Notes on Backward Stochastic Differential Equations for Computing XVA
aa r X i v : . [ q -f i n . M F ] J un Notes on Backward Stochastic Differential Equations forComputing XVA
Jun Sekine ∗† and Akihiro Tanaka ‡ Abstract
The X-valuation adjustment (XVA) problem, which is a recent topic in mathematicalfinance, is considered and analyzed. First, the basic properties of backward stochastic dif-ferential equations (BSDEs) with a random horizon in a progressively enlarged filtration arereviewed. Next, the pricing/hedging problem for defaultable over-the-counter (OTC) deriva-tive securities is described using such BSDEs. An explicit sufficient condition is given to ensurethe non-existence of an arbitrage opportunity for both the seller and buyer of the derivativesecurities. Furthermore, an explicit pricing formula is presented in which XVA is interpretedas approximated correction terms of the theoretical fair price.
Keywords : BSDE, XVA, derivative pricing, defaultable security, arbitrage-free price
Backward stochastic differential equations (BSDEs) have been studied intensively from both theo-retical and application viewpoints. Bismut (1976, 1978) studied BSDEs related to stochastic controlproblems, and Pardoux and Peng (1990) introduced general nonlinear
BSDEs driven by Brownianmotion as a noise process. After those early pioneering studies and since the late 1990s, the fieldof mathematical finance has provided various interesting research topics to develop the theory andapplication of BSDEs (e.g., El Karoui et al., 2000). In the present paper, we are interested in onesuch recent research topic in mathematical finance, namely, the X-valuation adjustment (XVA)problem. The pricing and hedging methodology for over-the-counter (OTC) financial derivativesecurities for practitioners in financial institutions has been modified since the global financial crisisin 2008. The pre-crisis pricing was based on the Black–Scholes–Merton paradigm, andp RN := E [DF r ( T ) ξ T ]was regarded as the “fair” price of the derivative security ( T, ξ T ). Here, ξ T is a random vari-able representing the payoff at the maturity date T ∈ R ++ (:= (0 , ∞ )) of the derivative security, ∗ Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama-cho, Toyonaka, Osaka, 560-8531, Japan Email: [email protected] † Jun Sekine’s research is supported by a Grant-in-Aid for Scientific Research (C), No. 19K03636, from the JapanSociety for the Promotion of Science. ‡ Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama-cho, Toyonaka, Osaka, 560-8531, Japan / Sumitomo Mitsui Banking Corporation, 1-1-2, Marunouchi, Chiyoda-ku, Tokyo, 100-0005, Japan,Email: [email protected] r ( T ) := exp n − R T r ( u ) du o is a suitable discounting factor, where r := ( r ( t )) t ≥ is a risk-freeinterest rate process, and E [( · )] represents the expectation with respect to the so-called risk-neutralprobability measure. By contrast, the post-crisis pricing formula used by practitioners in financialinstitutions is now described as ¯p RN + X x x VA (1)for the derivative security (
T, ξ T ). Here,¯p RN := E [DF ¯ r ( T ) ξ T ] , employing ¯ r := (¯ r ( t )) t ≥ as a risk-free interest rate process, which is different from r used in thepre-crisis model, and X x x VA = CVA − DVA + FVA + ColVA + · · · represents various valuation adjustments (e.g., credit valuation adjustment, debt valuation ad-justment, funding valuation adjustment, collateral valuation adjustment). We may interpret thepost-crisis modification as reflecting the following current situations.(a) The credit risk (default risk) of investors and their counterparties and the liquidity risk (ofassets and cash) are widely recognized and and now considered seriously.(b) As a consequence of (a), the differences in various interest rates (e.g., risk-free rate, reporate, funding rate, collateral rate) can no longer be neglected.In this paper, we aim to understand the post-crisis pricing formula (1) in a better way from atheoretical viewpoint. Using BSDEs, which model the value processes of hedging portfolios, weinterpret (1) as an approximate value of the fair price (i.e., the replication cost) of a derivativesecurity. Concretely, this paper is organized as follows. • In Section 2, we prepare a BSDE with a random horizon, where two random times τ , τ andthe progressively enlarged filtration by these random times are introduced, and the horizonis set as τ ∧ τ ∧ T ( T ∈ R ++ ). We review some basic properties of such a BSDE, that is,the existence of a unique solution and its construction, using a reduced BSDE defined on asmaller filtration (see Theorems 1–3). These results are then used in Section 3. • In Section 3, we construct a financial market model that generalizes the model given byBichuch et al. (2018). On it, we derive BSDEs for pricing and hedging derivative securities,which express nonlinear dynamic hedging portfolio values of the seller and buyer. Here, wemodel the default time of the hedger (i.e., the seller of a derivative security) τ and that ofher counterparty (i.e., the buyer of the derivative security) τ , each of which are defined byrandom times. The contract between the hedger and her counterparty expires if the hedgeror the counterparty defaults. Hence, τ ∧ τ ∧ T is interpreted as the (random) horizon of thecontract, where T is the prescribed fixed maturity, and we naturally have BSDEs consideredin Section 2. The London Interbank Offered Rate (LIBOR) was a popular choice as the risk-free rate in pre-crisis models,whereas the Overnight Index Swap (OIS) rate is now recognized as a suitable candidate as the risk-free rate inpost-crisis models. In Section 4, working with the BSDEs introduced in Section 3, we obtain the following.(i) An explicit sufficient condition is presented to ensure the non-existence of an arbitrageopportunity for both the seller and buyer of the derivative security (see Theorem 4).We note that a rather restrictive condition is necessary to ensure the existence of anarbitrage-free price (see Remark 14).(ii) The pricing formula (1) used by practitioners is interpreted as an approximation ofthe theoretical fair price of the derivative security: XVA is regarded as certain “zero-th” order approximated correction terms. (see Theorem 5, Corollary 1, Proposition 3,and Remark 16). Furthermore, we mention a higher first-order approximation (seeSubsection 4.3).We intend to write this paper in an expository manner generally: Section 2 is devoted for reviewingknown results and some results in Section 4 (that is, Theorem 4 and Proposition 1 and 2) are ratherstraightforward extensions of existing results of the closely related work by Bichuch et al. (2015,2018) and Tanaka (2019). For other parts, we regard the following as being the contributions ofthe paper in comparison with Bichuch et al. (2015, 2018) and Tanaka (2019).1) The market model is generalized: our model treats(i) a multiple risky asset model, and(ii) a stochastic factor model that includes a stochastic volatility, a stochastic interest rate,and a stochastic hazard rate.2) Different definitions of arbitrages and admissible trading strategies are employed (see Subsec-tion 3.5). Because we analyze the pricing/hedging problem of derivative securities by usingBSDEs, our choices seem to be natural and clear.3) For XVA, an interpretation of pricing formula (1) is given as well as its arbitrage-free prop-erty (see Theorem 5, Corollary 1, and Proposition 3 with the following Remark 16 in Sub-section 4.2, and cf. the results in [24]).4) Regarding the lending-borrowing spreads of interest rates as “small parameters”, the firstorder perturbed BSDEs are derived and the associated approximated valuation adjustmentterms are computed (see Proposition 4 in Subsection 4.3).
Let (Ω , F , P ) be a complete probability space and let W := ( W ( t )) t ≥ , W ( t ) := ( W ( t ) , . . . , W n ( t )) ⊤ be an n -dimensional Brownian motion on it. Define the filtration by F t := σ ( W ( s ); s ∈ [0 , t ]) ∨ N , t ≥ , N is the totality of null sets. Let E , E be exponentially distributed random variables,assuming that W , E , and E are mutually independent. Using nonnegative F t -progressivelymeasurable processes h i := ( h i ( t )) t ≥ , ( i = 1 , τ , τ by τ i := inf (cid:26) t ≥ (cid:12)(cid:12)(cid:12) Z t h i ( u ) du ≥ E i (cid:27) . (2)The indicator processes for τ i ( i = 1 , N i ( t ) := 1 { t ≥ τ i } , t ≥ , are submartingales with respect to the filtration H t := σ ( N ( s ) , N ( s ); s ∈ [0 , t ]) , t ≥ , and their Doob–Meyer decompositions are written as N i ( t ) = M i ( t ) + Z t { − N i ( s ) } h i ( s ) ds, t ≥ i = 1 ,
2, where M i ( t ) := N i ( t ) − Z t { − N i ( s ) } h i ( s ) ds, t ≥ i = 1 ,
2) are two independent martingales with respect to ( H t ) t ≥ . Moreover, ( W, M , M ) remainas martingales with respect to the progressively enlarged filtration, G t := F t ∨ H t , t ≥ ≤ s ≤ t , P (cid:0) τ i > s (cid:12)(cid:12) F t (cid:1) = P (cid:0) τ i > s (cid:12)(cid:12) F ∞ (cid:1) = exp (cid:26) − Z s h i ( u ) du (cid:27) , where F ∞ := σ ( ∪ t ≥ F t ). From this, we see that for ds ≪ P (cid:0) τ i ≤ s + ds (cid:12)(cid:12) τ i > s, F ∞ (cid:1) = P ( s < τ i ≤ s + ds |F ∞ ) P ( τ i > s |F ∞ )=1 − exp ( − Z s + dss h i ( u ) du ) ≈ h i ( s ) ds, and h i is called the hazard rate (or intensity) process for τ i . Following Pham (2010), we employthe notation below. Notation 1. • F := ( F t ) t ≥ , G := ( G t ) t ≥ , and H := ( H t ) t ≥ . • P ( F ) (resp. P ( G ) ): σ -algebra generated by F (resp. G )-predictable measurable subsets on R + × Ω . Equivalently, σ -algebra on R + × Ω generated by F -adapted left-continuous processes. O ( F ) (resp. O ( G ) ): σ -algebra generated by F (resp. G )-optional measurable subsets on R + × Ω .Equivalently, σ -algebra on R + × Ω generated by F -adapted right-continuous processes. • P F (resp. P G ): the space of F (resp. G )-predictable processes. • O F (resp. O G ): the space of F (resp. G )-optional processes. • P ( k ) F : the space of the parametrized processes, f : R + × Ω × R k + ∋ ( t, ω, u ) f t ( ω, u ) ∈ R ,which is P ( F ) ⊗ B ( R k + ) / B ( R ) -measurable. • O ( k ) F : the space of the parametrized processes, f : R + × Ω × R k + ∋ ( t, ω, u ) f t ( ω, u ) ∈ R ,which is O ( F ) ⊗ B ( R k + ) / B ( R ) -measurable. • Denote by P F ,t := (cid:8) f [0 ,t ] | f ∈ P F (cid:9) , O F ,t := (cid:8) f [0 ,t ] | f ∈ O F (cid:9) , P ( k ) F ,t := n f ( · )1 [0 ,t ] | f ∈ P ( k ) F o ,and O ( k ) F ,t := n f ( · )1 [0 ,t ] | f ∈ O ( k ) F o , for example. We recall the following basic properties of stochastic processes under the progressively enlargedfiltration G . Lemma 1 (Lemmas 5.1 and 2.1 of Pham, 2010) . (1) Any G t -predictable process ( P ( t )) t ≥ has the expression that P ( t ) = p ( t )1 { t ≤ τ ∧ τ } + p t ( τ )1 { τ
We call the quadruplet ( Y, Z, U , U ) : [0 , T ] × Ω → R × R n × R × R a solution toBSDE (3) if it satisfies the following conditions. (a) Y := ( Y ( t )) t ∈ [0 ,T ] is a G -adapted RCLL (i.e., right continuous and having left limit) process(which is an element of O G ,T ), and ( Z, U , U ) ∈ ( P G ,T ) n +2 . (b) For t ∈ [0 , T ] , it holds that Y ( t )1 { τ ∧ τ ≤ t } = (cid:8) φ ( τ )1 { τ <τ } + φ ( τ )1 { τ <τ } (cid:9) { τ ∧ τ ≤ t } ,Z ( t )1 { τ ∧ τ ≤ t } =0 ,U i ( t )1 { τ ∧ τ ≤ t } =0 , i = 1 , . (c) For t ∈ [0 , T ] , it holds that Y ( t ) = φ ( τ )1 { τ <τ ,τ ≤ T } + φ ( τ )1 { τ <τ ,τ ≤ T } + ξ T { τ ∧ τ >T } + Z T ∧ τ ∧ τ t ∧ τ ∧ τ f ( s, Y ( s ) , Z ( s ) , U ( s ) , U ( s )) ds − Z T ∧ τ ∧ τ t ∧ τ ∧ τ (cid:8) Z ( s ) ⊤ dW ( s ) + U ( s ) dM ( s ) + U ( s ) dM ( s ) (cid:9) . S β,T := (cid:8) Y ∈ O G ,T (cid:12)(cid:12) k Y k β,T < ∞ (cid:9) , H ,dβ,T := n Z ∈ ( P G ,T ) d (cid:12)(cid:12) k Z k β,T < ∞ o , letting β ∈ R and denoting k Y k β,T := E "Z T e βt | Y ( t ) | dt . We then obtain the following.
Theorem 1.
Under Assumption 1, BSDE (3) admits a unique solution ( Y, Z, U , U ) ∈ S β,T × H ,n +2 β,T for any sufficiently large β > .Sketch. The method of proof is standard, although the horizon is random, which is rather “non-standard”. We consider a Picard-type iteration, that is, for a given (cid:0) ¯ Y , ¯ Z, ¯ U , ¯ U (cid:1) ∈ S β,T × H ,n +2 β,T ,we construct the solution to BSDE − dY ( t ) = f (cid:0) t, ¯ Y ( t ) , ¯ Z ( t ) , ¯ U ( t ) , ¯ U ( t ) (cid:1) dt − Z ( t ) ⊤ dW ( t ) − U ( t ) dM ( t ) − U ( t ) dM ( t ) ,t ∈ [0 , τ ] ,Y ( τ ) = ζ, (4)where we denote τ := τ ∧ τ , τ := τ ∧ T,ζ := φ ( τ )1 { τ <τ ∧ T } + φ ( τ )1 { τ <τ ∧ T } + ξ T { T <τ ∧ τ } . Indeed, using the G -martingale representation M ( t ) := E (cid:20) ζ + Z τ f (cid:0) u, ¯ Y ( u ) , ¯ Z ( u ) , ¯ U ( u ) , ¯ U ( u ) (cid:1) du (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) = E (cid:20) ζ + Z τ f (cid:0) u, ¯ Y ( u ) , ¯ Z ( u ) , ¯ U ( u ) , ¯ U ( u ) (cid:1) du (cid:21) + Z t φ ( u ) ⊤ dW ( u ) + Z t ψ ( u ) dM ( u ) + Z t ψ ( u ) dM ( u ) , t ∈ [0 , T ]for some ( φ, ψ , ψ ) ∈ H ,n +2 β,T (e.g., see Section 5.2 of Bielecki and Rutkowski, 2004), we define˜ Y t := E (cid:20) ζ + Z τt ∧ τ f (cid:0) u, ¯ Y u , ¯ Z u , ¯ U u , ¯ U u (cid:1) du (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) , t ∈ [0 , T ] , ˜ Z : ≡ φ, ˜ U : ≡ ψ , ˜ U : ≡ ψ . Note that the martingale ( M t ) t ∈ [0 ,T ] with respect to the right-continuous filtration G admits anRCLL modification. Hence,˜ Y ( t ) = M ( t ) − Z t ∧ τ f (cid:0) u, ¯ Y ( u ) , ¯ Z ( u ) , ¯ U ( u ) , ¯ U ( u ) (cid:1) du (cid:16) ˜ Y ( t ) (cid:17) t ∈ [0 ,T ] again. Furthermore, we cancheck the integrability, ˜ Y ∈ S β,T . Hence, (cid:0) ˜ Y , ˜ Z, ˜ U , ˜ U (cid:1) is the solution to (4). Next, we show thatthe map Ψ : S β,T × H ,n +2 β,T ∋ (cid:0) ¯ Y , ¯ Z, ¯ U , ¯ U (cid:1) (cid:16) ˜ Y , ˜ Z, ˜ U , ˜ U (cid:17) ∈ S β,T × H ,n +2 β,T is a contraction for sufficiently large β >
0, and using the fixed point theorem for the contractionmap, we conclude that the fixed point of the map Ψ is the solution.
Remark 1.
We refer to Section 19 of Cohen and Elliott (2015) for the detail of such a Picard-typeiteration argument, where a more general semimartingale BSDE (driven by L´evy noise) is treatedwith a fixed constant time horizon.
Actually, we can construct the solution to BSDE (3) on the filtered probability space (Ω , F , P , G ),using another reduced BSDE on the smaller filtered probability space (Ω , F , P , F ). Assuming Assumption 2. h i ( i = 1 , ) are bounded, we obtain the following. Theorem 2.
Under Assumptions 1 and 2, the solution ( Y, Z, U , U ) ∈ S β,T × H ,n +2 β,T has therepresentation that Y ( t ) = ¯ Y ( t )1 { ≤ t<τ ∧ τ ∧ T } + n φ ( τ )1 { τ <τ ∧ T } + φ ( τ )1 { τ <τ ∧ T } + ξ T { T <τ ∧ τ } o { t = τ ∧ τ ∧ T } ,Z ( t ) = ¯ Z ( t ) ,U i ( t ) = φ i ( t ) − ¯ Y ( t ) , i = 1 , . (5) Here, (cid:0) ¯ Y , ¯ Z (cid:1) ∈ S β,T × H ,nβ,T is the solution to a BSDE on (Ω , F , P , F ) , namely, − d ¯ Y ( t ) = ¯ f (cid:0) t, ¯ Y ( t ) , ¯ Z ( t ) (cid:1) dt − ¯ Z ( t ) ⊤ dW ( t ) , t ∈ [0 , T ] ,Y T = ξ T , (6) where ¯ f ( t, y, z ) := f ( t, y, z, φ ( t ) − y, φ ( t ) − y ) + { φ ( t ) − y } h ( t ) + { φ ( t ) − y } h ( t ) . Remark 2.
Similar reduction results for BSDEs (into smaller filtrations) have been studied byCr´epey and Song (2016) and Pham (2010) in more-general settings.Sketch.
Note that BSDE (3) is rewritten as − dY ( t ) = ˜ f ( t, Y ( t ) , Z ( t ) , U ( t ) , U ( t )) dt − Z ( t ) ⊤ dW ( t )on { ≤ t < τ ∧ τ ∧ T } , ∆ Y ( t ) = U ( τ )1 { τ <τ ∧ T } + U ( τ )1 { τ <τ ∧ T } ,Y ( t ) = φ ( τ )1 { τ <τ ∧ T } + φ ( τ )1 { τ <τ ∧ T } + F T { T <τ ∧ τ } on { t = τ ∧ τ ∧ T } , (7)8here we use ∆ Y ( t ) := Y ( t ) − Y ( t − ) and˜ f ( t, y, z, u , u ) = f ( t, y, z, u , u ) + u h ( t ) + u h ( t ) . We show that if we define (
Y, Z, U , U ) by (5), then it actually satisfies (7). First, we see thatBSDE (6) on (Ω , F , P , F ) has a unique solution ( ¯ Y , ¯ Z ) ∈ S β,T × H ,nβ,T for any sufficiently large β > f is a standard driver (e.g., ¯ f ( t, y, z ) satisfies a globally Lipschitz condition withrespect to ( y, z )). Next, we can check that (5) indeed satisfies (7); for example, on { t = τ ∧ τ ∧ T } , ∆ Y ( t ) = φ ( τ )1 { τ <τ ∧ T } + φ ( τ )1 { τ <τ ∧ T } + ξ T { T <τ ∧ τ } − ¯ Y ( t − )= φ ( τ )1 { τ <τ ∧ T } + φ ( τ )1 { τ <τ ∧ T } + ξ T { T <τ ∧ τ } − (cid:0) ¯ Y ( τ ∧ τ )1 { τ ∧ τ ≤ T } + ξ T { τ ∧ τ >T } (cid:1) = U ( τ )1 { τ <τ ∧ T } + U ( τ )1 { τ <τ ∧ T } . Hence, the desired assertion follows as it is easy to see the integrabilities given by (5), (
Y, Z, U , U ) ∈ S β,T × H ,n +2 β,T . Remark 3.
We impose Assumption 2 to simplify the statement of Theorem 2. We can relax it byemploying a different solution space (from S β,T × H ,n +2 β,T ) associated with the so-called stochasticLipschitz BSDEs. For the study of such BSDEs, see El Karoui and Huang (1997) and Nagayama(2019), for example. When we treat BSDE (3) in a practical application, more-concrete modeling is preferable: In thissubsection, we consider BSDE (3) under Assumptions 1 and 2 and the following setting.(i) There is a Markovian state variable process X := ( X ( t )) t ≥ , which is governed by the fol-lowing Markovian forward stochastic differential equation (FSDE), namely, dX ( t ) = b ( t, X ( t )) dt + a ( t, X ( t )) dW ( t ) , X (0) ∈ R d , (8)on (Ω , F , P , ( F t ) t ≥ ), where a : R + × R d → R d × n and b : R + × R d → R d .(ii) h i ( t ) := ˜ h i ( X ( t )), i = 1 ,
2, where ˜ h i : R d → R + is bounded.(iii) The driver f : [0 , T ] × Ω × R × R n × R → R of BSDE (3) is written as f ( t, ω, y, z, u , u ) := g ( t, X ( t, ω ) , y, z, u , u ) , where g : [0 , T ] × R d × R × R n × R × R → R .(iv) ξ T := Ξ( X ( T )), where Ξ : R d → R .(v) φ i ( t ) := ϕ i ( X ( t )), i = 1 ,
2, where ϕ i : R d → R .In this case, the solution to BSDE (3) can be constructed as follows using the solution to a second-order parabolic semilinear partial differential equation (PDE).9 heorem 3. Consider the second-order parabolic semilinear PDE − ∂ t V ( t, x ) = L t V ( t, x ) + ¯ g (cid:0) t, x, V ( t, x ) , a ( t, x ) ⊤ ∇ V ( t, x ) (cid:1) , ( t, x ) ∈ [0 , T ) × R d ,V ( T, x ) =Ξ( x ) , (9) where L t V := 12 tr (cid:0) aa ⊤ ( t, · ) ∇∇ V (cid:1) + b ⊤ ( t, · ) ∇ V (10) is the infinitesimal generator for X with the gradient ∇ V := ( ∂ x V, . . . , ∂ x d V ) ⊤ and the Hessianmatrix ∇∇ V := (cid:16) ∂ x i x j V (cid:17) ≤ i,j ≤ d , and ¯ g ( t, x, y, z ) := g ( t, x, y, z, ϕ ( x ) − y, ϕ ( x ) − z ) + X i =1 { ϕ i ( x ) − y } ˜ h i ( x ) . Suppose that there exists a unique classical solution V ∈ C , ([0 , T ] × R d ) to (9). Then, the solutionto BSDE (3) is represented as Y ( t ) = V ( t, X ( t )) 1 { ≤ t<τ ∧ τ ∧ T } + n ϕ ( X ( τ )) 1 { τ <τ ∧ T } + ϕ ( X ( τ )) 1 { τ <τ ∧ T } + Ξ ( X ( T )) 1 { T <τ ∧ τ } o { t = τ ∧ τ ∧ T } ,Z ( t ) = a ( t, X ( t )) ⊤ ∇ V ( t, X ( t )) ,U i ( t ) = ϕ i ( X ( t )) − V ( t, X ( t )) , i = 1 , . Sketch.
Associated with BSDE (6), we consider the (decoupled) forward-backward stochastic dif-ferential equation (FBSDE) dX ( t ) = b ( t, X ( t )) dt + a ( t, X ( t )) dW ( t ) ,X (0) ∈ R d , − d ¯ Y ( t ) =¯ g (cid:0) t, X ( t ) , ¯ Y ( t ) , ¯ Z ( t ) (cid:1) dt − ¯ Z ( t ) ⊤ dW ( t ) , ¯ Y ( T ) =Ξ( X ( T )) . (11)By the nonlinear Feynman–Kac formula (e.g., see El Karoui et al., 2000 or Zhang, 2017), thesolution to (11) is expressed as¯ Y ( t ) := V ( t, X ( t )) , ¯ Z ( t ) := a ( t, X ( t )) ⊤ ∇ V ( t, X ( t )) , t ∈ [0 , T ] . The desired assertion follows by using Theorem 2.
Remark 4.
In the study of credit risk modeling in mathematical finance, similar techniques,namely the reduction of a BSDE (onto a Brownian filtration) combined with the (nonlinear)Feynman–Kac formula, have been utilized: see Bichuch et al. (2015), Bielecki et al. (2005), andCr´epey (2015), for example. XVA Calculation via BSDE
In this section, we introduce a “post-crisis” financial market model and a hedger’s model forpricing OTC financial derivative securities, which generalize those employed by Bichuch et al.(2015, 2018) and Tanaka (2019). We then derive BSDEs that describe the self-financing hedgingportfolio values of the hedger (seller) and her counterparty (buyer). After preparing mathematicalmodels of a financial market, a hedger, and her counterparty, we formulate hedging problems andgive the definition of the arbitrage-free price of a derivative security. Throughout this section, wecontinue to use the mathematical setup introduced in Section 2.
Let T ∈ R ++ be a fixed time horizon, and consider a frictionless financial market model in con-tinuous time. In it, there are price processes of n non-defaultable risky assets S := ( S , . . . , S n ) ⊤ , S i := ( S i ( t )) t ∈ [0 ,T ] , one defaultable risky asset P I := ( P I ( t )) t ∈ [0 ,T ] issued by an investor’s firm, andone defaultable risky asset P C := ( P C ( t )) t ∈ [0 ,T ] issued by the firm of a counterparty of the investor.They are governed by the following stochastic differential equations (SDEs) on (Ω , F , P , G ): dS ( t ) =diag ( S ( t )) { σ ( t ) dW ( t ) + r D ( t ) dt } , S (0) ∈ R n ++ , (12) dP I ( t ) = P I ( t − ) { σ I ( t ) dW ( t ) − dM ( t ) + r D ( t ) dt } , P I (0) ∈ R ++ , (13) dP C ( t ) = P C ( t − ) { σ C ( t ) dW ( t ) − dM ( t ) + r D ( t ) dt } , P C (0) ∈ R ++ . (14)Here, σ ∈ ( P F ,T ) n × n , σ i ∈ ( P F ,T ) × n , i ∈ { I, C } , and r D ∈ P F ,T , which are assumed to be bounded,and σ ( t, ω ) is invertible for a.e. ( t, ω ) ∈ [0 , T ] × Ω. Furthermore, we denote diag( x ) = ( x i δ ij ) ≤ i,j ≤ n for x := ( x , . . . , x n ) ⊤ ∈ R n and := (1 , . . . , ⊤ ∈ R n . Remark 5.
We regard the process r D as the risk-free interest rate process in the market, whichdoes not contain credit risk spread. Define the cash account process B D := ( B D ( t )) t ≥ associatedwith the risk-free rate r D by dB D ( t ) = B D ( t ) r D ( t ) dt, B D (0) = 1 , or equivalently B D ( t ) = exp (cid:26)Z t r D ( u ) du (cid:27) . We then see that S i B D , i = 1 , . . . , n, P j B D , j = 1 , are G -local martingales. These mean that we are starting with the probability space (Ω , F , P ) witha risk-neutral (pricing) probability P , not with the real-world (physical) probability. A typical example of such an interest rate in a real financial market is the OIS rate. More precisely, P is an equivalent martingale measure (EMM). See Remark 13 in Subsection 3.5. τ and τ defined by (2) are interpreted as the default times of the investorwho issues P I and the counterparty who issues P C , respectively. We solve (13) as P I ( t ) = P I (0) exp (cid:20)Z t σ I ( u ) dW ( u ) + Z t (cid:18) r D ( u ) + h ( u ) − | σ I ( u ) | (cid:19) du (cid:21) { − N ( t ) } , for example. Recall that the price becomes zero when defaults occur, i.e., P I ( τ ) = 0. Remark 6.
As concrete examples of P I and P C , we can consider T -maturity zero coupon bondswithout recoveries, namely P I ( t ) = E " exp ( − Z Tt ( r D ( u ) + h ( u )) du ) (cid:12)(cid:12)(cid:12)(cid:12) F t { − N ( t ) } ,P C ( t ) = E " exp ( − Z Tt ( r D ( u ) + h ( u )) du ) (cid:12)(cid:12)(cid:12)(cid:12) F t { − N ( t ) } . The volatility terms ( σ j ( t )) t ∈ [0 ,T ] ( j ∈ { I, C } ) are described by using the ( P , F t ) -Brownian martin-gale representation: For example, in the j = I case, ( σ I ( t )) t ∈ [0 ,T ] is determined to satisfy E " exp ( − Z T ( r D ( u ) + h ( u )) du ) (cid:12)(cid:12)(cid:12)(cid:12) F t = P I (0) exp (cid:26)Z t σ I ( s ) dW ( s ) − Z t | σ I ( s ) | ds (cid:27) for t ∈ [0 , T ] . We treat the following derivative security in our financial market model.
Definition 2.
A European derivative security is described as ( T, τ , τ , ξ T , φ , φ ) , where ξ T ∈ L (Ω , F T , P ) and φ i ∈ n φ ∈ O F ,T (cid:12)(cid:12) E h sup t ∈ [0 ,T ] | φ ( t ) | i < ∞ o ( i = 1 , . Here, • τ ∧ τ ∧ T is the maturity, • ξ T is the payoff at the maturity when no default occurs, • φ ( τ ) is the payoff at the maturity when the investor defaults, • φ ( τ ) is the payoff at the maturity when the counterparty defaults.This means that at the maturity, H := ξ T { T <τ ∧ τ } + φ ( τ )1 { τ <τ ,τ ≤ T } + φ ( τ )1 { τ <τ ,τ ≤ T } (15) is paid to the counterparty (buyer) from the investor (seller, writer). emark 7. A typical example of the payoff ( ξ T , φ , φ ) is ξ T := h (cid:0) ( S ( t )) t ∈ [0 ,T ] (cid:1) with h : C ([0 , T ] , R n ++ ) → R and, for i = 1 , , φ i ( t ) := ϕ i (cid:16) ˆ V ( t ) (cid:17) with some nonlinear (piecewise-linear) ϕ i : R → R and ˆ V ( t ) := E " exp ( − Z Tt r D ( u ) du ) ξ T (cid:12)(cid:12)(cid:12)(cid:12) F t , t ∈ [0 , T ] . (16) (16) is interpreted as the reference value process of the derivative ( T, ξ T ) with the payoff ξ T at thematurity T in a default-free market. In Bichuch et al. (2018), ϕ ( v ) := v − L I ( v − αv ) + and ϕ ( v ) := v + L C ( v − αv ) − (17) are employed, where x + := max( x, , x − := max( − x,
0) = − min( x, , ≤ L I , L C , α ≤ . Theconstant L I (resp. L C ) is called the loss rate upon default of the investor (resp. the counterparty),and α is called the collateralization level. For a more detailed explanation, see Sections 3.2 and3.4 of Bichuch et al. (2018). For hedging purposes, the writer (seller) of the derivative security given in Definition 2 constructsa dynamic portfolio, which is denoted by (cid:0) π, π I , π C , π f , π r , π col (cid:1) . Here, π := ( π , . . . , π n ) ⊤ ∈ ( P G ,T ) n , π j := ( π j ( t )) t ∈ [0 ,T ] is an investment strategy for the risky assets S := ( S , . . . , S n ) ⊤ , π j := ( π j ( t )) t ∈ [0 ,T ] ∈ P G ,T , j ∈ { I, C } are investment strategies for the risky assets P I and P C , respectively, and π j := ( π j ( t )) t ∈ [0 ,T ] ∈ P G ,T , j ∈ { f, r, col } are investment strategies for the cash accounts B f , B r , and B col , which are called the fundingaccount, the repo account, and the collateral account, respectively. They are defined by dB j ( t ) = B j ( t ) (cid:8) r − j ( t )1 { π j ( t ) < } + r + j ( t )1 { π j ( t ) > } (cid:9) dt, B j (0) = 1 (18)with r − j := ( r − j ( t )) t ∈ [0 ,T ] ∈ P F ,T , r + j := ( r + j ( t )) t ∈ [0 ,T ] ∈ P F ,T , and j ∈ { f, r, col } , where r ± f , r ± r and r ± col are called the funding rate, the repo rate, and the collateral rate, respectively.13 emark 8. The cash account process B f represents the cumulative amount of cash that the hedgerborrows from (or lends to) her treasury desk. The rate r − f is called the funding borrowing rateand the rate r + f is called the funding lending rate. The cash account process B r represents thecumulative amount of cash that the investor borrows from (or lends to) a repo market. The rate r − r is called the repo borrowing rate, which is applied when the hedger borrows money from therepo market and implements a long position for the non-defaultable risky assets S . The rate r + r is called the repo lending rate, which is applied when the hedger lends money to the repo marketand implements a short-selling position for the non-defaultable risky assets S . The cash accountprocess B col represents the cumulative amount of cash that the investor receives from (or posts to)the counterparty as the collateral of the derivative security. The rate r − col is paid by the hedger tothe counterparty if he/she has received the collateral. The rates r + col is received by the hedger ifhe/she has posted the collateral. These rates can differ because different markets may be used todetermine the contractual rates earned by cash collateral. For r ± f and r ± r , it is natural and realistic to assume that2 ǫ j : ≡ r − j − r + j ≥ j ∈ { f, r } . (19)For j ∈ { f, r } , denoting the “mid-rate” by r j : ≡ r − j + r + j , we see that r ± j ≡ r j ∓ ǫ j . The value process Y := ( Y ( t )) t ∈ [0 ,T ] associated with a given dynamic portfolio strategy (cid:0) π, π I , π C , π f , π r , π col (cid:1) is governed by an SDE on (Ω , F , P , G ), namely, dY ( t ) = π ( t ) ⊤ dS ( t ) + π I ( t ) dP I ( t ) + π C ( t ) dP C ( t )+ π f ( t ) dB f ( t ) + π r ( t ) dB r ( t ) + π col ( t ) dB col ( t ) ,Y (0) = y, (20)subject to Y ( t ) = π ( t ) ⊤ S ( t ) + π I ( t ) P I ( t ) + π C ( t ) P C ( t )+ π f ( t ) B f ( t ) + π r ( t ) B r ( t ) + π col ( t ) B col ( t ) , (21) π ( t ) ⊤ S ( t ) + π r ( t ) B r ( t ) = 0 , (22) π col ( t ) B col ( t ) − α ˆ V ( t ) = 0 . (23)Here, (21) corresponds to the so-called self-financing condition, (22) implies that the hedger accessesthe repo market to purchase/sell non-defaultable risky assets (stocks), and (23) implies that α ˆ V ( t ) For example, the choice of currency (USD, Euro, etc.). We refer the interested reader to Fujii and Takahashi(2011), where the impact of the choice of currency of collateral is studied.
14s regarded as the collateral value at time t , where α ∈ [0 ,
1] is the collateral level, which is thesame as the one given in Remark 7. From (21)–(23), recall that the relations π r ( t ) = − B r ( t ) − π ( t ) ⊤ S ( t ) , (24) π col ( t ) = B col ( t ) − α ˆ V ( t ) , (25) π f ( t ) = B f ( t ) − n Y ( t − ) − π I ( t ) P I ( t − ) − π C ( t ) P C ( t − ) − α ˆ V ( t ) o (26)hold. Hence, we can interpret that ( y, Π) ∈ R × ( P G ,T ) n +2 , where Π := (cid:0) π, π I , π C (cid:1) , is a portfoliostrategy that determines the portfolio value process (20), and we sometimes write Y : ≡ Y ( y, Π) , emphasizing the portfolio strategy ( y, Π). Combining (20) with (12)–(14), (18), and (24)–(26), wesee that dY ( t ) = π ( t ) ⊤ diag ( S ( t )) [ σ ( t ) dW ( t ) + { r D ( t ) − r r ( t ; π r ( t )) } dt ]+ π I ( t ) P I ( t − ) (cid:2) σ I ( t ) dW ( t ) − dM ( t ) + (cid:8) r D ( t ) − r f ( t ; π f ( t )) (cid:9) dt (cid:3) + π C ( t ) P C ( t − ) (cid:2) σ C ( t ) dW ( t ) − dM ( t ) + (cid:8) r D ( t ) − r f ( t ; π f ( t )) (cid:9) dt (cid:3) + n Y ( t ) − α ˆ V ( t ) o r f ( t ; π f ( t )) dt + α ˆ V ( t ) r col ( t ; π col ( t )) dt, (27)where we denote r j ( t ; p ) := r − j ( t )1 { p< } + r + j ( t )1 { p> } , j ∈ { f, r, col } . Remark 9.
Suppose that r D ≡ r ± f ≡ r ± r ≡ r ± col . Then (27) becomes dY ( t ) = π ( t ) ⊤ diag ( S ( t )) σ ( t ) dW ( t ) + π I ( t ) P I ( t − ) { σ I ( t ) dW ( t ) − dM ( t ) } + π C ( t ) P C ( t − ) { σ C ( t ) dW ( t ) − dM ( t ) } + r D ( t ) Y ( t ) dt, which is solved as Y ( y, Π) ( t ) = B D ( t ) " y + Z t B D ( s ) − π ( s ) ⊤ diag ( S ( s )) σ ( s ) dW ( s )+ Z t B D ( s ) − π I ( s ) P I ( s − ) { σ I ( s ) dW ( s ) − dM ( s ) } + Z t B D ( s ) − π C ( s ) P C ( s − ) { σ C ( s ) dW ( s ) − dM ( s ) } . (28) That is, the discounted value process
Y /B D is a local martingale, which is a standard result sharedin a classical framework with “one risk-free rate world.” For the derivative security given in Definition 2, we call the portfolio strategy (ˆ y, ˆΠ) ∈ R × ( P G ,T ) n +2 that satisfies Y (ˆ y, ˆΠ) τ ∧ τ ∧ T = H (29)15he replicating portfolio strategy for the hedger.Furthermore, for pricing purposes, we next consider a dynamic portfolio strategy (cid:0) − ˜ π, − ˜ π I , − ˜ π C , ˜ π f , ˜ π r , ˜ π col (cid:1) and the associated value process ˜ Y of the buyer (counterparty). We define d ˜ Y ( t ) = − ˜ π ( t ) ⊤ dS ( t ) − ˜ π I ( t ) dP I ( t ) − ˜ π C ( t ) dP C ( t )+ ˜ π f ( t ) dB f ( t ) + ˜ π r ( t ) dB r ( t ) + ˜ π col ( t ) dB col ( t ) , ˜ Y (0) = − ˜ y subject to ˜ Y ( t ) = − ˜ π ( t ) ⊤ S ( t ) − ˜ π I ( t ) P I ( t ) − ˜ π C ( t ) P C ( t )+ ˜ π f ( t ) B f ( t ) + ˜ π r ( t ) B r ( t ) + ˜ π col ( t ) B col ( t ) , (30) − ˜ π ( t ) ⊤ S ( t ) + ˜ π r ( t ) B r ( t ) = 0 , (31)˜ π col ( t ) B col ( t ) + α ˆ V ( t ) = 0 , (32)where ˜ π ∈ ( P G ,T ) n and ˜ π i ∈ P G ,T for i ∈ { I, C, f, r, col } . Here, as we see in (32), the collateralvalue at time t is regarded as − α ˆ V ( t ), the opposite value of that for the writer (hedger). Becausewe see that ˜ π r ( t ) = B r ( t ) − ˜ π ( t ) ⊤ S ( t ) , ˜ π col ( t ) = − B col ( t ) − α ˆ V ( t ) , ˜ π f ( t ) = B f ( t ) − n ˜ Y ( t − ) + ˜ π I ( t ) P I ( t − ) + ˜ π C ( t ) P C ( t − ) + α ˆ V ( t ) o from (30)–(32), we regard (cid:0) − ˜ y, − ˜Π (cid:1) ∈ R × ( P G ,T ) n +2 with ˜Π := (cid:0) ˜ π, ˜ π I , ˜ π C (cid:1) as the portfoliostrategy, and we rewrite the dynamics of ˜ Y : ≡ ˜ Y ( − ˜ y, − ˜Π) as d ˜ Y ( t ) = − ˜ π ( t ) ⊤ diag ( S ( t )) [ σ ( t ) dW ( t ) + { r D ( t ) − r r ( t ; π r ( t )) } dt ] − ˜ π I ( t ) P I ( t − ) (cid:2) σ I ( t ) dW ( t ) − dM ( t ) + (cid:8) r D ( t ) − r f ( t ; π f ( t )) (cid:9) dt (cid:3) − ˜ π C ( t ) P C ( t − ) (cid:2) σ C ( t ) dW ( t ) − dM ( t ) + (cid:8) r D ( t ) − r f ( t ; π f ( t )) (cid:9) dt (cid:3) + n ˜ Y ( t ) + α ˆ V ( t ) o r f ( t ; π f ( t )) dt − α ˆ V ( t ) r col ( t ; π col ( t )) dt. (33) Remark 10.
We have assumed that the funding rate r ± f,I for the investor (writer) and the fundingrate r ± f,C for the counterparty (buyer) are identical, i.e., r ± f ≡ r ± f,I ≡ r ± f,C , which is a restrictivesituation. However, without such an assumption, it looks difficult and complicated to derive anexplicit sufficient condition to ensure the no-arbitrage property (see Theorem 4 and its proof ). Remark 11.
Suppose that r D ≡ r ± f ≡ r ± r ≡ r ± col . Using a similar calculation to that in Remark 9,we solve (33) to see that ˜ Y ( − y ′ , − ˜Π) ≡ − Y ( y ′ , ˜Π) , where the right-hand side Y ( y ′ , ˜Π) is given by (28)by letting y := y ′ and Π : ≡ ˜Π . If the portfolio strategy ( − ˜ y, − ˜Π) ∈ R × ( P G ,T ) n +2 satisfies˜ Y ( − ˜ y, − ˜Π) τ ∧ τ ∧ T = − H (34)for the derivative security given in Definition 2, then we call it the replicating portfolio strategyfor the buyer. 16 .4 Deriving BSDE The replicating portfolio (ˆ y, ˆΠ) that satisfies (29) is represented using the solution to a BSDE. Let Y + : ≡ Y (ˆ y, ˆΠ) ,U +1 ( t ) := − π I ( t ) P I ( t − ) ,U +2 ( t ) := − π C ( t ) P C ( t − ) ,Z + ( t ) := σ ( t ) ⊤ diag ( S ( t )) π ( t ) − U +1 ( t ) σ I ( t ) ⊤ − U +2 ( t ) σ C ( t ) ⊤ . Recalling π f ( t ) B f ( t ) = Y + ( t ) + U +1 ( t ) + U +2 ( t ) − α ˆ V ( t ) , we see that π f ( t ) ≥ ≤
0) is equivalent to Y + ( t ) + U +1 ( t ) + U +2 ( t ) − α ˆ V ( t ) ≥ , (resp. ≤ − π r ( t ) B r ( t ) = π ( t ) ⊤ diag( S ( t )) = (cid:8) Z + ( t ) ⊤ + U +1 ( t ) σ I ( t ) + U +2 ( t ) σ C ( t ) (cid:9) σ ( t ) − , we see that π r ( t ) ≥ ≤
0) is equivalent to (cid:8) Z + ( t ) ⊤ + U +1 ( t ) σ I ( t ) + U +2 ( t ) σ C ( t ) (cid:9) σ ( t ) − ≤ ≥ . Using these relations, we then rewrite (27) as dY + ( t ) = − f + (cid:16) t, Y + ( t ) , Z + ( t ) , U +1 ( t ) , U +2 ( t ); ˆ V ( t ) (cid:17) dt + Z + ( t ) ⊤ dW ( t ) + U +1 ( t ) dM ( t ) + U +2 ( t ) dM ( t ) , where f + ( t, y, z, u , u ; ˆ v ) := f ( t, y, z, u , u ) + α (cid:8) r f ( t )ˆ v − r + col ( t )ˆ v + + r − col ( t )ˆ v − (cid:9) + ǫ f ( t ) | y + u + u − α ˆ v | + ǫ r ( t ) (cid:12)(cid:12)(cid:8) z ⊤ + u σ I ( t ) + u σ C ( t ) (cid:9) σ ( t ) − (cid:12)(cid:12) , (35)with f (cid:0) t, y, z, u , u (cid:1) := − r f ( t ) y + (cid:8) r r ( t ) − r D ( t ) (cid:9) z ⊤ σ ( t ) − + (cid:2) − (cid:8) r f ( t ) − r D ( t ) (cid:9) + (cid:8) r r ( t ) − r D ( t ) (cid:9) σ I ( t ) σ ( t ) − (cid:3) u + (cid:2) − (cid:8) r f ( t ) − r D ( t ) (cid:9) + (cid:8) r r ( t ) − r D ( t ) (cid:9) σ C ( t ) σ ( t ) − (cid:3) u . (36)So, we consider the BSDE on the filtered probability space (Ω , F , P , G ), namely − dY + ( t ) = f + (cid:16) t, Y + ( t ) , Z + ( t ) , U +1 ( t ) , U +2 ( t ); ˆ V ( t ) (cid:17) dt − Z + ( t ) ⊤ dW ( t ) − U +1 ( t ) dM ( t ) − U +2 ( t ) dM ( t )for 0 ≤ t ≤ τ ∧ τ ∧ T,Y + ( τ ∧ τ ∧ T ) = H. (37)17sing the solution to (37), the replicating portfolio (cid:16) ˆ y, ˆΠ (cid:17) that satisfies (29) is constructed asˆ y := Y + (0) , ˆ π ( t ) :=diag( S t ) − (cid:0) σ ( t ) ⊤ (cid:1) − (cid:8) Z + ( t ) + U +1 ( t ) σ ⊤ I ( t ) + U +2 ( t ) σ ⊤ C ( t ) (cid:9) , ˆ π I ( t ) := − P I ( t − ) − U +1 ( t ) , ˆ π C ( t ) := − P C ( t − ) − U +2 ( t )for 0 ≤ t ≤ τ ∧ τ ∧ T . Similarly, the replicating portfolio ( − ˜ y, − ˜Π) that satisfies (34) can berepresented using the solution to a BSDE. Let Y − : ≡ − ˜ Y ( − ˜ y, − ˜Π) ,U − ( t ) := − ˜ π I ( t ) P I ( t − ) ,U − ( t ) := − ˜ π C ( t ) P C ( t − ) ,Z − ( t ) := σ ( t ) ⊤ diag ( S ( t )) ˜ π ( t ) − U − ( t ) σ I ( t ) ⊤ − U − ( t ) σ C ( t ) ⊤ . Recalling − ˜ π f ( t ) B f ( t ) = ˜ Y − ( t ) + U − ( t ) + U − ( t ) − α ˆ V ( t ) , we see that π f ( t ) ≥ ≤
0) is equivalent to Y − ( t ) + U − ( t ) + U − ( t ) − α ˆ V ( t ) ≤ , (resp. ≥ π r ( t ) B r ( t ) =˜ π ( t ) ⊤ diag( S ( t )) = (cid:8) Z − ( t ) ⊤ + U − ( t ) σ I ( t ) + U − ( t ) σ C ( t ) (cid:9) σ ( t ) − , we see that π r ( t ) ≥ ≤
0) is equivalent to (cid:8) Z − ( t ) ⊤ + U − ( t ) σ I ( t ) + U − ( t ) σ C ( t ) (cid:9) σ ( t ) − ≥ ≤ . Using these relations, we then rewrite (33) as dY − ( t ) = − f − (cid:16) t, Y − ( t ) , Z − ( t ) , U − ( t ) , U − ( t ); ˆ V ( t ) (cid:17) dt + Z − ( t ) ⊤ dW ( t ) + U − ( t ) dM ( t ) + U − ( t ) dM ( t ) , where f − ( t, y, z, u , u ; ˆ v ) := − f + ( t, − y, − z, − u , − u ; − ˆ v )= f ( t, y, z, u , u ) + α (cid:8) r f ( t )ˆ v + r + col ( t )ˆ v − − r − col ( t )ˆ v + (cid:9) − ǫ f ( t ) | y + u + u − α ˆ v |− ǫ r ( t ) (cid:12)(cid:12)(cid:8) z ⊤ + u σ I ( t ) + u σ C ( t ) (cid:9) σ ( t ) − (cid:12)(cid:12) . (38)18o, we consider the BSDE on the filtered probability space (Ω , F , P , G ) − dY − ( t ) = f − (cid:16) t, Y − ( t ) , Z − ( t ) , U − ( t ) , U − ( t ); ˆ V ( t ) (cid:17) dt − Z − ( t ) ⊤ dW ( t ) − U − ( t ) dM ( t ) − U − ( t ) dM ( t )for 0 ≤ t ≤ τ ∧ τ ∧ T,Y − ( τ ∧ τ ∧ T ) = H. (39)The replicating portfolio (cid:16) − ˜ y, − ˜Π (cid:17) that satisfies (34) is now constructed as˜ y := Y − (0) , ˜ π ( t ) :=diag( S t ) − (cid:0) σ ( t ) ⊤ (cid:1) − (cid:8) Z − ( t ) + U − ( t ) σ ⊤ I ( t ) + U − ( t ) σ ⊤ C ( t ) (cid:9) , ˜ π I ( t ) := − P I ( t − ) − U − ( t ) , ˜ π C ( t ) := − P C ( t − ) − U − ( t )for 0 ≤ t ≤ τ ∧ τ ∧ T , using the solution to (39). Remark 12.
BSDEs (37) and (39) with (15) and (16) can be seen as the system of BSDEs − dY ± ( t ) = f ± (cid:16) t, Y ± ( t ) , Z ± ( t ) , U ± ( t ) , U ± ( t ); ˆ V ( t ) (cid:17) dt − Z ± ( t ) ⊤ dW ( t ) − U ± ( t ) dM ( t ) − U ± ( t ) dM ( t ) , for ≤ t ≤ τ ∧ τ ∧ T,Y ± ( τ ∧ τ ∧ T ) = H, − d ˆ V ( t ) = − r D ( t ) ˆ V ( t ) dt − ∆( t ) ⊤ dW ( t ) for ≤ t ≤ T, ˆ V ( T ) = ξ T , (40) in which (cid:16) Y ± , Z ± , U ± , U ± , ˆ V , ∆ (cid:17) are solutions. To study the hedging problem via BSDEs (37) and (39) with (15) and (16), it is natural to employthe following space of admissible hedging strategies A β,T := n(cid:0) π, π I , π C (cid:1) ∈ ( P G ,T ) d +2 (cid:12)(cid:12)(cid:12) (cid:0) σ ⊤ diag( S ) π, π I P − I , π C P − C (cid:1) ∈ H ,n +2 β,T o , where β > P − i ( t ) := P i ( t − ) for t > P − i (0) := P i (0). We then formulate the minimal superhedging price (i.e., the maximal price forthe writer) and the maximal subhedging price (i.e., the minimal price for the buyer) as follows. Definition 3.
For the derivative security given in Definition 2, ¯ p := inf n y ∈ R (cid:12)(cid:12) − H + Y ( y, Π) ( τ ∧ τ ∧ T ) ≥ for some ( y, Π) ∈ R × A β,T o s called the minimal superhedging price, which is the maximal price of the writer (seller), and p := sup n y ∈ R (cid:12)(cid:12) H + ˜ Y ( − y, − Π) ( τ ∧ τ ∧ T ) ≥ for some ( y, Π) ∈ R × A β,T o is called the maximal subhedging price, which is the minimal price of the buyer. If there exists ¯Π ∈ A β,T such that − H + Y ( ¯ p, ¯Π )( τ ∧ τ ∧ T ) ≥ , then the pair (cid:0) ¯ p, ¯Π (cid:1) is called the minimal superhedging strategy, and if there exists Π ∈ A β,T suchthat H + ˜ Y ( − p, − Π )( τ ∧ τ ∧ T ) ≥ , then the pair (cid:0) − p, − Π (cid:1) is called the maximal subhedging strategy. Associated with the hedging problem, we give the following definition.
Definition 4.
Consider the derivative security given in Definition 2. Suppose that a writer sellsthe derivative security with price p ∈ R at time . If it holds that − H + Y ( p, Π) ( τ ∧ τ ∧ T ) ≥ and P (cid:16) − H + Y ( p, Π) ( τ ∧ τ ∧ T ) > (cid:17) > for some Π ∈ A β,T , then we say that an arbitrage opportunity for the writer occurs. Similarly,suppose that a buyer purchases the derivative security with price p ∈ R at time . If it holds that H + ˜ Y ( − p, − Π) ( τ ∧ τ ∧ T ) ≥ and P (cid:16) H + ˜ Y ( − p, − Π) ( τ ∧ τ ∧ T ) > (cid:17) > for some Π ∈ A β,T , then we say that an arbitrage opportunity for the buyer occurs. Moreover, ifthe price ˆ p ∈ R at time does not admit arbitrage opportunities for both writer and buyer, then ˆ p is called an arbitrage-free price. Remark 13.
In our financial market model, we assume implicitly that the probability measure P is an EMM. Hence, P ∼ P , where P is a real-world (physical) probability measure given in thesame measurable space (Ω , F ) . Therefore, in Definition 3, the P -a.s. statement can be replaced bythe P -a.s. statement. Also, in Definition 4, P can be replaced by P to claim that P ( · · · ) > . The following Markovian model is typical and popularly treated in practice. Let the coefficientsof the market model be described as σ ( t ) :=˜ σ ( t, F ( t )) , r D ( t ) := ˜ r D ( t, F ( t )) ,σ i ( t ) :=˜ σ j ( t, F ( t )) , i ∈ { I, C } ,h j ( t ) :=˜ h i ( t, F ( t )) , j ∈ { , } ,r k ( t ) :=˜ r k ( t, F ( t )) , ǫ k ( t ) := ˜ ǫ k ( t, F ( t )) , k ∈ { f, r } , and r ± col ( t ) :=˜ r ± col ( t, F ( t )) , σ : [0 , T ] × R m → R n × n , ˜ r D , ˜ σ i , ˜ h j , ˜ r k , ˜ ǫ k , ˜ r ± col : [0 , T ] × R m → R , and ( F ( t )) t ∈ [0 ,T ] is calledthe stochastic factor process, which can be interpreted as a model of economic factors and affectsthe market model through the coefficients σ, σ i ( i ∈ { I, C } ), h j ( j = 1 , r k , ǫ k ( k ∈ { f, r } ), and r ± col . It is given by the solution to the SDE dF ( t ) = µ F ( t, F ( t )) dt + σ F ( t, F ( t )) dW ( t ) , F (0) ∈ R m on (Ω , F , P , F ), where µ F : [0 , T ] × R m → R m and σ F : [0 , T ] × R m → R m × n . Let X ⊤ : ≡ (cid:0) X ⊤ , X ⊤ (cid:1) : ≡ (cid:0) S ⊤ , F ⊤ (cid:1) and define, for x := ( x , x ) ∈ R n × R m , b ( t, x ) := (cid:18) diag( x ) r D ( t, x ) µ F ( t, x ) (cid:19) , a ( t, x ) := (cid:18) diag( x ) σ ( t, x ) σ F ( t, x ) (cid:19) . Then, the SDE for X is written as (8) with d = n + m . Furthermore, we set ξ T := Ξ ( X ( T )) and φ i ( t ) := ϕ i ( ˆ V ( t )) for i ∈ { , } ,where Ξ : R n + m → R and ϕ i : R → R . In this situation, we can apply Theorem 3 to represent thesolution to BSDEs (40), using the solutions to the associated PDEs (see Proposition 2 in Section 4). Throughout this section, we always assume that σ i ( i ∈ { I, C } ), σ, σ − , r D , r ± j ( j ∈ { f, r, col } ),and h k ( k = 1 ,
2) are bounded. Applying the results in Section 2 and a comparison theorem forBSDEs, the following claims are straightforward to see.
Proposition 1.
For any sufficiently large β > , there exist unique solutions (cid:0) Y ± , Z ± , U ± , U ± (cid:1) ∈ S β,T × H ,n +2 β,T to BSDEs (37) and (39) with (15) and (16). Moreover, the solutions have therepresentations that Y ± ( t ) = ¯ Y ± ( t )1 { ≤ t<τ ∧ τ ∧ T } + n φ ( τ )1 { τ <τ ∧ T } + φ ( τ )1 { τ <τ ∧ T } + ξ T { T <τ ∧ τ } o { t = τ ∧ τ ∧ T } ,Z ± ( t ) = ¯ Z ± ( t ) ,U ± i ( t ) = φ i ( t ) − ¯ Y ± ( t ) , i = 1 , . (41) Here, (cid:0) ¯ Y ± , ¯ Z ± (cid:1) ∈ S β,T × H ,nβ,T are the solutions to BSDEs on (Ω , F , P , F ) , namely − d ¯ Y ± ( t ) = ¯ f ± (cid:16) t, ¯ Y ± ( t ) , ¯ Z ± ( t ); ˆ V ( t ) , φ ( t ) , φ ( t ) (cid:17) dt − ¯ Z ± ( t ) ⊤ dW ( t ) for ≤ t ≤ T, ¯ Y ± ( T ) = ξ T , − d ˆ V ( t ) = − r D ( t ) ˆ V ( t ) dt − ∆( t ) ⊤ dW ( t ) for ≤ t ≤ T, ˆ V ( T ) = ξ T , (42)21 here we define ¯ f ± ( t, y, z ; ˆ v, p , p ) := f ± ( t, y, z, p − y, p − y ; ˆ v ) + ( p − y ) h ( t ) + ( p − y ) h ( t ) . (43) In addition to Condition (19), assume that r − col ≥ r + col . (44) Then, it always holds that Y − ≤ Y + and ¯ Y − ≤ ¯ Y + . (45) Sketch.
Using (19) and (44), we see that¯ f + ( t, y, z ; ˆ v, p , p ) − ¯ f − ( t, y, z ; ˆ v, p , p )= α (cid:8) r − col ( t ) − r + col ( t ) (cid:9) | ˆ v | + 2 ǫ f ( t ) | y + ( p − y ) + ( p − y ) − α ˆ v | + 2 ǫ r ( t ) (cid:12)(cid:12)(cid:8) z ⊤ + ( p − y ) σ I ( t ) + ( p − y ) σ C ( t ) (cid:9) σ ( t ) − (cid:12)(cid:12) ≥ . Hence, (45) follows from a comparison theorem of BSDEs. Other assertions follow from the resultsin Section 2.Next, consider the Markovian model given in Subsection 3.6. Then, corresponding to (42), wehave the Markovian system of BSDEs (decoupled FBSDEs) dX ( t ) = b ( t, X ( t )) dt + a ( t, X ( t )) dW ( t ) , X (0) ∈ R n + m , − d ¯ Y ± ( t ) =¯ g ± (cid:16) t, X ( t ) , ¯ Y ± ( t ) , ¯ Z ± ( t ); ˆ V ( t ) , ϕ ( ˆ V ( t )) , ϕ ( ˆ V ( t )) (cid:17) dt − ¯ Z ± ( t ) ⊤ dW ( t ) , ¯ Y ± ( T ) =Ξ ( X ( T )) , − d ˆ V ( t ) = − ˜ r D ( t, X ( t )) ˆ V ( t ) dt − ∆( t ) ⊤ dW ( t ) , ˆ V ( T ) =Ξ ( X ( T )) . (46)Here, the relation ¯ g ± ( t, X ( t, ω ) , y, z ; ˆ v, p , p ) = ¯ f ± ( t, ω, y, z ; ˆ v, p , p )holds, and the functions ¯ g ± : [0 , T ] × R m × R × R n × R → R are written as¯ g ± ( t, x , y, z ; ˆ v, p , p ) := ¯ g ( t, x , y, z ; p , p )+ α (cid:8) ˜ r f ( t, x )ˆ v ∓ ˜ r + col ( t, x )ˆ v ± ± ˜ r − col ( t, x )ˆ v ∓ (cid:9) ± ˜ ǫ f ( t, x ) | y + ( p − y ) + ( p − y ) − α ˆ v |± ˜ ǫ r ( t, x ) (cid:12)(cid:12)(cid:8) z ⊤ + ( p − y )˜ σ I ( t, x ) + ( p − y )˜ σ C ( t, x ) (cid:9) ˜ σ ( t, x ) − (cid:12)(cid:12) with ¯ g ( t, x , y, z ; p , p ) := z ⊤ (cid:8) (˜ r r − ˜ r D )˜ σ − (cid:9) ( t, x ) − n (2˜ r D − ˜ r f + ˜ h + ˜ h ) + (˜ r r − ˜ r D )(˜ σ I + ˜ σ C )˜ σ − o ( t, x ) y + n ˜ h − (˜ r f − ˜ r D ) + (˜ r r − ˜ r D )˜ σ I ˜ σ − o ( t, x ) p + n ˜ h − (˜ r f − ˜ r D ) + (˜ r r − ˜ r D )˜ σ C ˜ σ − o ( t, x ) p . Proposition 2.
Denote d := n + m and consider the system of second-order parabolic semilinearPDEs − ∂ t V = {L t − ˜ r D ( t, x ) } V, ( t, x ) ∈ [0 , T ) × R d ,V ( T, x ) =Ξ( x ) , − ∂ t U ± = L t U ± + ¯ g ± (cid:0) t, x , U, a ⊤ ∇ U ± ; V, ϕ ( V ) , ϕ ( V ) (cid:1) , ( t, x ) ∈ [0 , T ) × R d ,U ± ( T, x ) =Ξ( x ) , (47) where L t ( · ) is the infinitesimal generator for X given by (10). Suppose that there exists a uniqueclassical solution ( V, U ± ) ∈ (cid:0) C , ([0 , T ] × R d ) (cid:1) to (47). Then the solution to BSDE (46) isrepresented as ¯ Y ± ( t ) = U ± ( t, X ( t )) , ¯ Z ± ( t ) = (cid:0) a ∇ U ± (cid:1) ( t, X ( t )) , t ∈ [0 , T ] . Theorem 4.
In addition to Conditions (19) and (44), assume the following: h ≥ r − f − r D − (cid:0) r + r − r D (cid:1) ( σ I σ − ) + + (cid:0) r − r − r D (cid:1) ( σ I σ − ) − ,h ≥ r − f − r D − (cid:0) r + r − r D (cid:1) ( σ C σ − ) + + (cid:0) r − r − r D (cid:1) ( σ C σ − ) − , (48) and r + f ≥ r − col . (49) Then it holds that p = Y − (0) ≤ Y + (0) = ¯ p . Hence, for the derivative security given in Definition 2,any price p ∈ [ Y − (0) , Y + (0)] at time is arbitrage-free. Remark 14.
The conditions imposed in Theorem 4 to ensure the arbitrage-free property look tobe rather strong: violating (44), (48), or (49) seems to be realizable in real situations. Relaxingthe arbitrage-free condition by admitting “certain” arbitrage opportunities might be an interestingresearch direction for this bilateral hedging scheme with collateralizations. We refer to Thoednithi(2015) and Nie and Rutkowski (2018) as related studies.Sketch.
Using (35), (36), (38), and (43), we see that¯ f ± ( t, y, z ; ˆ v, p , p )= z ⊤ (cid:8) ( r r − r D ) σ − (cid:9) ( t ) − (cid:8) (2 r D − r f + h + h ) + ( r r − r D )( σ I + σ C ) σ − (cid:9) ( t ) y + (cid:8) h − ( r f − r D ) + ( r r − r D ) σ I σ − (cid:9) ( t ) p + (cid:8) h − ( r f − r D ) + ( r r − r D ) σ C σ − (cid:9) ( t ) p + α (cid:8) r f ( t )ˆ v ∓ r + col ( t )ˆ v ± ± r − col ( t )ˆ v ∓ (cid:9) ± ǫ f ( t ) | y + ( p − y ) + ( p − y ) − α ˆ v |± ǫ r ( t ) (cid:12)(cid:12)(cid:8) z ⊤ + ( p − y ) σ I ( t ) + ( p − y ) σ C ( t ) (cid:9) σ ( t ) − (cid:12)(cid:12) . δ , δ , δ ≥
0, we see that¯ f + ( · , y, z ; ˆ v + δ , p + δ , p + δ ) − ¯ f + ( · , y, z ; ˆ v, p , p )= (cid:8) h − ( r f − r D ) + ( r r − r D ) σ I σ − (cid:9) δ + (cid:8) h − ( r f − r D ) + ( r r − r D ) σ C σ − (cid:9) δ + α (cid:2) r f δ − r + col (cid:8) (ˆ v + δ ) + − ˆ v + (cid:9) + r − col (cid:8) (ˆ v + δ ) − − ˆ v − (cid:9)(cid:3) + ǫ f {| p + p − α ˆ v − y + ( δ + δ − αδ ) | − | p + p − α ˆ v − y |} + ǫ r h(cid:12)(cid:12)(cid:8) z ⊤ + ( p − y ) σ I + ( p − y ) σ C (cid:9) σ − + { δ σ I + δ σ C } σ − (cid:12)(cid:12) − (cid:12)(cid:12)(cid:8) z ⊤ + ( p − y ) σ I + ( p − y ) σ C (cid:9) σ − (cid:12)(cid:12)i . (50)Using the inequality | x + y | − | x | ≥ −| y | and the relation r + col (cid:8) (ˆ v + δ ) + − ˆ v + (cid:9) − r − col (cid:8) (ˆ v + δ ) − − ˆ v − (cid:9) ≤ (cid:0) r + col ∨ r − col (cid:1) δ , we see that (50) ≥ (cid:8) h − ( r f − r D ) + ( r r − r D ) σ I σ − (cid:9) δ + (cid:8) h − ( r f − r D ) + ( r r − r D ) σ C σ − (cid:9) δ + α (cid:0) r f − r − col (cid:1) δ − ǫ f ( δ + δ + αδ ) − ǫ r (cid:8) | σ I σ − | δ + | σ C σ − | δ (cid:9) = n h − r − f + r D + ( r r − r D ) σ I σ − − ǫ r | σ I σ − | o δ + n h − r − f + r D + ( r r − r D ) σ C σ − − ǫ r | σ C σ − | o δ + α (cid:16) r + f − r − col (cid:17) δ ≥ , (51)where we use (48) and (49). Consider the system of BSDEs (42) and write the solution as¯ Y ± ( t ; ξ T , φ , φ ) , ¯ Z ± ( t ; ξ T , φ , φ ) t ∈ [0 , T ]by emphasizing the parameters ( ξ T , φ , φ ). Take other payoff parameters (cid:16) ˜ ξ T , ˜ φ , ˜ φ (cid:17) such that ˜ ξ T ≥ ξ T , ˜ φ ≥ φ , and ˜ φ ≥ φ . Using the comparison theorem for BSDEstwice (for ˆ V and ¯ Y + ), and using relations (50) and (51), we deduce that¯ Y + (cid:16) ˜ ξ T , ˜ φ , ˜ φ (cid:17) ≥ ¯ Y + ( ξ T , φ , φ )and that Y + (cid:16) ˜ ξ T , ˜ φ , ˜ φ (cid:17) ≥ Y + ( ξ T , φ , φ ) . This implies the minimality of Y + ( ξ T , φ , φ ) and the equality,¯ p = Y + (0; ξ T , φ , φ ) . The equality, p = Y − (0; ξ T , φ , φ ) , can be seen similarly. 24 emark 15. We have that for k ≥ , Y ± ( t ; kξ T , kφ , kφ ) ≡ kY ± ( t ; ξ T , φ , φ ) for t ∈ [0 , T ] .This positive homogeneity is seen from those of the drivers of BSDEs (42), namely ¯ f ± ( t, ky, kz ; k ˆ v, kφ , kφ ) = k ¯ f ± ( t, y, z ; ˆ v, φ , φ ) , − r D ( t ) ( k ˆ v ) = k {− r D ( t )ˆ v } . See Jiang (2008) for the details.
In this subsection, we assume that ǫ f ∨ ǫ r ≤ ǫ (52)with some (small) positive constant ǫ ≪
1. Consider the system of BSDEs − dY , ± ( t ) = f , ± (cid:16) t, Y , ± ( t ) , Z , ± ( t ) , U , ± ( t ) , U , ± ( t ); ˆ V ( t ) (cid:17) dt − Z , ± ( t ) ⊤ dW ( t ) − U , ± ( t ) dM ( t ) − U , ± ( t ) dM ( t ) , for 0 ≤ t ≤ τ ∧ τ ∧ T,Y , ± ( τ ∧ τ ∧ T ) = H, − d ˆ V ( t ) = − r D ( t ) ˆ V ( t ) dt − ∆( t ) ⊤ dW ( t ) for 0 ≤ t ≤ T, ˆ V ( T ) = ξ T (53)on (Ω , F , P , G ), where f , ± ( t, y, z, u , u ; ˆ v ) := f ( t, y, z, u , u ) + α (cid:8) r f ( t )ˆ v ∓ r + col ( t )ˆ v ± ± r − col ( t )ˆ v ∓ (cid:9) . Associated with (53), consider the reduced system of BSDEs − d ¯ Y , ± ( t ) = ¯ f , ± (cid:16) t, ¯ Y , ± ( t ) , ¯ Z , ± ( t ); ˆ V ( t ) , φ ( t ) , φ ( t ) (cid:17) dt − ¯ Z , ± ( t ) ⊤ dW ( t ) for 0 ≤ t ≤ T, ¯ Y , ± ( T ) = ξ T , − d ˆ V ( t ) = − r D ( t ) ˆ V ( t ) dt − ∆( t ) ⊤ dW ( t ) for 0 ≤ t ≤ T, ˆ V ( T ) = ξ T (54)on (Ω , F , P , F ), where¯ f , ± ( t, y, z ; ˆ v, p , p ) := f , ± ( t, y, z, p − y, p − y ; ˆ v ) + ( p − y ) h ( t ) + ( p − y ) h ( t ) . We obtain the following. 25 heorem 5.
Assume Conditions (19) and (44). For ( ¯ Y ± , ¯ Z ± ) , ( ¯ Y , ± , ¯ Z , ± ) , which are solutionsto BSDEs (42) and (54), respectively, it holds that ¯ Y − ≤ ¯ Y , − ≤ ¯ Y , + ≤ ¯ Y + (55) and that (cid:13)(cid:13) ¯ Y ± − ¯ Y , ± (cid:13)(cid:13) β,T + (cid:13)(cid:13) ¯ Z ± − ¯ Z , ± (cid:13)(cid:13) β,T = O ( ǫ ) (56) as ǫ → in both + and − cases.Sketch. The relation (55) is easily seen from the comparison theorem of BSDEs. To see (56),we can apply the continuity (and the differentiability) results with their proofs with respect toparameterized BSDEs, shown in El Karoui et al. (2000) (see Proposition 2.4 and its proof in [15]for the details).Combining Theorems 4 and 5, we see the following.
Corollary 1.
Assume Conditions (19), (44), (48), and (49). Then Y , − (0) and Y , + (0) arearbitrage-free prices at time for the derivative security given in Definition 2. The above corollary implies that Y , ± (0) may be regarded as approximated prices of thederivative security for the writer and her counterparty, which prohibit the existence of an arbitrageopportunity. Because BSDEs for ( Y , ± , Z , ± ) are linear, we obtain the closed-form expressionsfor Y , ± as follows. Let us introduce the probability measure ˜ P T on (Ω , F T ) by d ˜ P T (cid:12)(cid:12) F t = E ( t ) d P (cid:12)(cid:12) F t , t ∈ [0 , T ] , where E ( t ) := exp (cid:20)Z t (cid:8) r r ( u ) − r D ( u ) (cid:9) ⊤ ( σ ( u ) − ) ⊤ dW ( u ) − Z t (cid:8) r r ( u ) − r D ( u ) (cid:9) (cid:12)(cid:12) σ ( u ) − (cid:12)(cid:12) du (cid:21) . We denote the expectation with respect to ˜ P T conditioned by F t by ˜ E t [( · · · )] = ˜ E [( · · · ) |F t ]. Recallthat ˜ W ( t ) := W ( t ) − Z t (cid:8) r r ( u ) − r D ( u ) (cid:9) σ ( u ) − du, t ∈ [0 , T ]is a (˜ P T , F )-Brownian motion by the Maruyama–Girsanov theorem, and on (cid:16) Ω , F , ˜ P T , F (cid:17) the riskyasset price process S has the dynamics dS ( t ) = diag( S ( t )) n σ ( t ) d ˜ W ( t ) + r r ( t ) dt o , S (0) ∈ R n ++ . Also, we denote DF r ( t, u ) := exp (cid:26) − Z ut r ( s ) ds (cid:27) for the process r := ( r ( t )) t ∈ [0 ,T ] . We then obtain the following. That is, the drivers f , ± ( t, y, z, u , u ; ˆ v ) are linear with respect to ( y, z, u , u ). roposition 3. The following representation holds: ¯ Y , ± ( t ) = V( t ) + VA ( t ) + VA ( t ) + VA ( t ) + VA ( t ) + VA ± ( t ) . (57) Here, V( t ) :=˜ E t h DF r f ( t, T ) ξ T i , VA ( t ) :=˜ E t "Z Tt DF R ( t, u ) h ( u ) ˆ φ ( u ) du , VA ( t ) :=˜ E t "Z Tt DF R ( t, u ) h ( u ) ˆ φ ( u ) du , VA ( t ) := − ˜ E t "Z Tt DF R ( t, u ) n ( r f − r D ) (cid:16) ˆ φ + ˆ φ (cid:17)o ( u ) du , VA ( t ) :=˜ E t "Z Tt DF R ( t, u ) n ( r r − r D ) (cid:16) ˆ φ σ I + ˆ φ σ C (cid:17) σ − o ( u ) du , VA ± ( t ) := α ˜ E t "Z Tt DF R ( t, u ) n(cid:0) r f − r ± col (cid:1) ˆ V + − (cid:0) r f − r ∓ col (cid:1) ˆ V − o ( u ) du , where we define ˆ φ i := φ i − V for i = 1 , , and R := r D − (cid:0) r f − r D (cid:1) + (cid:8) ( r r − r D ) ( σ I + σ C ) ( σ ) − (cid:9) + h + h . Proof.
Using the representation formula for linear BSDE (e.g., see Proposition 2.2 of [15]), we seethat ¯ Y , ± ( t ) = ¯V( t ) + VA ( t ) + VA ( t ) + VA ( t ) + VA ( t ) + VA ± ( t ) , where ¯V( t ) :=˜ E t [DF R ( t, T ) ξ T ] , VA ( t ) :=˜ E t "Z Tt DF R ( t, u ) h ( u ) φ ( u ) du , VA ( t ) :=˜ E t "Z Tt DF R ( t, u ) h ( u ) φ ( u ) du , VA ( t ) := − ˜ E t "Z Tt DF R ( t, u ) (cid:8) ( r f − r D ) ( φ + φ ) (cid:9) ( u ) du , VA ( t ) :=˜ E t "Z Tt DF R ( t, u ) (cid:8) ( r r − r D ) ( φ σ I + φ σ C ) σ − (cid:9) ( u ) du . (cid:2) VA + VA + VA + VA − VA − VA − VA − VA (cid:3) ( t )= − ˜ E t "Z Tt DF R ( t, u )V( u ) (cid:8) R ( u ) − r f ( u ) (cid:9) du = − ˜ E t "Z Tt DF R ( t, u )˜ E u h DF r f ( u, T ) ξ T i (cid:8) R ( u ) − r f ( u ) (cid:9) du =˜ E t " DF r f ( t, T ) ξ T Z Tt ∂∂u DF R − r f ( t, u ) du =˜ E t h DF r f ( t, T ) n DF R − r f ( t, T ) − o ξ T i =˜ E t hn DF R ( t, T ) − DF r f ( t, T ) o ξ T i = ¯V( t ) − V( t ) , hence the proof is complete. Remark 16.
Suppose that r r ≡ r f ≡ r D holds. In this case, ˜ P T ≡ P and V ≡ ˆ V follow.Furthermore, consider φ i ( t ) := ϕ i (cid:16) ˆ V ( t ) (cid:17) , where (17) is employed for i = 1 , . Then, in (57), VA ≡ VA ≡ , and − VA , VA , and VA ± are called the debt valuation adjustment (DVA), thecredit valuation adjustment (CVA), and the collateral valuation adjustment (ColVA), respectively,which are popularly used XVA terms in practice for the valuation adjustment in the pricing ofderivative securities. Concretely, DVA, CVA, and ColVA at time t are written as DVA( t ) := − E t "Z Tt DF r D + h + h ( t, u ) h ( u ) ˆ φ ( u ) du , CVA( t ) := E t "Z Tt DF r D + h + h ( t, u ) h ( u ) ˆ φ ( u ) du , ColVA ± ( t ) := E t "Z Tt DF r D + h + h ( t, u ) n(cid:0) r D − r ± col (cid:1) α ˆ V + − (cid:0) r D − r ∓ col (cid:1) α ˆ V − o ( u ) du , respectively, where we denote E t [( · · · )] := E [( · · · ) |F t ] . Further, FVA( t ) := E t "Z Tt DF r D + h + h ( t, u ) (cid:8) ( r f − r D ) ( φ + φ ) (cid:9) ( u ) du , called the funding valuation adjustment (FVA) at time t , is another popularly used adjustmentterm in practice, which reflects the funding cost of uncollateralised derivatives above the riskfreerate of return. We can roughly relate these XVA terms with the correction terms in Proposition3 as follows: Let r r ≡ r D , which implies VA ≡ . Further, suppose r f ≈ r D . Then, we may In practice, the difference r r − r D seems to have been usually ignored. nterpret as DVA ≈ − VA , CVA ≈ VA , ColVA ± ≈ VA ± , and FVA ≈ VA , or FVA ≈ VA + (VA + DVA) + (VA − CVA) + (cid:0)
ColVA ± − VA ± (cid:1) . For other theoretical studies on the valuation adjustments and related interpretation of XVA usedin practice, we refer to Brigo et al. (2020) and the reference therein. Also, for comprehensiveinformation on XVA issue and expanding related issues (e.g., computational issue), see for exampleGregory (2015) and Glau et al. (2016), and the references therein, which are still nonexhaustive.
As we see in Theorem 5 and Corollary 1, under certain conditions, Y , + ( t )( < Y + ( t )), which is azeroth-order approximation of the minimal hedging cost Y + ( t ), is an arbitrage-free price for thewriter at time t . In this subsection, we try to improve our hedging strategy by using a first-orderapproximation. Using the solution to BSDE (53), consider the linear BSDE − dY , ± ( t ) = f (cid:16) t, Y , ± ( t ) , Z , ± ( t ) , U , ± ( t ) , U , ± ( t ) (cid:17) dt + f , ± (cid:16) t, Y , ± ( t ) , Z , ± ( t ) , U , ± ( t ) , U , ± ( t ) , ˆ V ( t ) (cid:17) dt − Z , ± ( t ) dW ( t ) − U , ± ( t ) dM ( t ) − U , ± ( t ) dM ( t ) ,Y , ± ( τ ∧ τ ∧ T ) =0 (58)on (Ω , F , P , G ), where f , ± ( t, y, z, u , u ; ˆ v ) := ± ǫ f ( t ) | y + u + u − α ˆ v |± ǫ r ( t ) (cid:12)(cid:12)(cid:8) z ⊤ + u σ I ( t ) + u σ C ( t ) (cid:9) σ ( t ) − (cid:12)(cid:12) . Furthermore, using the solution to BSDE (54), consider the linear BSDE − d ¯ Y , ± ( t ) = ¯ f (cid:0) t, ¯ Y , ± ( t ) , ¯ Z , ± ( t ); φ ( t ) , φ ( t ) (cid:1) dt + ¯ f , ± (cid:16) t, ¯ Y , ± ( t ) , ¯ Z , ± ( t ); ˆ V ( t ) , φ ( t ) , φ ( t ) (cid:17) dt − ¯ Z , ± ( t ) dW ( t ) , ¯ Y , ± ( T ) =0 (59)on (Ω , F , P , F ), where¯ f ( t, y, z ; p , p ) := f ( t, y, z, p − y, p − y ) , ¯ f , ± ( t, y, z ; ˆ v, p , p ) := ± ǫ f ( t ) | y + ( p − y ) + ( p − y ) − α ˆ v |± ǫ r ( t ) (cid:12)(cid:12)(cid:8) z ⊤ + ( p − y ) σ I ( t ) + ( p − y ) σ C ( t ) (cid:9) σ ( t ) − (cid:12)(cid:12) . Proposition 4.
It holds that for any sufficiently large β > , k ¯ Y ± − (cid:0) ¯ Y , ± + ¯ Y , ± (cid:1) k β,T + k ¯ Z ± − (cid:0) ¯ Z , ± + ¯ Z , ± (cid:1) k β,T = O ( ǫ ) as ǫ → , where we assume (52). Acknowledgements
The authors are grateful to an anonymous referee for valuable comments and helpful suggestions.
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