A new approximate CVA of interest rate swap in the SABR/LIBOR market model : an asymptotic expansion approach

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1 Avalable onlne at Proceda Computer Scence Internatonal Conference on Computatonal Scence, ICCS 11 A new approxmate CVA of nterest rate swap n the SABR/LIBOR market model : an asymptotc expanson approach Masahro Nsba 1, Tokyo Insttute of Technology Abstract The author presents a new approxmate prcng formula for the credt valuaton adjustment of nterest rate swap n the SABR/LIBOR market model usng an asymptotc expanson method. He compare t wth Monte Carlo smulaton. It s shown that the new method makes computatons extremely fast. Keywords: CVA, SABR/LIBOR market model, Asymptotc expanson 1. Introducton The counterparty rsk management s becomng ncreasngly mportant after the fnancal crss n 8 and the tghtenng of the regulatons [3], [1]. Many fnancal nsttutons are more nterested n the credt valuaton adjustmentcva of dervatves, but they have problems wth the CVA especally n the calculaton speed [1]. Most nsttutons cte nterest rate products as contrbutng the most to ther overall rsk [1]. In the nterest rate dervatves markets, the SABR model s wdely used by practtoners n the fnancal ndustry. The LIBOR market model has become the market standard, and the SABR model can capture the volatlty smle of nterest rates [4], [6]. In ths paper, we consder about the CVA of the nterest rate swap. It takes a lot of tme to calculate the CVA of the swap by Monte Carlo smulaton. Usng a new approach, ths paper derves an approxmate formula for the CVA of the swap. Ths new formula enable to calculate the CVA much faster than Monte Carlo method. To obtan the formula, we apply an asymptotc expanson method based on nfnte dmensonal analyss called the Watanabe-Yoshda theory and the Mallavn calculus [1], [11]. The method was appled to dervatve valuaton problems [7], [8], [5]. The asymptotc expanson requres a huge amount of manpulaton of symbolc expressons, so we develop a new lbrary for Maxma n Lsp. Wth ths lbrary, we can obtan hgh order asymptotc expansons of the soluton to SDEs. In the followng sectons, after an explanaton of the framework of an asymptotc expanson method n Secton, Secton 3 provde a bref descrpton of the new lbrary for maxma. Secton 4 apples the method to the forward swap rate. Secton 5 gves the new approxmate prcng formula for CVA of the nterest rate swap and Secton 6 provdes the accuracy valdaton by comparng the approxmate formula wth Monte Carlo smulaton. Secton 7 concludes. Emal address: nshba9@craft.ttech.ac.jp Masahro Nsba 1 Center for Research n Advanced Fnancal Technology, Tokyo Insttute of Technology, -1-1 Ookayama, Meguro-ku, Tokyo Japan Publshed by Elsever Ltd. Open access under CC BY-NC-ND lcense. Selecton and/or peer-revew under responsblty of Prof. Mtsuhsa Sato and Prof. Satosh Matsuoka do:1.116/j.procs

2 Masahro Nsba. / Proceda Computer Scence Asymptotc expanson We consder R d -valued dffuson process X ɛ that s the soluton to the followng stochastc dfferental equatons: dx ɛ = V X ɛ,ɛ dt + ɛv X ɛ dw t 1 X ɛ = x where x s constant, W = W 1,, W d s an d-dmensonal standard Brownan process, and ɛ [, 1] s a known parameter. The followng theorem s proved n [1]. Theorem.1. Suppose V : R d [, 1] R d and V : R d R d R m are smooth, and these dervatves of any order are bounded. Next, suppose that a functon g : R d R to be smooth and all dervatves have polynomal growth orders. Then, for ɛ,gx ɛ T has ts asymptotc expanson: gx ɛ T = g T + ɛg 1T + ɛ g T + ɛ 3 g 3T + oɛ 3. 3 We explan the calculatng method of asymptotc expanson. The coeffcents n the expanson, g T, g 1T, g T,, can be obtaned by Taylor s formula and represented by multple Wener-Ito ntegrals. For examples, let D t = Xɛ t ɛ ɛ=, E t = X ɛ t ɛ=, F ɛ t = 3 X ɛ t ɛ=, and g ɛ 3 T, g 1T, g T, g 3T are represented by g T = g X T 4 g 1T = g T = 1 g 3T = 1 6 =1 + 1 g X T D T 5, j=1, j,k=1, j=1 j g X T D T D j T + 1 =1 j k g X T D T D j T Dk T j g X T E T D j T =1 g X T E T 6 g X T F T 7 where g T s determnstc, the other terms are stochastc, D t, E t, F t, = 1,, d denote the -th elements of D t, E t, F t respectvely. D t, E t, F t are represented by D t = E t = F t = + +3 [ Y t Yu 1 ɛ V X u, du + V ] X u dwu Y t Yu 1 j k V X j,k=1 j=1 Y t Y 1 u j,k=1 ɛ j V X u, D j u du + j,k,l=1 u, DuD j k u du + V X u, du j=1 j V X u D j u dw u j k l V X u, D j ud k ud l u du + 3 j k ɛ V X u, D j ud k u du + 3 j=1 j,k=1 j ɛ V X u, Eu j du j k V X u, EuD j k u du 8 9

3 1414 Masahro Nsba. / Proceda Computer Scence j ɛ V X u, Du j du + 3 ɛ V X u, du j=1 +3 j,k=1 j k V X u D j u D k u dw u + 3 where Y denotes the soluton to the dfferental equaton: j=1 j V X u E j u dw u 1 dy t = V X t, Y t dt, 11 Y = I d. 1 Here, V denotes the d d matrx whose j, k-element s k V j, and I d denotes the d d dentty matrx. Next, normalze g X ɛ T to G ɛ = g X ɛ T gt ɛ for ɛ, 1]. Moreover, let a t = a t = t g [ X T YT Yt 1 V ] X t where t A denotes the nverse matrx of A. We make the followng assumpton: Assumpton 1 Σ T = a t t a t dt >. 15 Note that g 1T follows a normal dstrbuton wth varance Σ T and hence Assumpton 1 means that the dstrbuton of g 1T does not degenerate. Next, let ψ G ɛξ be a characterstc functon of G ɛ. Then, ψ G ɛξ s expanded around ɛ = as follows: ψ G ɛξ = E [ exp 1ξG ɛ] = E [ exp ] [ ] 1ξg 1T + ɛ 1ξ E exp 1ξg1T gt +ɛ 1ξ E [ exp ] ɛ [ ] 1ξg 1T g3t + 1ξ E exp 1ξg1T g T + oɛ 1ξ ΣT = exp + ɛ 1ξ E [ exp [ ]] 1ξg 1T E gt g 1T +ɛ 1ξ E [ exp [ ]] 1ξg 1T E g3t g 1T + ɛ 1ξ E [ exp 1ξg1T E [ g T g 1T ]] + oɛ. 16 Then, we obtan the below proposton. Lemma.1. We assume Assumpton 1. Then, approxmate probablty densty functon of G ɛ s represented as follows: [ f G ɛ = n[x;, Σ T ] + ɛ ] x {h xn[x;, Σ T ]} +ɛ [ ] x {h 3xn[x;, Σ T ]} + 1 [ ] ɛ x {h xn[x;, Σ T ]} + oɛ. 17 where G ɛ s defned by the equaton 13, h x = E [ g T g 1T = x ],h x = E [ g T g 1T = x ],h 3 x = E [ g 3T g 1T = x ], and n[x;, Σ T ] denotes the pdf of the normal dstrbuton wth average and varance Σ T.

4 Masahro Nsba. / Proceda Computer Scence We can calculate the condtonal expectatons n the equaton 17 by applyng Ito s lemma and the followng proposton whch n [9]. Proposton.1. Let J n f n denote the n-tmes terated Ito ntegral of f n L T n : J n f n = 1 n 1 for n Z +,J f = f where f s constant. Then, ts expectaton condtonal on J 1 q = x s gven by E [ J n f n J 1 q = x ] 1 n 1 = f n t 1, t,, t n dw tn dw t dw t1 18 Hn x; q f n t 1, t,, t n qt 1 qt qt n dt n dt dt 1 q L T where T >, T = [, T], {1,,, n},t T and H n x; Σ s the Hermte polynomal of degree n, that s to say H n x; Σ = Σ n e x /Σ d n /Σ dx n. e x 3. Asymptotc expanson lbrary for Maxma We develop a lbrary for the asymptotc expanson of the soluton to SDEs n Maxma usng lsp, the programmng language. So, we explan about Maxma, a computer algebra system, and the lbrary n ths secton Maxma Maxma s a system for the manpulaton of symbolc and numercal expressons, ncludng dfferentaton and ntegraton, etc [13]. 3.. Asymptotc expanson lbrary To obtan the symbolc expressons of the asymptotc expanson, we extend Maxma to manpulate Ito s ntegral, Ito s formula, Fubn s theorem, the formula to calculate condtonal expectatonsproposton.1, formulas to solve SDEs, and the nverson formula of characterstc functons, etc. We defne these theorems and formulas n the lbrary as follows. Defnton 3.1. Ito s formula n the lbrary L T n 19 f 1 tdw1t f tdw t = + + f 1 t f t f sdw sdw 1 t f 1 sdw 1 sdw t f 1 t f tdw 1 tdw t where dw 1 tdw t = Defnton 3.. Fubn s theorem n the lbrary { W1 W dt W 1 = W. f 1 t f sdwsdt = f 1 tdt f tdwt f t f 1 sdsdwt.

5 1416 Masahro Nsba. / Proceda Computer Scence Defnton 3.3. The formula to calculate condtonal expectatons n the lbrary [ 1 n 1 ] E f n t 1, t,, t n dw tn dw t dw t1 qtdw t = x = 1 n 1 Hn x; q f n t 1, t,, t n qt 1 qt qt n dt n dt dt 1 q L T where T = [, T]T >,t T = 1,,, n and H n x; Σ s defned as H n x; Σ = Σ n e x /Σ Defnton 3.4. The formulas to solve SDEs n the lbrary We consder followng SDE: d n dx n e x /Σ. ds t = f t, S t dt + gt, S t dw, S = s. Formula 1. If gt, x = t, x R then, S t s the soluton to the followng ODE: ds t = f t, S t dt, S = s. Formula. If f t, x = at x and gt, x = bt x t, x R then, S t = s exp as 1 bs dt + Formula 3. If f t, x = at x and gt, x = bt t, x R then, S t = s + exp asds exp s bsdw. audu bsds L T n 4. Asymptotc expanson of forward swap rate Accordng to [6], forward swap rate n SABR/LMM market model s represented by the soluton to the followng stochastc dfferental equaton : ds t = S t B σ t dwt 1 1 dσ t = v 1 σ t dwt 1 + v σ t dwt where W = W 1, W s -dmensonal standard Wener process, and B [, 1]. Let the correlaton between forward swap rate and ts volatlty be ρ [ 1, 1], then v 1 = vρ, v = v 1 ρ where v. Now, we calculate the asymptotc expanson of the soluton to above SDE. 1, are represented as follows: S ɛ T = S + ɛ σ ɛ T = σ + ɛ S ɛ t B σ ɛ tdw 1 t 3 v 1 σ ɛ tdwt 1 + ɛ v σ ɛ tdwt 4

6 Masahro Nsba. / Proceda Computer Scence Then, for ɛ, S ɛ T has ts asymptotc expanson: ɛ n S ɛ T = The coeffcents, S n Tn =, 1,, 3,..., are represented as follows: m ɛ n S n T + oɛ m 5 n= S T = S 6 S 1 T = S B σ S T = Bσ S B 1 + σv 1 S B +σv S B dw 1 t 7 dw 1 sdw 1 t S 3 T = B σ 3 S 3B + Bσ v 1 S B 1 + σv 1 S B dw sdw 1 t 8 s dw 1 udw 1 sdw 1 t 1 + B σ 3 S 3B 1 Bσ3 S 3B + Bσ v 1 S B 1 dw s 1 dw 1 t + Bσ v S B 1 + σv 1 v S B +Bσ v S B 1 +σv S B +σv 1 v S B s s s dw 1 s dw sdw 1 t dw udw sdw 1 t dw udw 1 sdw 1 t dw 1 udw sdw 1 t 9 The coeffcents of hgher order are represented n a same way. By lemma.1, the approxmate pdf of G ɛ SABR = S ɛ T S T ɛ s represented as follows: f G ɛ x = n[x;, Σ T ] + ɛ SABR [ x {E [S T S 1 T = x] n[x;, Σ T ]} +ɛ [ ] x {E [S 3T S 1 T = x] n[x;, Σ T ]} + 1 [ { [ ɛ E S x T S 1 T = x ] n[x;, Σ T ] } ] + oɛ 3 where Σ T = E [ S 1 T ] and n[x;, Σ T ] denotes the pdf of the normal dstrbuton wth average and varance Σ T. ] 5. CVA of nterest rate swap Consder a payer swap such that a fxed nterest rate K s pad and a floatng nterest rate FT ; T, T +1, = a,...,b 1, s receved at the consecutve dates T a+1,...,t b Basc setup Ths subsecton defnes basc concepts such as tenor structures, dscount bond prce, the money market account, forward Lbor rates, the spot measure, the forward measure and the forward swap measure. Frst a tenor structure s gven by a fnte set date: = T < T 1 < < T N

7 1418 Masahro Nsba. / Proceda Computer Scence where T N s a pre-specfed date and δ = T j T j 1 for j = 1,,...,N. Pt, T j denotes the prce of the the dscount bound wth maturty T j at tme t, where PT j, T j = 1 and Pt, T j = for t T j, T N ]. The forward LIBOR rate at tme t T j wth term [T j 1, T j ] s defned as Ft; T j 1, T J = 1 Pt; T j 1 1, 31 δ Pt; T j for any j = 1,,...,N. The money market account s prce Bt s defned as Bt = Pt; T γt Π γt j=1 PT j 1; T j 3 where T γt denotes the frst tenor after tme t. Let Ω, F, Q denote a complete probablty space satsfyng the usual condtons where Q s the spot measure whch s the rsk neutral measure wth numerare Bt. Then, let Q T and Q a,b be the T-forward measure and the forward swap measure whch are equvalent martngale measures wth numerare Pt; T and Nt = b =a+1 Pt, T respectvely. 5.. Interest rate swapton Let S a,b t denote the forward rate wth term [T a, T b ] at tme t t [, T a ], and t s represented as follows: S a,b t = Pt, T a Pt, T b δ b =a+1 Pt, T. 33 The prce of a payer swapton wth nomnal 1, strke K, maturty t and underlyng tenor T a, T a+1...t b T a s the frst reset date and T b the maturty of the underlyng swap s represented as follows: π; t, T a, T b = δ = δ b P, T E [ Qa,b S t K +] =a+1 b =a+1 P, T ɛx K S t f K S t G ɛ xdx + oɛ SABR ɛ 34 where E Qa,b s expectaton under Q a,b CVA Defnton 5.1. The CVA of a dervatve whch value at tme t s Vt s defned as follows: [b ] CVA := 1 RE Q Bt 1 max Vt, ddpt 35 where R R [, 1] and DPt denote the recovery rate and default probablty respectvely CVA of nterest rate swap We defne the default probablty DPtas DPt = 1 exp λ dt = 1 exp λt. 36

8 Masahro Nsba. / Proceda Computer Scence Let Vt denote the prce of the nterest rate swap at tme t. Accordng to [1], ts CVA s represented as follows: [b ] CVA = 1 RE Q Bt 1 max Vt, ddpt b 1 R π; t, T γt, T b λe λt dt a = 1 R π; t, T a, T b λe λt dt + b 1 =a+1 π; t, T, T b λe λt dt T 1 where R R [, 1] denotes the recovery rate. Hence, we assumed the correlaton between the nterest rate swap and default s. 6. Numercal results Ths secton provdes the accuracy valdaton of the approxmate CVA of the swap n secton 5. Benchmark values are computed by Monte Carlo smulatons. Let ɛ be 1, B =.5, δ =.5, a = 1, b =, K = 3.5%, 4%, 4.5% and R =. Other parameters used n the test are gven n Table 1. The parameters are calbrated from market date n Table usng the method of [6]. 37 Table 1: Parameters frst reset date S σ v ρ In Monte Carlo smulatons for benchmark values, we use Euler-Maruyama scheme as a dscretzaton scheme wth 1, tme steps and generate 5,, paths n each smulaton. Table : Euro cap prces n bass ponts on 18 November 8 Yyear-K%

9 14 Masahro Nsba. / Proceda Computer Scence Table 3: Numercal result fxed rate MC A.E.nd A.E.3rd A.E.4th A.E.5th 3.5% % % Table 4: Dfference between MC and Asymptotc expanson fxed rate A.E.nd-MC A.E.3rd-MC A.E.4th-MC A.E.5th-MC 3.5% E E E E-5 4.% E E E E-5 4.5% -3.14E E E E-5 Table 5: #Partton, #Sample, and CPU tme requred for 3 dgts accuracy n the case of K = 3.5% Method #Partton #Sample CPU tmesec E-M + MC AE nd AE 3rd AE 4th AE 5th The results Table 3, Table 4 show the approxmaton error of the asymptotc expansons are relatvely small to nomnal value 1. Moreover, the computatonal tme of the new formula s much faster than that of the Monte Carlo smulaton Table Concluson We have presented a new approxmate formula for the CVA of the nterest rate swap n the SABR/LIBOR market model usng an asymptotc expanson method. Numercal results show that our approxmate formula s reasonablyaccurate compared wth Monte Carlo method. And new formula can be calculated much faster then Monte Carlo method. Our future research topcs are as follows: Frst, we apply ths approach to more complcated default probablty models. Second, we apply the asymptotc expanson to the forward swap rate wthout freezng technques. Thrd, we derve the formulas for the other dervatves such as an Amercan opton. References [1] Cesar, G., Aqulna, J., Charpllon, N., Flpovc, Z., Lee, G., and Manda, I. Modellng, Prcng, and Hedgeng Counterparty Credt Exposure. Sprnger, 9. [] Flpovc, D. Term-Structure Models. A Graduate Course. Sprnger, 9. [3] Gregory, J. Counterparty rsk credt rsk, Wley, 1. [4] Hagan, P. and A. Lesnewsk. LIBOR market model wth SABR style stochastc volatlty, 8. Avalable onlne at: [retreved 19 January 11] [5] Osajma, Y., The Asymptotc Expanson Formula of Impled Volatlty for Dynamc SABR Model and FX Hybrd Model, preprnt, The Unversty of Tokyo, 6.

10 Masahro Nsba. / Proceda Computer Scence [6] Rebonato, R., McKay, K., and Whte, R. The SABR/LIBOR Market Model: Prcng, Calbraton and Hedgng for Complex Interest-Rate Dervatves, Wley, 9. [7] Takahash, A., An Asymptotc Expanson Approach to Prcng Con- tngent Clams, Asa-Pacc Fnancal Markets, Vol. 6, , [8] Takahash, A., On an Asymptotc Expanson Approach to Nu- mercal Problems n Fnance, Sugaku, Mathematcal Socety of Japan, Vol.59-1, 75-91, 7. n Japanese. [9] Takahash, A., Takehara, K., and Toda, M. Computaton n an asymptotc expanson method. CARF Workng Paper Seres CARF-F-149, 9. [1] Watanabe, S., Analyss of Wener functonals Mallavn calculus and ts applcaton to heat kernels, Ann. Probab, 15, 1-39, [11] Yoshda, N., Asymptotc Expansons of Maxmum Lkelhood Estmators for Small Dusons va the Theory of Mallavn-Watanabe, Probablty Theory and Related Felds, Vol 9, 199. [1] Credt Value Adjustment: and the changng envronment for prcng and managng counterparty rsk. Algorthmcs, December 9. [13] Maxma Manual Ver. 5.. Avalable onlne at: [retreved 7 January 11]