Risk Management Methods for the Libor Market Model Using. Semidefinite Programming. Monday, June 24, A. d Aspremont

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1 Risk Management Methods for the Libor Market Model Usg Semidefite Programmg. A. d Aspremont Monday, June 24, 2002 Research done under the direction of Nicole El Karoui. Thanks to F.I.R.S.T. Swaps, BNP Paribas (London). Ecole Polytechnique: alexandre.daspremont@polytechnique.org

2 2 Risk Management Methods for the LMM Usg Semidefite Programmg. 1.1 Interest rate model calibration All Heath, Jarrow & Morton (1992) based models are fully parametrized by the curve today and a covariance matrix. The natural variable the calibration problem is a covariance matrix, i.e. a positive semidefite matrix. Classic calibration methods are heavily parametrized and only describe a small, often non-convex subsef the sef semidefite matrices. When usg these techniques, sensitivity analysis has to be done by bumpg the data and recalibratg.

3 3 Risk Management Methods for the LMM Usg Semidefite Programmg. 1.2 Results on the LMM model calibration We can express the swap rate as a baskef Forwards with very stable coefficients. European Caplets and Swaptions can be priced usg the Black (1976) market formula with a variance that is lear the coefficients of the Forward rates covariance matrix.. The calibration problem is a semidefite program and the dual solution naturally provides the local sensitivity to all market movements. The dual solution also provides the coefficients of an upper (lower) hedgg portfolio the sense of El Karoui & Quenez (1991) and Avellaneda & Paras (1996).

4 4 Risk Management Methods for the LMM Usg Semidefite Programmg. 1.3 Related literature Works by Nesterov & Nemirovskii (1994) and Vandenberghe & Boyd (1996) on semidefite programmg Brace, Gatarek & Musiela (1997) and Musiela & Rutkowski (1997) on the Libor market model. Rebonato (1998), Brace, Dun & Barton (1999) and Sgleton & Umantsev (2001) on Swaps as baskets of Forwards. Rebonato (1999) on a calibration method parametrized by factors. Parallel work by Brace & Womersley (2000) on the calibration of the BGM by semidefite programmg and the evaluation of the Bermudan Swaption.

5 5 Risk Management Methods for the LMM Usg Semidefite Programmg. 2 Swaption pricg 2.1 The Swap rate We write the swap rate as a baskef Forwards: swap(t, T0, Tn) = n i=0 ωi(t)k(t, Ti) where K(t, Ti) are the Forward Rates with maturities Ti,i =1,..., n and the weights ωi(t) are given by ωi(t) = float coverage(t i,t float float i+1 )B(t, T i+1 ) Level(t, T fixed 0,T n fixed )

6 6 Risk Management Methods for the LMM Usg Semidefite Programmg. 2.2 BGM Swaption price In practice, the weights ωi(t) are very stable (see Rebonato (1998)) and followg Jamshidian (1997), we can write the price of the Swaption with strike k as a thaf a Call on a Swap rate: Ps(t) =Level(t, T, T N )E Q LV L t n i=0 ωi(t )K(T,Ti) k + where Q LV L is the swap forward martgale probability measure.

7 7 Risk Management Methods for the LMM Usg Semidefite Programmg. In fact, as detailed Huynh (1994) or Brace et al. (1999) the Swaption price can be very efficiently approximated by the Black (1976) formula: Swaption = Level(t, T, T N ) ( swap(t, T, T N )N(h) κn(h V 1/2 T ) ) with h = ( ( ) swap(t,t,tn ) ln κ V 1/2 T where the cumulative variance is computed by: V T = T t N i=1 ˆωi(t)γ(s, Ti s) V T ) K(t, Ti) ds and ˆωi(t) = ωi(t) swap(t, T, T N ) with dk(s, Ti) =γ(s, Ti s)k(s, Ti)dW Q T i+1 s.

8 8 Risk Management Methods for the LMM Usg Semidefite Programmg. 2.3 BGM approximation precision We plot the difference between two distct sets of Swaption prices the Libor Market Model. One is obtaed by Monte-Carlo simulation usg enough steps to make the 95% confidence marg of error always less than 1bp. The second sef prices is computed usg the order zero approximation. The plots are based on the prices obtaed by calibratg the model to EURO Swaption prices on November We have used all Cap volatilities and the followg Swaptions: 2 to 5, 5 to 5, 5 to 2, 10 to 5, 7 to 5, 10 to 2, 10 to 7, 2 to 2, 1 to 9.

9 9 Risk Management Methods for the LMM Usg Semidefite Programmg. Absolute pricg error ( basis pots) 5 4 ) s t n i o p s i s a b ( r o r E i n t o i n t o i n t o i n t o i n t o 1 0 Swaption (Maturity, Underlyg) Figure 1: Absolute error ( bp) for various ATM Swaptions.

10 10 Risk Management Methods for the LMM Usg Semidefite Programmg. Error the 10 to 2 Swaption price vs moneyness ) s t n i o p s i s a b ( r o r E Moneyness ( Delta) Figure 2: Absolute error ( bp) on the 10 to 2.

11 11 Risk Management Methods for the LMM Usg Semidefite Programmg. Error the 10 to 7 Swaption price vs moneyness 4 3 ) s t n i o p s i s a b ( r o r E Moneyness ( Delta) Figure 3: Absolute error ( bp) on the 10 to 7.

12 12 Risk Management Methods for the LMM Usg Semidefite Programmg. 3 Calibration We have approximated the Swaption (Tm,Tu+m) price by: P = Level(t, Tm, Tu+m)BS(T, swap(t, Tm, Tu+m), V(Tm, Tu+m)) where BS is the Black (1976) formula with V (Tm,Tu+m) = Tm t u i=m ˆωi(t)γ(s, Ti s) Suppose that we need to impose a sequence of M market pricg constrats. We express these constrats terms of the market variance puts σ k 2 : 2 ds V (Tm k,tu k +m k )=σ 2 k T m k for k =1,..., M

13 13 Risk Management Methods for the LMM Usg Semidefite Programmg. We can rewrite the cumulative variance: = = Tm t Tm t Tm t u i=m u u i=m j=m ˆωi(t)γ(s, Ti s) Tr(ΩXs) ds 2 ds ˆωi(t)ˆωj(t) γ(s, Ti s),γ(s, Tj s) ds where Tr is the trace, Xs is the Forward rate covariance matrix, with (Xs) i,j = γ(s, Ti s),γ(s, Tj s) and and (ˆω(t)ˆω T (t) ) is a rank one matrix with (ˆω(t)ˆω T (t) ) i,j =ˆω i(t)ˆωj(t).

14 14 Risk Management Methods for the LMM Usg Semidefite Programmg. This means that the calibration constrats are lear Xs and can be written: Tm k Tr(Ω k Xs) ds = σ k 2 T m k for k =1,..., M t If we discretize time we can write the above constrats as: Tr(Ω k X)=σ 2 k T m k for k =1,..., M where Ω k is a block diagonal matrix.

15 15 Risk Management Methods for the LMM Usg Semidefite Programmg. 3.1 Semidefite programmg The calibration problem can fally be stated as: fd X s.t. Tr(Ω k X)=σ k 2 T m k for k =1,..., M X 0 where X 0 stands for X semidefite positive. If we choose an objective matrix Ω0, this becomes a semidefite program: m Tr(Ω0X) s.t. Tr(Ω k X)=σ k 2 T m k for k =1,..., M X 0 which can be solved very efficiently (see Nesterov & Nemirovskii (1994), Vandenberghe & Boyd (1996) for the theory and Sturm (1999) for a MAT- LAB code).

16 16 Risk Management Methods for the LMM Usg Semidefite Programmg Z X Figure 4: The semidefite cone dim 3: {m(eig[x,y;y,z])=0}

17 17 Risk Management Methods for the LMM Usg Semidefite Programmg. 1 y x Figure 5: A typical SDP feasible set dimension 3.

18 18 Risk Management Methods for the LMM Usg Semidefite Programmg. 3.2 Defite advantages The calibration program has a unique solution computed polynomial time, with a certificate of optimality or feasibility. Bid-Ask spread data, smoothness or other prices can be cluded the puts and objective. The algorithms provide both primal and dual solutions. The primal gives the calibrated Forward rate covariance matrix, while the dual provides local sensitivity results.

19 19 Risk Management Methods for the LMM Usg Semidefite Programmg. 3.3 Smooth calibration We calibrate the model to EURO Swaption prices on November We use all Caplet volatilities and the followg Swaptions: 2 to 5, 5 to 5, 5 to 2, 10 to 5, 7 to 5, 10 to 2, 10 to 7, 2 to 2, 1 to 9 (data courtesy of BNP Paribas, London). We add a smoothness constrat (mimum surface), this acts as a Tikhonov (1963) stabilization of the solution and reduces purely the number of numerical hedgg transactions.

20 20 Risk Management Methods for the LMM Usg Semidefite Programmg Figure 6: Forward rates covariance matrix

21 21 Risk Management Methods for the LMM Usg Semidefite Programmg. 3.4 The dual program When the origal program is given by: the dual becomes: max Tr(Ω0X) s.t. Tr(Ω k X)=σ k 2 T m k for k =1,..., M X 0 m M k=1 y k σ 2 k T m k s.t. Ω0 M k=1 y k Ω k most S.D.P. solvers (such as SEDUMI by Sturm (1999) for example) compute both solutions at the same time.

22 22 Risk Management Methods for the LMM Usg Semidefite Programmg Local sensitivity We can study the impacn the solution of a (small) change market conditions given by u k for k =1,..., M, the calibration program becomes: maximize Tr(CX) s.t. Tr(Ω k X)=σ k 2 T k + u k for k =1,..., M X 0 Usg Todd & ildirim (1999) we can compute the new calibrated matrix X + X with: [ ( X = E 1 FA AE 1 FA ) ] 1 u where E,F and A are lear operators computed from (X,y ) the primal and dual solutions to the origal calibration program (with u k =0).

23 23 Risk Management Methods for the LMM Usg Semidefite Programmg Super hedgg price As expected, we can terpret the dual solution to the calibration program terms of hedgg portfolio. As Avellaneda & Paras (1996) we suppose that the volatility is uncerta and we hedge by mixg a static portfolio of derivatives with a dynamic super-replication strategy. The price is obtaed by solvg the followg (formal) program: Price = M {Value of static hedge + Max (PV of residual liability)}

24 24 Risk Management Methods for the LMM Usg Semidefite Programmg Super hedgg portfolio Suppose we study an upper hedgg price on a particular Swaption (Ω0,T0). We can fd an approximate solution to the previous problem by solvg the followg problem: f λ M k=1 or its dual: λ k C k +sup X 0 BS(Tr(Ω0X)) M k=1 λ k BS (Tr(Ω k X)) maximize BS0(Tr(Ω0X)) s.t. BS k (Tr(Ω k X)) = C k for k =1,..., M X 0

25 25 Risk Management Methods for the LMM Usg Semidefite Programmg. We can write the KKT optimality conditions on this problem: hence if y i then Z = BS 0(Ω0X) v Ω0 + M BS k=1 λ k (Ω k X) k v Ω k XZ =0 BS k (Tr(Ω k X)) = Ci for k =1,..., M 0 X, Z solves the dual S.D.P: mimize Mk=1 y k σ k 2 T k s.t. 0 M k=1 y k Ω k C λ k = y i BS0 (Tr(Ω0X)) / v BS k (Tr(Ω k X)) / v will be the coefficients of a super replicatg portfolio the Swaptions (Ω k,t k ).

26 26 Risk Management Methods for the LMM Usg Semidefite Programmg. Sydney Opera House Effect 20% 18% 16% 14% l. o v. g l o 12% 10% 8% 6% to to to to to to Swaption Figure 7: Upper and lower bounds for various Swaption (EUR, 11/6/2000) Sup mkt Inf

27 27 Risk Management Methods for the LMM Usg Semidefite Programmg. 3.5 Low rank solution There is no way to efficiently guarantee that the solution will be of given rank. But there are some excellent heuristical methods. For example, as Boyd, Fazel & Hdi (2000), we can use another semidefite positive matrix the objective to get a low rank solution.

28 28 Risk Management Methods for the LMM Usg Semidefite Programmg Figure 8: Low rank solution

29 29 Risk Management Methods for the LMM Usg Semidefite Programmg Figure 9: Eigenvalues of the low rank solution (semilog).

30 30 Risk Management Methods for the LMM Usg Semidefite Programmg Figure 10: Eigenvalues of the smooth solution (semilog).

31 31 Risk Management Methods for the LMM Usg Semidefite Programmg. 4 Conclusion Semidefite programmg provides a fast, reliable calibration method for the LMM model. The improvement the solution s stability should reduce unnecessary hedgg costs. The dual solution provides all the essential local sensitivity results. The fal trade-off the calibration problem is low rank vs. stability. c

32 32 Risk Management Methods for the LMM Usg Semidefite Programmg. References Avellaneda, M. & Paras, A. (1996), Managg the volatility risk of portfolios of derivative securities: the lagrangian uncerta volatility model, Applied Mathematical Fance 3, Black, F. (1976), The pricg of commodity contracts., Journal of Fancial Economics 3, Boyd, S. P., Fazel, M. & Hdi, H. (2000), A rank mimization heuristic with application to mimum order system approximation., Workg paper. American Control Conference, September c

33 33 Risk Management Methods for the LMM Usg Semidefite Programmg. Brace, A., Dun, T. & Barton, G. (1999), Towards a central terest rate model, Workg Paper. FMMA. Brace, A., Gatarek, D. & Musiela, M. (1997), The market model of terest rate dynamics, Mathematical Fance 7(2), Brace, A. & Womersley, R. S. (2000), Exact fit to the swaption volatility matrix usg semidefite programmg, Workg paper, ICBI Global Derivatives Conference. El Karoui, N. & Quenez, M. (1991), Programmation dynamique et evaluation des actifs contgents en marches complets., Compte Rendu de l Academie des Sciences de Paris, Srie I 313, c

34 34 Risk Management Methods for the LMM Usg Semidefite Programmg. Heath, D., Jarrow, R. & Morton, A. (1992), Bond pricg and the term structure of terest rates: A new methodology, Econometrica 61(1), Huynh, C. B. (1994), Back to baskets, Risk 7(5). Jamshidian, F. (1997), Libor and swap market models and measures, Fance and Stochastics 1(4), Musiela, M. & Rutkowski, M. (1997), Martgale methods fancial modellg, Applications of mathematics 36, Sprger, Berl ; New ork. Marek Musiela, Marek Rutkowski. Includes bibliographical references (p. [471]-506) and dex. c

35 35 Risk Management Methods for the LMM Usg Semidefite Programmg. Nesterov, I. E. & Nemirovskii, A. S. (1994), Interior-pot polynomial algorithms convex programmg, Society for Industrial and Applied Mathematics, Philadelphia. Rebonato, R. (1998), Interest-Rate Options Models, Fancial Engeerg, Wiley. Rebonato, R. (1999), On the simultaneous calibration of multi-factor lognormal terest rate models to black volatilities and to the correlation matrix., Workg paper. Sgleton, K. J. & Umantsev, L. (2001), Pricg coupon-bond options and swaptions affe term structure models, Workg paper, Stanford University Graduate School of Busess.. c

36 36 Risk Management Methods for the LMM Usg Semidefite Programmg. Sturm, J. F. (1999), Usg sedumi 1.0x, a matlab toolbox for optimization over symmetric cones., Workg paper, Departmenf Quantitative Economics, Maastricht University, The Netherlands.. Tikhonov, A. N. (1963), Solution of correctly formulated problems and the regularization method, Soviet Math. Dokl. (4). Todd, M. & ildirim, E. A. (1999), Sensitivity analysis lear programmg and semidefite programmg usg terior-pots methods., Workg paper, School of Operation Research and Industrial Engeerg, Cornell University.. Vandenberghe, L. & Boyd, S. (1996), Semidefite programmg, SIAM Review 38, c