Calibration of BGM models by semidefinite programming. Friday, May 31, A. d Aspremont

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1 Calibraion of BGM models by semidefie programmg. A. d Aspremon Friday, May 31, 2002 Research done under he direcion of Nicole El Karoui. Thanks o F.I.R.S.T. Swaps, BNP Paribas (London). Ecole Polyechnique: alexandre.daspremon@polyechnique.org

2 2 Calibraion of BGM models by semidefie programmg. 1.1 Ineres rae model calibraion All Heah, Jarrow & Moron (1992) based models are fully paramerized by he curve oday and a covariance funcion. If we discreize his covariance funcion, he naural variable he calibraion problem is a covariance marix, i.e. a posiive semidefie marix. Classic calibraion mehods are heavily paramerized and only describe a small, ofen non-convex subsef he sef semidefie marices. When usg hese echniques, sensiiviy analysis has o be done by recalibrag. A. d Aspremon Fronières en Fance Paris, Friday May

3 3 Calibraion of BGM models by semidefie programmg. 1.2 Resuls on he BGM model calibraion We can express he swap rae as a baskef Forwards wih very sable coefficiens. European Caples and Swapions can be priced usg he Black (1976) marke formula wih an appropriaely chosen variance. This marke variance is lear he coefficiens of he Forward raes covariance marix. This allows us o solve he calibraion problem as a semidefie program. A. d Aspremon Fronières en Fance Paris, Friday May

4 4 Calibraion of BGM models by semidefie programmg. 1.3 Relaed lieraure Works by Neserov & Nemirovskii (1994) and Vandenberghe & Boyd (1996) on semidefie programmg Brace, Gaarek & Musiela (1997) and Musiela & Rukowski (1997) on he Libor marke model. Rebonao (1998), Brace, Dun & Baron (1999) and Sgleon & Umansev (2001) on Swaps as baskes of Forwards. Rebonao (1999) on a calibraion mehod paramerized by facors. Parallel work by Brace & Womersley (2000) on he calibraion of he BGM by semidefie programmg and he evaluaion of he Bermudan Swapion. A. d Aspremon Fronières en Fance Paris, Friday May

5 5 Calibraion of BGM models by semidefie programmg. 2 Swapion pricg 2.1 The Swap rae The Swap is defed here as he rae ha equals he PV of a fixed and a floag leg: where swap(, T0,Tn) = Level(, T fixed 0,T fixed n )= floag B(, T 0 ) B(, T floag n+1 ) Level(, T fixed 0,T n fixed ) n+1 i=1 coverage(t fixed i 1,T fixed i )B(, T fixed i ) A. d Aspremon Fronières en Fance Paris, Friday May

6 6 Calibraion of BGM models by semidefie programmg. This rae can aga be wrien: swap(, T0, Tn) = n i=0 ωi()k(, Ti) where K(, Ti) are he Forward Raes wih mauriies Ti,i =1,..., n and he weighs ωi() are given by ωi() = floa coverage(t i,t floa floa i+1 )B(, T i+1 ) Level(, T fixed 0,T n fixed ) In pracice, hese weighs are very sable (see Rebonao (1998)). A. d Aspremon Fronières en Fance Paris, Friday May

7 7 Calibraion of BGM models by semidefie programmg. This sabiliy has been sudied Hamy (1999) of which we repor, wih he auhor s permission, some summary saisics: Currency USD USD GBP GBP EUR EUR Swap M raio Max raio Variance Sample raio of volailiy beween weighs and correspondg Forwards. Here, M raio and Max raio are he mimum (resp. maximum) volailiy raio among he weighs of a paricular Swap. Compued usg he sandard quadraic variaion esimaor wih exponenially decayg weighs ( period, marke daa couresy of BNP-Paribas London). A. d Aspremon Fronières en Fance Paris, Friday May

8 8 Calibraion of BGM models by semidefie programmg. 2.2 BGM Swapion price Followg Jamshidian (1997), we can wrie he price of he Swapion wih srike k as a haf a Call on a Swap rae: Ps() =Level(, T, T N )E Q LV L n i=0 ωi(t )K(T,Ti) k + where Q LV L is he swap forward margale probabiliy measure. In wha follows, we will make wo approximaions: We replace he weighs ωi(t ) by heir value oday ωi(). We suppose ha n i=0 ωi()k(t,ti) isasumofq LV L lognormal margale. A. d Aspremon Fronières en Fance Paris, Friday May

9 9 Calibraion of BGM models by semidefie programmg. 2.3 (A remark on he) Gaussian HJM Swapion price We can also express he price of he Swapion as haf a Bond Pu: Ps() =B(, T )E Q T 1 B(, T N+1 ) kδ N i=i T B(, Ti) + In he Gaussian H.J.M. model (see El Karoui & Lacose (1992), Musiela & Rukowski (1997) or Duffie & Kan (1996)), his expression defes he price of a Swapion as haf a Pun a baskef lognormal zero-coupon prices. A. d Aspremon Fronières en Fance Paris, Friday May

10 10 Calibraion of BGM models by semidefie programmg. 2.4 Baskepion pricg We have seen ha we can reduce he problem of pricg a Swapion o haf pricg a classic Black & Scholes (1973) baskepion. In generic erms, he problem becomes haf compug: C = E [ (S ω T k)+] wih S ω T = n i=1 ωik i s and dki s = Ki s σi s dw s i=1,...,n Rn n de- where W is a n-dimensional Q T -BM and σs = ( σ s i scribes he volailiy marix. ) A. d Aspremon Fronières en Fance Paris, Friday May

11 11 Calibraion of BGM models by semidefie programmg. We can wrie he dynamics of he baske as: ds ω u S ω u = ( ni=1 ωi,uσ i u ) dwu d ωi,s ωi,s = ( nj=1 ωj,s ( σ i s σ j s )) (dws + n j=1 ωj,sσ j sds ) where we have noed: ωi,s = ω ik i s ni=1 ωik i s We noice ha 0 ωi,s 1 wih n j=1 ωi,s = 1. We also se: σ i s = σ i s σω s wih σ ω s = n ωi,σ s j j=1 noe ha σ ω s = n j=1 ωi,σ j s is F measurable. A. d Aspremon Fronières en Fance Paris, Friday May

12 12 Calibraion of BGM models by semidefie programmg. We can develop hese dynamics around small values of σ s i and n j=1 ωj,s σ s j paricular. For some ε>0, we wrie: ds s ω,ε = S s ω,ε ( σ s ω + ε n j=1 ωj,s σ s j ) dws d ω i,s ε = ω i,s ε ( σ s i ε n j=1 ω j,s σj ε s )(dws + σ s ω ds + ε n j=1 ωj,s σ sds j ) As Fournie, Lebuchoux & Touzi (1997) and Lebuchoux & Musiela (1999) we compue: [ ( C ε = E S ω,ε T k ) ] + (S ω, ω) and approximae i around ε =0by: C ε = C 0 + C (1) ε + o(ε) Boh C 0 and C (1) (as well as C (2) ) can be compued explicily. A. d Aspremon Fronières en Fance Paris, Friday May

13 13 Calibraion of BGM models by semidefie programmg. In fac, C 0 is given by he BS formula: C 0 = BS(T,S ω,v T )=S ω N(h(V T )) κn ( h(v T ) V 1/2 ) T wih h (V T )= and we ge C (1) as: C (1) = S ω n ( ln ( S ω κ T ) V 1/2 T n j= V T ωj, ) σ j s,σ ω s V 1/2 T and V T = exp ln Sω K + s σ j u,σ ω u du V T V 1/2 T T σ ω s 2 ds ( s 2 σ j ) u,σω u du ds A. d Aspremon Fronières en Fance Paris, Friday May

14 14 Calibraion of BGM models by semidefie programmg. 2.5 The BGM Swapion pricg formula We can wrie he order zero price approximaion for Swapions: ( Swapion = Level(, T, T N ) swap(, T, T N )N(h) κn(h V 1/2 ) T ) wih where V T = T N i=1 h = ( ( ) swap(,t,tn ) ln κ ˆωi()γ(s, Ti s) 2 V 1/2 T V T ) K(, Ti) ds and ˆωi() = ωi() swap(, T, T N ) and dk(s, Ti) =γ(s, Ti s)k(s, Ti)dW Q T i+1 s. A. d Aspremon Fronières en Fance Paris, Friday May

15 15 Calibraion of BGM models by semidefie programmg. 2.6 BGM approximaion precision We plo he difference beween wo disc ses of Swapion prices he Libor Marke Model. One is obaed by Mone-Carlo simulaion usg enough seps o make he 95% confidence marg of error always less han 1bp. The second sef prices is compued usg he order zero approximaion. The plos are based on he prices obaed by calibrag he model o EURO Swapion prices on November We have used all Cap volailiies and he followg Swapions: 2 o 5, 5 o 5, 5 o 2, 10 o 5, 7 o 5, 10 o 2, 10 o 7, 2 o 2, 1 o 9. A. d Aspremon Fronières en Fance Paris, Friday May

16 16 Calibraion of BGM models by semidefie programmg. Absolue pricg error ( basis pos) 5 4 ) s n i o p s i s a b ( r o r E i n o i n o i n o i n o i n o 1 0 Swapion (Mauriy, Underlyg) Figure 1: Absolue error ( bp) for various ATM Swapions. A. d Aspremon Fronières en Fance Paris, Friday May

17 17 Calibraion of BGM models by semidefie programmg. Error he 10 o 2 Swapion price vs moneyness ) s n i o p s i s a b ( r o r E Moneyness ( Dela) Figure 2: Absolue error ( bp) on he 10 o 2. A. d Aspremon Fronières en Fance Paris, Friday May

18 18 Calibraion of BGM models by semidefie programmg. Error he 10 o 7 Swapion price vs moneyness 4 3 ) s n i o p s i s a b ( r o r E Moneyness ( Dela) Figure 3: Absolue error ( bp) on he 10 o 7. A. d Aspremon Fronières en Fance Paris, Friday May

19 19 Calibraion of BGM models by semidefie programmg. 3 Calibraion We have approximaed he Swapion (Tm,Tu+m) price by: P = Level(, Tm, Tu+m)BS(T, swap(, Tm, Tu+m), V(Tm, Tu+m)) where BS is he Black (1976) formula wih V (Tm,Tu+m) = Tm u i=m ˆωi()γ(s, Ti s) Suppose ha we need o impose a sequence of M marke pricg consras. We express hese consras erms of he marke variance pus σ k 2 : 2 ds V (Tm k,tu k +m k )=σ 2 k T m k for k =1,..., M A. d Aspremon Fronières en Fance Paris, Friday May

20 20 Calibraion of BGM models by semidefie programmg. We can rewrie he cumulaive variance: = = Tm Tm Tm u i=m u u i=m j=m ˆωi()γ(s, Ti s) Tr(ΩXs) ds 2 ds ˆωi()ˆωj() γ(s, Ti s),γ(s, Tj s) ds where Tr is he race, Xs is he Forward rae covariance marix, wih (Xs) i,j = γ(s, Ti s),γ(s, Tj s) and and (ˆω()ˆω T () ) is a rank one marix wih (ˆω()ˆω T () ) i,j =ˆω i()ˆωj(). A. d Aspremon Fronières en Fance Paris, Friday May

21 21 Calibraion of BGM models by semidefie programmg. This means ha he calibraion consras are lear Gs and can be wrien: Tm k Tr(Ω k Xs) ds = σ k 2 T m k for k =1,..., M Suppose, for example, ha he volailiy of he slidg mauriy Libors is saionary and discreized yearly, wih γ(s, Ti s) =γ( Ti s ), we can rewrie he pricg consras: Tr(Ω k X)=σ 2 k T m k for k =1,..., M where Ω k = T m k i=1 Ω k,i, wih Ω k,i a marix equal o zero everywhere excep for he submarix Ω k sarg a posiion (i, i). A. d Aspremon Fronières en Fance Paris, Friday May

22 22 Calibraion of BGM models by semidefie programmg. 3.1 Semidefie programmg The calibraion problem can fally be saed as: fd X s.. Tr(Ω k X)=σ k 2 T m k for k =1,..., M X 0 where X 0 sands for X semidefie posiive. If we choose an objecive marix Ω0, his becomes a semidefie program: m Tr(Ω0X) s.. Tr(Ω k X)=σ k 2 T m k for k =1,..., M X 0 which can be solved very efficienly (see Neserov & Nemirovskii (1994), Vandenberghe & Boyd (1996) for he heory and Surm (1999) for a MAT- LAB code). A. d Aspremon Fronières en Fance Paris, Friday May

23 23 Calibraion of BGM models by semidefie programmg Z X Figure 4: The semidefie cone dim 3: {m(eig[x,y;y,z])=0} A. d Aspremon Fronières en Fance Paris, Friday May

24 24 Calibraion of BGM models by semidefie programmg. 1 y x Figure 5: A ypical SDP feasible se dimension 3. A. d Aspremon Fronières en Fance Paris, Friday May

25 25 Calibraion of BGM models by semidefie programmg. 3.2 Defie advanages The calibraion program has a unique soluion compued polynomial ime, wih a cerificae of opimaliy or feasibiliy. The dual soluion provides he local sensiiviy (see Todd & ildirim (1999)) o all marke price movemens (no more bump and recalibrae ). Bid-Ask spread daa, smoohness or oher prices can be cluded he pus and objecive. As Con (2001), we can use Tikhonov regularizaion o sabilize he soluion (hence reduce hedgg ransacion coss). A. d Aspremon Fronières en Fance Paris, Friday May

26 26 Calibraion of BGM models by semidefie programmg. 3.3 Example: Swapion price bounds We can use a Swapion marix as he objecive and compue is maximum or mimum price given a sef oher Caple and Swapion prices: m/max Tr(Ω0X) s.. Tr(Ω k X)=σ k 2 T m k for k =1,..., M X 0 In he nex figure, we look a he evoluion of hese price bounds on he 5 o 3 Swapion as more and more Swapions are added o he calibraion se (which cludes all Caple prices). We use he same saionary slidg dynamics, wih dk(s, Ti) =γ(s, Ti s)k(s, Ti)dW Q T i+1 s where γ(s, Ti s) =γ( Ti s ). A. d Aspremon Fronières en Fance Paris, Friday May

27 27 Calibraion of BGM models by semidefie programmg. Bounds convergence on he 5 o % 13.5% 13.0% d e d 12.5% a n io p a 12.0% w S Upper bound Marke Lower bound 11.5% 11.0% 10.5% 2 o 5 5 o 5 5 o 2 10 o 5 7 o 5 10 o 2 10 o 7 2 o 2 Volailiy Figure 6: Upper and lower price bounds convergence as more Swapions are cluded he calibraion se. A. d Aspremon Fronières en Fance Paris, Friday May

28 28 Calibraion of BGM models by semidefie programmg. Sydney Opera House Effec 20% 18% 16% 14% l. o v. g l o 12% 10% 8% 6% o o o o o o Swapion Figure 7: Upper and lower bounds for various Swapion (EUR, 11/6/2000) A. d Aspremon Fronières en Fance Paris, Friday May Sup mk Inf

29 29 Calibraion of BGM models by semidefie programmg. 3.4 Smooh calibraion example We calibrae he model o EURO Swapion prices on November We use all Caple volailiies and he followg Swapions: 2 o 5, 5 o 5, 5 o 2, 10 o 5, 7 o 5, 10 o 2, 10 o 7, 2 o 2, 1 o 9 (daa couresy of BNP Paribas, London). We add a smoohness consra (mimum surface). A. d Aspremon Fronières en Fance Paris, Friday May

30 30 Calibraion of BGM models by semidefie programmg Figure 8: Forward raes covariance marix A. d Aspremon Fronières en Fance Paris, Friday May

31 31 Calibraion of BGM models by semidefie programmg Figure 9: Eigenvalues of he smooh soluion (semilog). A. d Aspremon Fronières en Fance Paris, Friday May

32 32 Calibraion of BGM models by semidefie programmg. 3.5 Low rank soluion There is no way o efficienly guaranee ha he soluion will be of given rank. Bu here are some excellen heurisical mehods. For example, as Boyd, Fazel & Hdi (2000), we can use anoher semidefie posiive marix he objecive o ge a low rank soluion. A. d Aspremon Fronières en Fance Paris, Friday May

33 33 Calibraion of BGM models by semidefie programmg Figure 10: Low rank soluion A. d Aspremon Fronières en Fance Paris, Friday May

34 34 Calibraion of BGM models by semidefie programmg Figure 11: Eigenvalues of he low rank soluion (semilog). A. d Aspremon Fronières en Fance Paris, Friday May

35 35 Calibraion of BGM models by semidefie programmg. 4 Conclusion We oba a fas, reliable calibraion mehod for he BGM model. The improvemen he soluion s sabiliy should reduce unnecessary hedgg coss. The fal rade-off he calibraion program becomes low rank vs. sabiliy. c A. d Aspremon Fronières en Fance Paris, Friday May

36 36 Calibraion of BGM models by semidefie programmg. References Black, F. (1976), The pricg of commodiy conracs., Journal of Fancial Economics 3, Black, F. & Scholes, M. (1973), The pricg of opions and corporae liabiliies, Journal of Poliical Economy 81, Boyd, S. P., Fazel, M. & Hdi, H. (2000), A rank mimizaion heurisic wih applicaion o mimum order sysem approximaion., Workg paper. American Conrol Conference, Sepember Brace, A., Dun, T. & Baron, G. (1999), Towards a cenral eres rae model, Workg Paper. FMMA. c A. d Aspremon Fronières en Fance Paris, Friday May

37 37 Calibraion of BGM models by semidefie programmg. Brace, A., Gaarek, D. & Musiela, M. (1997), The marke model of eres rae dynamics, Mahemaical Fance 7(2), Brace, A. & Womersley, R. S. (2000), Exac fi o he swapion volailiy marix usg semidefie programmg, Workg paper, ICBI Global Derivaives Conference. Con, R. (2001), Inverse problems fancial modelg: heoreical and numerical aspecs of model calibraion., Lecure Noes, Prceon Universiy, Deparmenf Operaions Research and Fancial Engeerg. Duffie, D. & Kan, R. (1996), A yield facor model of eres raes, Mahemaical Fance 6(4). c A. d Aspremon Fronières en Fance Paris, Friday May

38 38 Calibraion of BGM models by semidefie programmg. El Karoui, N. & Lacose, V. (1992), Mulifacor analysis of he erm srucure of eres raes, Proceedgs, AFFI. Fournie, E., Lebuchoux, J. & Touzi, N. (1997), Small noise expansion and imporance samplg, Asympoic Analysis 14, Hamy, O. (1999), Saisics of he weighs., F.I.R.S.T. workg paper, Paribas Capial Markes, London. Heah, D., Jarrow, R. & Moron, A. (1992), Bond pricg and he erm srucure of eres raes: A new mehodology, Economerica 61(1), c A. d Aspremon Fronières en Fance Paris, Friday May

39 39 Calibraion of BGM models by semidefie programmg. Jamshidian, F. (1997), Libor and swap marke models and measures, Fance and Sochasics 1(4), Lebuchoux, J. & Musiela, M. (1999), Marke models and smile effecs caps and swapions volailiies., Workg paper, Paribas Capial Markes.. Musiela, M. & Rukowski, M. (1997), Margale mehods fancial modellg, Applicaions of mahemaics 36, Sprger, Berl ; New ork. Marek Musiela, Marek Rukowski. Includes bibliographical references (p. [471]-506) and dex. Neserov, I. E. & Nemirovskii, A. S. (1994), Inerior-po polynomial algorihms convex programmg, Sociey for Indusrial and Applied Mahemaics, Philadelphia. c A. d Aspremon Fronières en Fance Paris, Friday May

40 40 Calibraion of BGM models by semidefie programmg. Rebonao, R. (1998), Ineres-Rae Opions Models, Fancial Engeerg, Wiley. Rebonao, R. (1999), On he simulaneous calibraion of muli-facor lognormal eres rae models o black volailiies and o he correlaion marix., Workg paper. Sgleon, K. J. & Umansev, L. (2001), Pricg coupon-bond opions and swapions affe erm srucure models, Workg paper, Sanford Universiy Graduae School of Busess.. Surm, J. F. (1999), Usg sedumi 1.0x, a malab oolbox for opimizaion over symmeric cones., Workg paper, Deparmenf Quaniaive Economics, Maasrich Universiy, The Neherlands.. c A. d Aspremon Fronières en Fance Paris, Friday May

41 41 Calibraion of BGM models by semidefie programmg. Todd, M. & ildirim, E. A. (1999), Sensiiviy analysis lear programmg and semidefie programmg usg erior-pos mehods., Workg paper, School of Operaion Research and Indusrial Engeerg, Cornell Universiy.. Vandenberghe, L. & Boyd, S. (1996), Semidefie programmg, SIAM Review 38, c A. d Aspremon Fronières en Fance Paris, Friday May