) > (where the generic density process Z is replaced bythe constructed process Z ′ ∈ Z (0 , Z ∗ ) ). This contradicts the assumption that Z ∗ is aworst case martingale density.Conversely, given a process Z ∗ ∈ M + (1) , suppose C Z ∗ (0 , · ) : Ω → R takes a constantvalue, say c , Q Z ∗ a.s., and C Z ∗ (0 , · ) c P a.s. Given any t ∈ [0 , T ] and any Z ∈Z ( t, Z ∗ ) , C Z ∗ (0 , · ) c Q Z a.s. due to the absolute continuity of Q Z w.r.t. P . Theseproperties lead to conditional expectations E Q Z ∗ Ä C Z ∗ (0 , · ) | F t ä = c and E Q Z Ä C Z ∗ (0 , · ) | F t ä c Noticing that C Z ∗ ( t, · ) = C Z ∗ (0 , · ) − ℓ ( t, · ) where ℓ ( t, · ) is F t measurable, We have E Q Z Ä CZ ∗ ( t, · ) | F t ä c − ℓ ( t, · ) = E Q Z ∗ Ä C Z ∗ ( t, · ) | F t ä According to Eq. 4.9, K ′ (0) . Because K ′′ ( λ ) < (Eq. 4.8) for all λ ∈ [0 , , K (0) > K (1) . According to the definition of K ( λ ) (Eq. 4.6), we have E Q Z ∗ Ä b ℓ ( T, Z ∗ ) − b ℓ ( t, Z ∗ ) (cid:12)(cid:12)(cid:12) F t ä = K (0) > K (1) = E Q Z Ä b ℓ ( T, Z ) − b ℓ ( t, Z ) (cid:12)(cid:12)(cid:12) F t ä This inequality applies to every t ∈ [0 , T ] and every Z ∈ Z ( t, Z ∗ ) . As a result, Z ∗ solvesEq. 4.5 for all t ∈ [0 , T ] and is indeed a worst case martingale density. (cid:3) It is noted that Proposition. 3.2 is a general result that works for any F adaptedprocess ( b ℓ ( t, Z )) t ∈ [0 ,T ] , irrespective of its actual formulation (Eq. 2.3). On the otherhand, Proposition. 4.2 makes use of the formulation, thus specifying the condition of aworst case martingale density w.r.t the function f ( x ) . Note that any worst case densityprocess Z ∗ ∈ M + (1) solves the original problem formulated in Eq. 2.5. Assuming theexistence of such Z ∗ , we regard Eq. 2.5 as the initial value (at t = 0 ) of a particularprocess, termed as the value process. In general, we define three F adapted processesas below. ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 13
Definition 4.3.
Given ϑ ∈ (0 , ∞ ) and a worst case martingale density Z ∗ ∈ M + (1) ,the value process, U : [0 , T ] × Ω → R , the worst case risk, V : [0 , T ] × Ω → R , andthe budget process η : [0 , T ] × Ω → R , regarded as non anticipative functionals, aredefined by U ( t, · ) := “ L ( t, Z ∗ ) + ℓ ( t, · ) V ( t, · ) := “ L ( t, Z ∗ ) + b ℓ ( t, Z ∗ ) + F ( t, Z ∗ ) η ( t, · ) := ϑ ( V ( t, · ) − U ( t, · )) where ( F ( t, Z ∗ )) t ∈ [0 ,T ] is the Q Z ∗ martingale that satisfies F ( T, Z ∗ ) = f ( Z ∗ ( T )) /Z ∗ ( T ) .Intuitively, U ( t, · ) gives the worst case expected loss, subtracting the on going costof perturbing the nominal model from time t to T . According to the definition of theworst case martingale density (Eq. 3.3), U ( t, · ) = E Q Z ∗ Ä b ℓ ( T, Z ∗ ) − b ℓ ( t, Z ∗ ) (cid:12)(cid:12)(cid:12) F t ä + ℓ ( t, · )= E Q Z ∗ Ä ℓ ( T, · ) (cid:12)(cid:12)(cid:12) F t ä − ϑ − Z ∗ ( t ) − E Ä f ( Z ∗ ( T )) − f ( Z ∗ ( t )) (cid:12)(cid:12)(cid:12) F t ä Z ∗ ( t ) > (4.11)The second term is the penalization term for perturbing the nominal model from time t onwards. For continuity it is defined to be zero in the limiting case of Z ∗ ( t ) = 0 .According to Definition 4.3, V ( t, · ) is the worst case expected loss, V ( t, · ) = E Q Z ∗ Ä b ℓ ( T, Z ∗ ) (cid:12)(cid:12)(cid:12) F t ä + ϑ − Z ∗ ( t ) − E Q Ä f ( Z ∗ ( T )) (cid:12)(cid:12)(cid:12) F t ä Z ∗ ( t ) > = E Q Z ∗ Ä ℓ ( T, · ) (cid:12)(cid:12)(cid:12) F t ä The difference between V ( t, · ) and U ( t, · ) gives the cost for perturbing the nominalmodel (measured by the f divergence), characterized by the process η : η ( t, · ) = Z ∗ ( t ) − E Ä f ( Z ∗ ( T )) − f ( Z ∗ ( t )) (cid:12)(cid:12)(cid:12) F t ä Z ∗ ( t ) > We may further consider the terminal and initial values of the three processes. Thevalue process, U ( t, · ) , measures the target formulated in Eq. 2.5 from backwards, inthe sense that U ( T ) = ℓ ( T, · ) and U (0) = E Q Z ∗ Ä b ℓ ( T, Z ∗ ) ä = max Z ∈ M + (1) E Q Z Ä b ℓ ( T, Z ) ä (4.12)The worst case risk process measures the model risk, Eq. 2.2, from backwards. Ac cording to Lemma. 2.1(4), the worst case density Z ∗ solves the primal problem with η := η (0 , · ) = E ( f ( Z ∗ ( T ))) . Therefore V ( T ) = ℓ ( T, · ) and V (0) = E Q Z ∗ ( ℓ ( T, · )) = sup Z ∈Z η E Q Z ∗ Ä ℓ ( T, · ) ä The cumulative budget η (i.e. relative entropy budget in Glasserman and Xu (2014)) ismeasured by the budget process from backwards, η ( T ) = 0 and η (0) = E ( f ( Z ∗ ( T ))) = η To solve the problem formulated in Eq. 2.5, Eq. 4.12 suggests solving the process U bybackward induction. In a similar way, the model risk, Eq. 2.2, and its correspondingcumulative budget, η , may be quantified by solving the processes V and η by backwardinduction. The full procedure is given by the following theorem. We name it the budget process as it measures the remaining budget of the fictitious adversary(Glasserman and Xu 2014). η (0 , · ) is referred as the relative entropy budget in Glasserman and Xu(2014). Theorem 4.4.
Given ϑ ∈ (0 , ∞ ) , suppose there exists a function z : R → R + thatsatisfies x − f ′ ( z ( x )) ϑ = c if x ∈ I c := { ϑ − y + c | y ∈ range ( f ′ ) } (4.13) z ( x ) = 0 if x / ∈ I c (4.14) where c ∈ R is a constant such that E Ä z ◦ ℓ ( T, · ) ä = 1 and P ( ℓ ( T, · ) < sup I c ) = 1 . Thenthe value process, U , the worst case risk, V , and the budget process, η , satisfy thefollowing equations U ( t, · ) = M ( t ) + f ( Z ( t )) ϑZ ( t ) + cV ( t, · ) = W ( t ) Z ( t ) (4.15) η ( t, · ) = ϑW ( t ) − M ( t ) − f ( Z ( t )) Z ( t ) − ϑc for all t ∈ [0 , T ] and a.a. ω ∈ { Z ( t ) > } , where ( Z, M, W ) is a F adapted P martingalethat satisfies the following terminal condition: Ö ZMW è ( T ) = Ö z ◦ ℓ ( T, · ) f ′ ◦ z ◦ ℓ ( T, · ) × z ◦ ℓ ( T, · ) − f ◦ z ◦ ℓ ( T, · ) z ◦ ℓ ( T, · ) × ℓ ( T, · ) è (4.16) Proof.
The function z defined by Eq. 4.13 provides a martingale density Z ∈ M + (1) bycomposition: Z ( t ) = E Ä z ◦ ℓ ( T, · ) | F t ä (4.17)for all t ∈ [0 , T ] . Z is exactly the first element of the vectorized process defined inEq. 4.16. It is indeed an element of M + (1) , for Z ( T ) = z ◦ ℓ ( T, · ) > and Z (0) = E Ä z ◦ ℓ ( T, · ) ä = 1 . The random variable C Z (0 , · ) := ℓ ( T, · ) − f ′ ( Z ( T, · )) ϑ = ℓ ( T, · ) − f ′ ◦ z ◦ ℓ ( T, · ) ϑ (4.18)is equal to the constant c Q Z a.s. In fact, c ∈ R is selected such that E Ä z ◦ ℓ ( T, · ) ä = E Ä z ◦ ℓ ( T, · ) ℓ ( T, · ) ∈ I c ä + E Ä z ◦ ℓ ( T, · ) ℓ ( T, · ) / ∈ I c ä = E Ä z ◦ ℓ ( T, · ) ℓ ( T, · ) ∈ I c ä by virtue of z ( x ) = 0 for all x / ∈ I c . Since C Z (0 , ω ) = c for all ω ∈ Ω satisfying ℓ ( T, ω ) ∈ I c , we have Q Z Ä C Z (0 , · ) = c ä = E Ä Z ( T ) C Z (0 , · )= c ä > E Ä z ◦ ℓ ( T, · ) ℓ ( T, · ) ∈ I c C Z (0 , · )= c ä = E Ä z ◦ ℓ ( T, · ) ℓ ( T, · ) ∈ I c ä = 1 Next we need to show that C Z (0 , · ) c P a.s. Notice that the function f ′ : (0 , ∞ ) → R is continuous and strictly increasing due to the convexity of f , implying that range ( f ′ ) =( f ′ (0 + ) , f ′ ( ∞ − )) . We conclude that range ( f ′ ) is an open interval and denote it by ( a, b ) ,where a and b can be either real numbers or ±∞ . According to the assumption, wehave P ( ℓ ( T, · ) < sup I c ) = P Ä ℓ ( T, · ) ∈ I c [ ℓ ( T, · ) { ϑ − a + c } ä ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 15
We extend the function f ′ continuously to zero by assigning f ′ (0) = a . P Ä C Z (0 , · ) c ä = E Ä ℓ ( T, · ) ∈ I c C Z (0 , · ) c ä + E Ä ℓ ( T, · ) / ∈ I c C Z (0 , · ) c ä = E Ä ℓ ( T, · ) ∈ I c ä + E Ä ℓ ( T, · ) / ∈ I c ℓ ( T, · ) − ϑ − f ′ (0) c ä = E Ä ℓ ( T, · ) ∈ I c ä + E Ä ℓ ( T, · ) / ∈ ( ϑ − a + c, ϑ − b + c ) ℓ ( T, · ) ϑ − a + c ä = P Ä ℓ ( T, · ) ∈ I c [ ℓ ( T, · ) ϑ − a + c ä = 1 We conclude that C Z (0 , · ) = c Q Z a.s. and C Z (0 , · ) c P a.s. According to Proposi tion. 4.2, Z defined in Eq. 4.17 is a worst case density process.The second component of Eq. 4.16 is a P martingale given by M ( t ) = E Ä f ′ ◦ z ◦ ℓ ( T, · ) × z ◦ ℓ ( T, · ) − f ◦ z ◦ ℓ ( T, · ) | F t ä = E Ä Z ( T ) f ′ ( Z ( T )) − f ( Z ( T )) | F t ä for all t ∈ [0 , T ] . Substituting Eq. 4.18 into Eq. 4.11, we have U ( t, · ) = E Q Z Ç C Z (0 , · ) + f ′ ( Z ( T, · )) ϑ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t å − E Ç f ( Z ( T )) − f ( Z ( t )) ϑZ ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t å = c + E Ç Z ( T ) f ′ ( Z ( T )) − f ( Z ( T )) + f ( Z ( t )) ϑZ ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t å = M ( t ) + f ( Z ( t )) ϑZ ( t ) + c By virtue of C Z (0 , · ) = c Q Z a.s., the equation above holds Q Z a.s. More precisely, itholds for a.a. ω ∈ { Z ( t ) > } . The third element of Eq. 4.16, W ( t ) = E Ä z ◦ ℓ ( T, · ) × ℓ ( T, · ) | F t ä = E Ä Z ( T ) ℓ ( T, · ) | F t ä , characterizes the worst case risk by V ( t, · ) = E Q Z (cid:16) ℓ ( T, · ) (cid:12)(cid:12)(cid:12) F t (cid:17) = E Ç Z ( T ) Z ( t ) ℓ ( T, · ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t å = W ( t ) Z ( t ) for all ω ∈ { ω ∈ Ω | Z ( t )( ω ) > } . Thus the equation above holds Q Z a.s. Following theexpressions for U ( t, · ) and V ( t, · ) , we get the formula for the budget process η ( t, · ) = ϑ Ä V ( t, · ) − U ( t, · ) ä = ϑW ( t ) − M ( t ) − f ( Z ( t )) Z ( t ) − ϑc (cid:3) In the proof above, we propose the inverse of the function f ′ , denoted by g : range ( f ′ ) → (0 , ∞ ) . Using this inverse function, we have the following propositionwhich states that certain integrability conditions guarantee the existence of the solu tion, given by Theorem 4.4, to the problem of model risk quantification. According to the definition of C Z (0 , · ) (Eq. 4.18), C Z (0 , ω ) = c for all ω ∈ Ω satisfying ℓ ( T, ω ) ∈ I c . Itfollows from Z ( T )( ω ) = z ◦ ℓ ( T, ω ) = 0 for a.a ω ∈ { ω ∈ Ω | ℓ ( T, ω ) / ∈ I c } that E Q Z ( C Z (0 , · ) | F t ) = E Q Z ( C Z (0 , · ) ℓ ( T, · ) ∈ I c | F t ) + E Q Z ( C Z (0 , · ) ℓ ( T, · ) / ∈ I c | F t )= Z ( t ) − E ( Z ( T, · ) C Z (0 , · ) ℓ ( T, · ) ∈ I c | F t ) + Z ( t ) − E ( Z ( T, · ) C Z (0 , · ) ℓ ( T, · ) / ∈ I c | F t )= cZ ( t ) − E ( Z ( T, · ) ℓ ( T, · ) ∈ I c | F t )= cZ ( t ) − E ( Z ( T, · ) | F t ) = c for a.a. ω ∈ { Z ( t ) > } . Proposition 4.5.
Denote g : ( a, b ) → (0 , ∞ ) as the inverse function of f ′ . If f ′ ( ∞ − ) = ∞ and for every c ∈ R g Ä ϑ ( ℓ ( T, · ) − c ) ä ℓ ( T, · ) ∈ I c is integrable under the reference measure P , then the assumptions in Theorem 4.4 hold.Proof. We need to prove the existence of c ∈ R and z : R → R + , such that Eq. 4.13 forall x ∈ I c and z ( x ) = 0 for all x / ∈ I c , E Ä z ◦ ℓ ( T, · ) ä = 1 and P ( ℓ ( T, · ) < sup I c ) = 1 .We have shown in the proof of Theorem 4.4 that range ( f ′ ) = ( a, b ) . Here b takes ∞ as the strictly increasing function f ′ diverges at infinity. For a given c ∈ R , the implicitequation Eq. 4.13 gives z ( x ) = g Ä ϑ ( x − c ) ä for all x ∈ I c = ( ϑ − a + c, ∞ ) . For all x / ∈ I c , z ( x ) = 0 which gives E Ä z ◦ ℓ ( T, · ) ä = E Ä z ◦ ℓ ( T, · ) ℓ ( T, · ) >ϑ − a + c ä + E Ä z ◦ ℓ ( T, · ) ℓ ( T, · ) ϑ − a + c ä = E Ä g Ä ϑ ( ℓ ( T, · ) − c ) ä ℓ ( T, · ) >ϑ − a + c ä We would like to show that the function K : R → R defined by K ( c ) := E Ä g Ä ϑ ( ℓ ( T, · ) − c ) ä ℓ ( T, · ) >ϑ − a + c ä (4.19)takes value of one for some c ∈ R .First we will show that K is continuous. Fix an arbitrary c ∈ R and ε ∈ (0 , ∞ ) .Resulted from the continuity of g , the function y ( · , ω ) : ( −∞ , c ] → R defined by y ( c, ω ) := Ä g Ä ϑ ( ℓ ( T, ω ) − c ) ä − g Ä ϑ ( ℓ ( T, ω ) − c ) ää ℓ ( T,ω ) >ϑ − a + c is continuous for every ω ∈ Ω . Therefore, the function Y : ( −∞ , c ] → R , defined by Y ( c ) := E ( y ( c, · )) , is continuous at c . Its continuity implies the existence of δ > such that | Y ( c ) | = | Y ( c ) − Y ( c ) | < ε/ for all c ∈ R satisfying c − δ < c c . Let δ − := min Ç δ, f ′ ( ε/ − aϑ å Then for all c − δ − < c c we have K ( c ) − K ( c ) = E Ä g Ä ϑ ( ℓ ( T, · ) − c ) ä ℓ ( T, · ) − ϑ − a ∈ ( c,c ] ä + Y ( c ) E Ä g Ä ϑ ( ϑ − a + c − c ) ä ℓ ( T, · ) − ϑ − a ∈ ( c,c ] ä + Y ( c ) < g ( a + ϑδ − ) + ε/ g Ä f ′ ( ε/ ä + ε/ ε It follows from the dominated convergence theorem that Y is continuous at c . In fact, the sequence, { y ( c − /n, · ) } ∞ n =1 , of real valued measurable functions converges pointwise to y ( c , · ) by virtue of itscontinuity. The sequence is dominated by y ( c − , · ) due to the fact that g increases monotonically. y ( c − , · ) is integrable as E (cid:0) | y ( c − , · ) | (cid:1) E (cid:0) g (cid:0) ϑ ( ℓ ( T, · ) − c + 1 (cid:1) ℓ ( T,ω ) >ϑ − a + c (cid:1) E (cid:0) g (cid:0) ϑ ( ℓ ( T, · ) − c + 1 (cid:1) ℓ ( T,ω ) >I c − (cid:1) < ∞ The dominated convergence theorem guarantees the convergence of the expectation lim n →∞ E (cid:0) y ( c − /n, · ) (cid:1) = E (cid:0) y ( c , · ) (cid:1) = 0 This means that given an arbitrary ε > , there exists n ∈ N such that (cid:12)(cid:12) E (cid:0) y ( c − /n, · ) (cid:1)(cid:12)(cid:12) < ε for all n > n . Due to the fact that g increases monotonically, for every c ∈ [ c − /n , c ] we have E (cid:0) y ( c, · ) (cid:1) − E (cid:0) y ( c , · ) (cid:1) = E (cid:0) y ( c, · ) (cid:1) E (cid:0) y ( c − /n, · ) (cid:1) < ε This proves that Y is continuous at c . ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 17
We may prove in a similar way that there exists δ + > such that K ( c ) − K ( c ) ∈ ( − ε, for all c < c < c + δ + . Combining the two arguments, | K ( c ) − K ( c ) | is less than ε forall c ∈ R satisfying | c − c | < min( δ + , δ − ) . This proves that the function K , defined inEq. 4.19, is continuous.Next we need to prove that there exist c + , c − ∈ R such that K ( c + ) and K ( c − ) > .In fact, the limit lim c →−∞ P Ä ℓ ( T, · ) > ϑ − a + c ä = 1 implies the existence of c ∈ R suchthat P Ä ℓ ( T, · ) > ϑ − a + c ä > /ξ for some ξ > . Defining c − := c − max Ä , f ′ ( ξ ) − a ä ϑ c we have K ( c − ) > E Ä g Ä ϑ ( ℓ ( T, · ) − c − ) ä ℓ ( T, · ) >ϑ − a + c ä > g Ä ϑ ( ϑ − a + c − c − ) ä E Ä ℓ ( T, · ) >ϑ − a + c ä > g Ä ϑ ( ϑ − a + ϑ − ( f ′ ( ξ ) − a ) ä E Ä ℓ ( T, · ) >ϑ − a + c ä = ξ P Ä ℓ ( T, · ) > ϑ − a + c ä > On the other hand, the following limit lim c →∞ E Ä g ( ϑℓ ( T, · )) ℓ ( T, · ) − ϑ − a ∈ (0 ,c ) ä = E Ä g ( ϑℓ ( T, · )) ℓ ( T, · ) >ϑ − a ä < ∞ implies the existence of c ∈ R such that E Ä g ( ϑℓ ( T, · )) ℓ ( T, · ) − ϑ − a ∈ (0 ,c ) ä > E Ä g ( ϑℓ ( T, · )) ℓ ( T, · ) >ϑ − a ä − Letting c + = max(0 , c ) , we have K ( c + ) E Ä g ( ϑℓ ( T, · )) ℓ ( T, · ) >ϑ − a + c + ä E Ä g ( ϑℓ ( T, · )) ℓ ( T, · ) >ϑ − a + c ä = E Ä g ( ϑℓ ( T, · )) ℓ ( T, · ) >ϑ − a ä − E Ä g ( ϑℓ ( T, · )) ℓ ( T, · ) − ϑ − a ∈ (0 ,c ) ä According to the intermediate value theorem, there exists c ∈ R such that the contin uous function K , defined in Eq. 4.19, takes the value of one. The condition P ( ℓ ( T, · ) < sup I c ) = 1 holds irrespective of the actual measure P , for [ x ∈ I c ℓ ( T, · ) x = [ x>ϑ − a + c { ω ∈ Ω | ℓ ( T, ω ) x } = { ω ∈ Ω | ℓ ( T, ω ) ∈ R } has probability one. As a result, the assumptions stated in Theorem 4.4 are valid,which guarantees the existence of the worst case solution provided by the theorem. (cid:3) We consider a special class of f divergence, including the renowned Kullback Leiblerdivergence, of which the function R ∋ x xf ′ ( x ) − f ( x ) is linear (or equivalently x xf ′′ ( x ) is constant). This type of f divergence has a particular advantage onapplying Theorem. 4.4, because the process M ( t ) = E Ä Z ( T ) f ′ ( Z ( T )) − f ( Z ( T )) | F t ä = E ( Z ( T ) | F t ) × f ′ Ä E ( Z ( T ) | F t ) ä − f Ä E ( Z ( T ) | F t ) ä The convergence is guaranteed by the dominated convergence theorem. See the footnote in the lastpage. Such c ∈ R is also unique by noticing that the function K is strictly decreasing. = Z ( t ) f ′ ( Z ( t )) − f ( Z ( t )) (4.20)can be calculated directly from Z ( t ) . Therefore in practice we only need to apply back ward induction to the two dimensional P martingale ( Z ( t ) , W ( t )) t ∈ [0 ,T ] . By substitutingEq. 4.20 into Eq. 4.15, we have the following proposition. Corollary 4.6.
Suppose in Theorem 4.4 there exists d ∈ (0 , ∞ ) such that xf ′′ ( x ) = d forall x ∈ R + . Then the value process, U , the worst case risk, V , and the budget process, η , satisfy the following equations U ( t, · ) = f ′ ( Z ( t )) ϑ + cV ( t, · ) = W ( t ) Z ( t ) (4.21) η ( t, · ) = ϑW ( t ) Z ( t ) − f ′ ( Z ( t )) − ϑc for all t ∈ [0 , T ] and all ω ∈ Ω such that Z ( t )( ω ) > , where ( Z, W ) is a F adapted P martingale that satisfies the following terminal condition: Ç ZW å ( T ) = Ç z ◦ ℓ ( T, · ) z ◦ ℓ ( T, · ) × ℓ ( T, · ) å Corollary 4.6 applies to the Kullback Leibler divergence. In particular, the calcu lation of the constant c is pretty straightforward. We illustrate this in the followingcorollary. Corollary 4.7.
Under the Kullback Leibler divergence, suppose E Ä e ϑℓ ( T, · ) ä < ∞ . Thenthere exists an unique solution to the problem of model risk quantification, given by U ( t, · ) = ln ˜ Z ( t ) ϑV ( t, · ) = ˜ W ( t )˜ Z ( t ) η ( t, · ) = ϑ ˜ W ( t )˜ Z ( t ) − ln ˜ Z ( t ) where Ä ˜ Z, ˜ W ä is a F adapted P martingale that satisfies the terminal condition: ˜ Z ˜ W ! ( T ) = Ç exp ( ϑℓ ( T, · )) ℓ ( T, · ) exp ( ϑℓ ( T, · )) å Proof.
The Kullback Leibler divergence adopts f ′ ( x ) = ( x ln x ) ′ = ln x + 1 for all x ∈ (0 , ∞ ) . f ′ diverges at ∞ . The inverse function g : R → (0 , ∞ ) is given by g ( x ) = e x − .Since E Ä e ϑℓ ( T, · ) ä < ∞ , we have E (cid:16)(cid:12)(cid:12)(cid:12) g Ä ϑ ( ℓ ( T, · ) − c ) ä ℓ ( T, · ) ∈ I c (cid:12)(cid:12)(cid:12)(cid:17) = e − ϑc − E Ä e ϑℓ ( T, · ) ä < ∞ for all c ∈ R . Proposition 4.5 guarantees the existence of a unique c ∈ R and z : R → R + satisfying E Ä z ◦ ℓ ( T, · ) ä = 1 , therefore a unique solution to the problem of model riskquantification.More specifically, we calculate the function z : R → R + from Eq. 4.13: z ( x ) = e ϑ ( x − c ) − ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 19 for all x ∈ R . The constant c ∈ R is given by E Ä z ◦ ℓ ( T, · ) ä = E Ä e ϑ ( ℓ ( T, · ) − c ) − ä ⇔ c = 1 ϑ ln E Ä e ϑℓ ( T, · ) − ä = ln ˜ Z (0) − ϑ The corollary defines two P martingales by ˜ Z ( t ) = E (cid:16) e ϑℓ ( T, · ) (cid:12)(cid:12)(cid:12) F t (cid:17) ˜ W ( t ) = E (cid:16) ℓ ( T, · ) e ϑℓ ( T, · ) (cid:12)(cid:12)(cid:12) F t (cid:17) The process Z and W in Corollary 4.6 are simply normalized versions of ˜ Z and ˜ W , Z ( t ) = E Ä z ◦ ℓ ( T, · ) | F t ä = E (cid:16) e ϑ ( ℓ ( T, · ) − c ) − (cid:12)(cid:12)(cid:12) F t (cid:17) = ˜ Z ( t )˜ Z (0) W ( t ) = E Ä z ◦ ℓ ( T, · ) × ℓ ( T, · ) | F t ä = E (cid:16) ℓ ( T, · ) e ϑ ( ℓ ( T, · ) − c ) − (cid:12)(cid:12)(cid:12) F t (cid:17) = ˜ W ( t )˜ Z (0) Substituting the equations above into Eq. 4.21, we have U ( t, · ) = ln( Z ( t )) + 1 ϑ + c = ln ˜ Z ( t ) ϑV ( t, · ) = W ( t ) Z ( t ) = ˜ W ( t )˜ Z ( t ) η ( t, · ) = ϑW ( t ) Z ( t ) − Ä ln( Z ( t )) − ä − ϑc = ϑ ˜ W ( t )˜ Z ( t ) − ln ˜ Z ( t ) Note that Z ( T )( ω ) = e ϑ ( ℓ ( T,ω ) − c ) − > for all ω ∈ Ω . Z ( t ) = E ( Z ( T ) | F t ) > , implyingthat the equations above hold for all t ∈ [0 , T ] and all ω ∈ Ω . (cid:3) Model Risk Measurement with Continuous Semimartingales
The last section provides the general theory on quantifying the model risk. In thissection, we focus on the class of continuous semimartingales. It has an importantproperty formulated by the functional Ito formula. To introduce the formula we needto briefly review the functional Ito calculus (Bally et al. 2016). First we define thehorizontal derivative and the vertical derivative of a non anticipative functional F :Λ dT → R . Its horizontal derivative at ( t, ω ) ∈ Λ dT is defined by the limit D F ( t, ω ) := lim h → + F ( t + h, ω ) − F ( t, ω ) h if it exists. Intuitively, it describes the rate of change w.r.t time, assuming no changeof the state variable from t onwards, and conditional to its history up to t given bythe stopped path ω t . On the other hand, the vertical derivative describes the rate ofchange w.r.t the state variable from t onwards. Formally, the vertical derivative at ( t, ω ) ∈ Λ dT , denoted by ∇ ω F ( t, ω ) , is defined as the gradient of the function R d ∋ x F Ä t, ω t + x [ t,T ] ä at , assuming its existence. The horizontal and vertical derivatives ofa non anticipative functional are also non anticipative functionals.We define the left continuous non anticipative functionals by noticing that the spaceof stopped paths, Λ dT , is endowed with a metric d ∞ . Suppose F : Λ dT → R is a non anticipative functional. F is left continuous if for every ( t, ω ) ∈ Λ dT and ε > , thereexists δ > such that | F ( t, ω ) − F ( t ′ , ω ′ ) | < ε for all ( t ′ , ω ′ ) ∈ Λ dT satisfying t ′ < t and d ∞ (( t, ω ) , ( t ′ , ω ′ )) < δ . We may further impose a boundedness condition to anon anticipative functional F . It states that for any compact K ⊂ R d and t < T ,there exists a C > such that | F ( t, ω ) | C for all t t and ω ∈ Ω . Suppose a non anticipative functional F is horizontally differentiable and vertically twice differentiablefor all ( t, ω ) ∈ Λ dT , and D F , ∇ ω F and ∇ ω F satisfy the boundedness condition above. Inaddition, F , ∇ ω F and ∇ ω F are left continuous, and D F is continuous for all ( t, ω ) ∈ Λ dT . Then we call F a regular functional.Suppose the canonical process X on Ω is a continuous semimartingale and F : Λ dT → R is a regular functional. The R valued process ( Y ( t )) t ∈ [0 ,T ] , defined by Y ( t ) = F ( t, · ) for all t ∈ [0 , T ] , follows the functional Ito formula P a.s.(Bally et al. 2016, pp. 190–191) Y ( t ) − Y (0) = Z t D F ( u, · ) du + Z t ∇ ω F ( u, · ) dX ( u ) + 12 Z t Tr Ä ∇ ω F ( u, · ) d [ X ]( u ) ä If we further impose the constraint that R T ξ ( t ) dX ( t ) = 0 for all bounded predictableprocesses ξ satisfying R T ξ ( t ) dt = 0 , then the canonical process X is a strong solutionto the SDE (Revuz and Yor 2013) dX ( t ) = µ ( t ) dt + σ ( t ) dW ( t ) (5.1)where ( W ( t )) t ∈ [0 ,T ] is a R d valued standard Wiener process on the underlying filteredprobability space (assuming its existence). ( µ ( t )) t ∈ [0 ,T ] is a R d valued predictable pro cess, and ( σ ( t )) t ∈ [0 ,T ] is a R d valued predictable process. We may identify their ele ments, say ( µ i ( t )) t ∈ [0 ,T ] and ( σ ij ( t )) t ∈ [0 ,T ] , with non anticipative functionals. The SDEEq. 5.1 may be regarded as a path dependent generalisation of the renowned Ito dif fusion process. The existence and uniqueness of its solutions have been given in theliterature by imposing various conditions (e.g. boundedness and Lipschitz proper ties, see Bally et al. (2016)). Now if X satisfies Eq. 5.1 P a.s., then it follows from thefunctional Ito formula that the process Y is a strong solution to the SDE dY ( t ) = Ñ D F ( t, · ) + µ ( t ) ∇ ω F ( t, · ) + Tr Ä σ ( t ) ∇ ω F ( t, · ) ä é dt + σ ( t ) ∇ ω F ( t, · ) dW ( t ) Note that the square of σ ( t ) is in the sense of matrix multiplication, i.e. σ ( t ) = σ ( t ) σ ( t ) T . For simplicity we may define a nonlinear differential operator A that sendsa regular functional to a non anticipative functional by A F := D F + µ ( t ) ∇ ω F + 12 Tr Ä σ ( t ) ∇ ω F ä (5.2)Then the process Y , defined by Y ( t ) = F ( t, · ) , is a strong solution to dY ( t ) = A F ( t, · ) dt + σ ( t ) ∇ ω F ( t, · ) dW ( t ) (5.3)Suppose Y is a P martingale, then the regular functional F satisfies A F = 0 P a.s.Applying this property, we may convert the martingale statement in Theorem 4.4 to ananalytical statement. This is formulated in the following corollary. Corollary 5.1.
Given ϑ ∈ (0 , ∞ ) , suppose there exist c ∈ R and z : R → R + definedin Theorem 4.4. If the canonical process X satisfies Eq. 5.1 for some R d valued pre dictable process ( µ ( t )) t ∈ [0 ,T ] and R d valued predictable process ( σ ( t )) t ∈ [0 ,T ] , then the ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 21 value process, U , the worst case risk, V , and the cost process, η , satisfy the followingequations U ( t, · ) = M ( t ) + f ( Z ( t )) ϑZ ( t ) + cV ( t, · ) = W ( t ) Z ( t ) η ( t, · ) = ϑW ( t ) − M ( t ) − f ( Z ( t )) Z ( t ) − ϑc for all t ∈ [0 , T ] and all ω ∈ Ω such that Z ( t )( ω ) > , where Z , M and W are identifiedby the solutions to the equation A F = 0 ( P a.s.), subject to their respective terminalconditions: Ö ZMW è ( T ) = Ö z ◦ ℓ ( T, · ) f ′ ◦ z ◦ ℓ ( T, · ) × z ◦ ℓ ( T, · ) − f ◦ z ◦ ℓ ( T, · ) z ◦ ℓ ( T, · ) × ℓ ( T, · ) è In practice, we are more interested in the type of f divergence that gives the constantfunction x xf ′′ ( x ) . Such f divergence allows us to solve U and V directly usingpath dependent partial differential equations. Proposition 5.2.
Suppose there exists d ∈ (0 , ∞ ) such that xf ′′ ( x ) = d for all x ∈ R + ,and the function f ′ diverges at infinity. In addition, the inverse function, g : Im f ′ → (0 , ∞ ) , provides a twice differentiable function R ∋ x g ( x ) x ∈ Im f ′ . The value processand the worst case risk, identified with the regular functionals U t := U ( t, · ) and V t := V ( t, · ) , solve the following path dependent partial differential equations Q Z a.s. A U t + θ g ′′ Ä ϑ ( U t − c ) ä g ′ Ä ϑ ( U t − c ) ä ( σ t ∇ ω U t ) = 0 A V t + ϑg ′ Ä ϑ ( U t − c ) ä ∇ ω U t σ t g Ä ϑ ( U t − c ) ä ∇ ω V t = 0 (5.4) subject to the terminal condition U T = V T = ℓ ( T, · ) . The cost process η t = ϑ ( V t − U t ) for all t ∈ [0 , T ] . Defining I c := { ϑ − y + c | y ∈ Im f ′ } , the solution exists if g Ä ϑ ( ℓ ( T, · ) − c ) ä ℓ ( T, · ) ∈ I c is integrable for every c ∈ R .Proof. It follows from Corollary 4.6 that Z ( t ) = g ( ϑ ( U t − c )) Z ( t ) > = g ( ϑ ( U t − c )) U t ∈ I c := g ( ϑ ( U t − c )) We have shown in the proof of Proposition 4.5 that f ′ diverges at infinity implies that Im f ′ is an openinterval in the form of ( a, ∞ ) . Then U ( t, ω ) = ϑ − f ′ ( Z ( t )( ω )) + c > ϑ − a + c ∈ I c for all ω ∈ { ω ∈ Ω | Z ( t )( ω ) > } . On the other hand, for all ω ∈ { ω ∈ Ω | Z ( t )( ω ) = 0 } , Z ( t )( ω ) = E Q Z ( Z ( T ) ℓ ( T, · ) ϑ − a + c | F t )( ω ) + E Q Z ( Z ( T ) ℓ ( T, · ) >ϑ − a + c | F t ) > E Q Z ( g ( ϑ ( ℓ ( T, · ) − c )) ℓ ( T, · ) >ϑ − a + c | F t )( ω ) This implies that E Q Z ( ℓ ( T, · ) >ϑ − a + c | F t )( ω ) = 0 by virtue of Im g = (0 , ∞ ) , which gives U ( t, ω ) = E Q Z ( ℓ ( T, · ) | F t )( ω ) = E Q Z ( ℓ ( T, · ) ℓ ( T, · ) ϑ − a + c | F t )( ω ) ϑ − a + c / ∈ I c for all t ∈ [0 , T ] , where g denotes the twice differentiable function R ∋ x g ( x ) x ∈ Im f ′ .Since ( Z ( t )) t ∈ [0 ,T ] is a P martingale that can be identified with a solution to the equation A F = 0 ( P a.s.), we have A g ( ϑ ( U t − c )) = g ′ Ä ϑ ( U t − c ) ä A U t + θ g ′′ Ä ϑ ( U t − c ) ä ( σ t ∇ ω U t ) For all ω ∈ Ω such that U ( t, ω ) ∈ I c , the equation is equivalent to A U t + θ g ′′ Ä ϑ ( U t − c ) ä g ′ Ä ϑ ( U t − c ) ä ( σ t ∇ ω U t ) = 0 (5.5)Noticing that { ω ∈ Ω | U ( t, ω ) ∈ I c } has measure one under Q Z , the equation aboveholds Q Z a.s.It follows from Eq. 5.3 that the P martingale ( Z ( t )) t ∈ [0 ,T ] solves the SDE dZ ( t ) = A g ( ϑ ( U t − c )) dt + σ t ∇ ω g ( ϑ ( U t − c )) dW ( t ) = ϑ g ′ ( ϑ ( U t − c )) σ t ∇ ω U t dW ( t ) We may define a process ( Y ( t )) t ∈ [0 ,T ] by the stochastic integral Y ( t ) := Z t Ç ϑg ′ ( ϑ ( U s − c )) g ( ϑ ( U s − c )) U s ∈ I c σ t ∇ ω U s å dW ( s ) for all t ∈ [0 , T ] . This transforms the SDE above into dZ ( t ) = ϑg ′ ( ϑ ( U t − c )) U t ∈ I c σ t ∇ ω U t dW ( t ) = g ( ϑ ( U t − c )) U t ∈ I c dY ( t ) = Z ( t ) dY ( t ) suggesting that the process ( Z ( t )) t ∈ [0 ,T ] is a Doleans Dade exponent, i.e. Z = E ( Y ) .Note that the SDE above ensures that ( Z ( t )) t ∈ [0 ,T ] is a local martingale. To guaranteethat it is indeed a martingale, we assume the Novikov’s condition, E exp Z T Ç ϑg ′ ( ϑ ( U t − c )) g ( ϑ ( U t − c )) U t ∈ I c σ t ∇ ω U t å dt !! < ∞ According to the Girsanov theorem, the Brownian motion under Q Z is given by addingan extra drift term. Noticing that U t ∈ I c Q Z a.s., the Girsanov theorem transforms theSDE of the canonical process under P (Eq. 5.1) to the following SDE (in the sense that ( X ( t )) t ∈ [0 ,T ] is a strong solution of the following under Q Z ), dX ( t ) = Ç µ t + ϑg ′ ( ϑ ( U t − c )) g ( ϑ ( U t − c )) σ t ∇ ω U t å dt + σ t dW Q Z ( t ) (5.6)The functional Ito formula, Eq. 5.2 5.3, applies to the alternative measure Q Z as well.Following the definition of the operator A , we have A Q Z F ( t, · ) = D F ( t, · ) + Ç µ t + ϑg ′ ( ϑ ( U t − c )) g ( ϑ ( U t − c )) ∇ ω U t σ t å ∇ ω F ( t, · ) + 12 Tr Ä σ ( t ) ∇ ω F ( t, · ) ä For all x ∈ ( a, ∞ ) , g ′ ( x ) = g ′ ( x ) > (due to the convexity of f ), and for all x ∈ ( −∞ , a ] , g ′ ( x ) = lim h → − g ( x ) − g ( x − h ) h = 0 Therefore, g ( x ) = g ( x ) x ∈ ( a, ∞ ) implies that g ′ ( x ) = g ′ ( x ) x ∈ ( a, ∞ ) , which in turns implies g ′′ ( x ) = g ′′ ( x ) x ∈ ( a, ∞ ) . For all ω ∈ { ω ∈ Ω | U ( t, ω ) ∈ I c } , ϑU ( t, ω ) − c ∈ ( a, ∞ ) and thus g ′ (cid:0) ϑ ( U ( t, ω ) − c ) (cid:1) = g ′ (cid:0) ϑ ( U ( t, ω ) − c ) (cid:1) > and g ′′ (cid:0) ϑ ( U ( t, ω ) − c ) (cid:1) = g ′′ (cid:0) ϑ ( U ( t, ω ) − c ) (cid:1) Q Z ( U ( t, · ) ∈ I c ) = Q Z ( Z ( t ) >
0) = E ( Z ( T ) Z ( t ) > ) = E ( Z ( t ) Z ( t ) > ) = E ( Z ( t )) = 1 ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 23 = A F ( t, · ) + g ′ ( ϑσ t ( U t − c )) ∇ ω U t σ t g ( ϑ ( U t − c )) ∇ ω F ( t, · ) for some regular functional F : Λ dT → R and all t ∈ [0 , T ] . The worst case modelrisk, E Q Z ( ℓ ( T, · ) | F t ) , is a Q Z martingale. Identified with the regular functional V , itsatisfies the following equation Q Z a.s. A Q Z V t = A V t + ϑg ′ Ä ϑ ( U t − c ) ä ∇ ω U t σ t g Ä ϑσ t ( U t − c ) ä ∇ ω V t (5.7)Combined with the terminal condition U T = V T = ℓ ( T, · ) , Eq. 5.8 and Eq. 5.7 providethe path dependent partial differential equations that govern the value process and theworst case risk, respectively. It follows from Proposition 4.5 that the solution indeedexists if g Ä ϑ ( ℓ ( T, · ) − c ) ä ℓ ( T, · ) ∈ I c is integrable for every c ∈ R . (cid:3) The renowned Kullback Leibler divergence provides us with much convenience onapplying Proposition 5.2 into practice. The function f ′ ( x ) = ln x + 1 diverges at ∞ , andits inverse g : R → (0 , ∞ ) given by g ( x ) = e x − is twice differentiable. In addition, theworst case martingale density Z ( T ) = e ϑ ( ℓ ( T, · ) − c ) − > supplies a measure Q Z that isequivalent to the reference measure P . Combining Corollary 4.7 with Proposition 5.2,and substituting g ( x ) = e x − into Eq. 5.4, we get the following corollary that applies tothe Kullback Leibler divergence. Corollary 5.3.
Under the Kullback Leibler divergence, suppose E Ä e ϑℓ ( T, · ) ä < ∞ . Thenthere exists an unique solution to the problem of model risk quantification. The valueprocess and the worst case risk, identified with regular functionals U t := U ( t, · ) and V t := V ( t, · ) , solve the following path dependent partial differential equations P a.s. A U t + ϑ σ t ∇ ω U t ) = 0 A V t + ϑ ∇ ω U t σ t ∇ ω V t = 0 (5.8) subject to the terminal condition U T = V T = ℓ ( T, · ) . The cost process η t = ϑ ( V t − U t ) forall t ∈ [0 , T ] . In practice, the path dependent partial differential equations, Eq. 5.8, are generallydifficult to solve. However, we may convert Eq. 5.8 into normal non linear partialdifferential equations for a special type of path dependency, formulated by ℓ ( T, · ) = h ( T, X ( T )) + Z T h ( t, X ( t )) dt + Z T h ( t, X ( t )) dX ( t ) (5.9)for some functions h : [0 , T ] × R d → R d and h i : [0 , T ] × R d → R ( i = 1 , . We furtherrestrict the canonical process X to the class of Ito diffusions. This means that theprocess is Markovian, and there exist functions µ : [0 , T ] × R d → R d and σ : [0 , T ] × R d → R d such that µ t = µ ( t, X ( t )) and σ t = σ ( t, X ( t )) . The path dependent partialdifferential equations, Eq. 5.8, degenerates to normal partial differential equations. Corollary 5.4.
Under the Kullback Leibler divergence, suppose E Ä e ϑℓ ( T, · ) ä < ∞ , thecanonical process ( X ( t )) t ∈ [0 ,T ] solves the SDE, dX ( t ) = µ ( t, X ( t )) dt + σ ( t, X ( t )) dW ( t ) ,and the cumulative loss ℓ ( T, · ) takes the form of Eq. 5.9. If there exists a function ˜ u : [0 , T ] × R d → R that solves the partial differential equation ∂ ˜ u ( t, x ) ∂t + µ ( t, x ) Ç ∂ ˜ u ( t, x ) ∂x + h ( t, x ) å + ϑ Ç σ ( t, x ) Ç ∂ ˜ u ( t, x ) ∂x + h ( t, x ) åå + 12 Tr Ç σ ( t, x ) ∂ ˜ u ( t, x ) ∂x å + h ( t, x ) = 0 (5.10) and a function ˜ v : [0 , T ] × R d → R that solves the partial differential equation ∂ ˜ v ( t, x ) ∂t + ϑ Ç ∂ ˜ u ( t, x ) ∂x + h ( t, x ) å σ ( t, x ) Ç ∂ ˜ v ( t, x ) ∂x + h ( t, x ) å + µ ( t, x ) Ç ∂ ˜ v ( t, x ) ∂x + h ( t, x ) å + 12 Tr Ç σ ( t, x ) ∂ ˜ v ( t, x ) ∂x å + h ( t, x ) = 0 (5.11) subject to the terminal condition ˜ u ( T, · ) = ˜ v ( T, · ) = h ( T, · ) , then the value process, theworst case risk and the cost process, identified with regular functionals, follow U t = ˜ u ( t, X ( t )) + Z t h ( s, X ( s )) ds + Z t h ( s, X ( s )) dX ( s ) V t = ˜ v ( t, X ( t )) + Z t h ( s, X ( s )) ds + Z t h ( s, X ( s )) dX ( s ) and η t = ϑ Ä ˜ v ( t, X ( t )) − ˜ u ( t, X ( t )) ä for all t ∈ [0 , T ] .Proof. We first define regular functionals ˜ U, ˜ V : Λ dT → R by ˜ U t := U t − Z t h ( s, X ( s )) ds − Z t h ( s, X ( s )) dX ( s )˜ V t := V t − Z t h ( s, X ( s )) ds − Z t h ( s, X ( s )) dX ( s ) (5.12)The horizontal and vertical derivatives can be derived from Eq. 5.12, D ˜ U t = D U t − h ( t, X ( t )) and D ˜ V t = D V t − h ( t, X ( t )) ∇ ω ˜ U t = ∇ ω U t − h ( t, X ( t )) and ∇ ω ˜ V t = ∇ ω V t − h ( t, X ( t )) ∇ ω ˜ U t = ∇ ω U t and ∇ ω ˜ V t = ∇ ω V t Substituting the equations above into Eq. 5.8, we transform Eq. 5.8 to D ˜ U t + µ ( t, X ( t )) Ä ∇ ω ˜ U t + h ( t, X ( s )) ä + ϑ Ä σ ( t, X ( t )) Ä ∇ ω ˜ U t + h ( t, X ( s )) ää + 12 Tr Ä σ ( t, X ( t )) ∇ ω ˜ U t ä + h ( t, X ( t )) = 0 (5.13)and D ˜ V t + ϑ Ä ∇ ω ˜ U t + h ( t, X ( s )) ä σ ( t, X ( t )) Ä ∇ ω ˜ V t + h ( t, X ( s )) ä + µ ( t, X ( t )) Ä ∇ ω ˜ V t + h ( t, X ( s )) ä + 12 Tr Ä σ ( t, X ( t )) ∇ ω ˜ V t ä + h ( t, X ( t )) = 0 (5.14)If there exists a function ˜ u : [0 , T ] × R d → R that solves the partial differential equation ∂ ˜ u ( t, x ) ∂t + µ ( t, x ) Ç ∂ ˜ u ( t, x ) ∂x + h ( t, x ) å + ϑ Ç σ ( t, x ) Ç ∂ ˜ u ( t, x ) ∂x + h ( t, x ) åå + 12 Tr Ç σ ( t, x ) ∂ ˜ u ( t, x ) ∂x å + h ( t, x ) = 0 ON PARAMETRIC ROBUST MODEL RISK MEASUREMENT WITH PATH DEPENDENT LOSS FUNCTIONS 25 and a function ˜ v : [0 , T ] × R d → R that solves ∂ ˜ v ( t, x ) ∂t + ϑ Ç ∂ ˜ u ( t, x ) ∂x + h ( t, x ) å σ ( t, x ) Ç ∂ ˜ v ( t, x ) ∂x + h ( t, x ) å + µ ( t, x ) Ç ∂ ˜ v ( t, x ) ∂x + h ( t, x ) å + 12 Tr Ç σ ( t, x ) ∂ ˜ v ( t, x ) ∂x å + h ( t, x ) = 0 then the regular functionals defined by ˜ U t := ˜ u ( t, X ( t )) and ˜ V t := ˜ v ( t, X ( t )) , for all t ∈ [0 , T ] , satisfy Eqs. 5.13 and 5.14. The terminal condition ˜ U T = ˜ V T = h ( T, X ( T )) is satisfied if ˜ u ( T, x ) = ˜ v ( T, x ) = h ( T, x ) holds for all x ∈ R . (cid:3) Note that Eq. 5.10 5.11 are non linear parabolic partial differential equations and ingeneral have to be solved numerically.6.
Concluding Remarks
This paper provides a theoretical framework of formulating and solving the problemof model risk quantification in a path dependent setting. We need several ingredi ents to formulate the problem, including terminal time T , a (path dependent) lossfunction ℓ , a nominal model (i.e. a canonical process ( X t ) t ∈ [0 ,T ] under a nominal mea sure P ) and some f divergence. The non parametric nature of this approach relieson the f divergence to restrict the set of proper alternative models. This is, however,only applicable to measures that are absolutely continuous w.r.t the nominal mea sure. More generic distance measure, such as the Wasserstein metric, may be appliedinstead (Feng and Schl¨ogl 2018). Despite of this incompleteness, f divergence, espe cially the Kullback Leibler divergence, is most tractable and yield simple results forpath dependent problems. References
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Yu Feng, Finance Discipline Group, University of Technology Sydney, P.O. Box 123, Broadway, NSW2007, Australia
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