# A numerical method to price European derivatives based on the one factor LIBOR Market Model of interest rates

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A numerical method to price European derivatives based on the one factor LIBOR Market Model of interest rates
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Pricing European derivatives based on the one factorLibor Market Model of interest rates Luca Vincenzo Ballestra Dipartimento di Scienze Sociali “D. Serrani”,Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,Ph. N. +39-071-2207251, FAX N. +39-071-2207150, E-mail: l.v.ballestra@univpm.it Graziella Pacelli Dipartimento di Scienze Sociali “D. Serrani”,Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,Ph. N. +39-071-2207050, FAX N. +39-071-2207150, E-mail: g.pacelli@univpm.it Francesco Zirilli Dipartimento di Matematica “G. Castelnuovo”,Universit`a di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy,Ph. N. +39-06-49913282, FAX N. +39-06-44701007, E-mail: f.zirilli@caspur.it The LIBOR Market Model (LMM) is a stochastic model that describes the dynamics of forward LIBOR interest rates. It has been introduced and developed by Brace, Gatarek andMusiela [3], Miltersen, Sandmann, and Sondermann [8], and Jamshidian [6].Let  t  denote time, and let  T  0  be the current time value. The so-called tenor structure isdeﬁned as a ﬁnite set of time values  T  i , i  = 1 , 2 ,... ,n  + 1 , such that  T  0  < T  1  < T  2  < ... <T  n +1 . For  i  = 1 , 2 ,... ,n  + 1 , let us consider a bond which matures at time  T  i , whose price attime  t  is denoted by  P  i ( t ) . Moreover we deﬁne  δ  i  =  T  i +1 − T  i ,  i  = 1 , 2 ,... ,n .For  i  = 1 , 2 ,... ,n  the forward LIBOR rate at time  t  denoted with  L i ( t )  is the interest rateﬁxed at time  t  for the period  [ T  i ,T  i +1 ] , and is given by the following function of the prices of the bonds: L i ( t ) = 1 δ  i   P  i ( t ) P  i +1 ( t )  − 1  , T  0  < t ≤ T  i , i  = 1 , 2 ,... ,n.  (1)According to the LMM, the forward LIBOR rates are described as stochastic processessatisfying the following stochastic differential equations: dL i ( t ) L i ( t ) = − n   j = i +1 δ   j ν  i ( t ) · ν   j ( t ) L  j ( t )1 +  δ   j L  j ( t )  dt  +  ν  i ( t ) · dW  n +1 ( t ) , T  0  < t ≤ T  1 , i  = 1 , 2 ... ,n,  (2) L i ( T  0 ) =  L i, 0 , i  = 1 , 2 ... ,n,  (3)1  where  W  n +1 ( t )  is a  p − dimensional standard Brownian motion under the so-called terminalmeasure (see [6]),  dW  n +1 ( t )  is its stochastic differential,  ν  i  is a  p − dimensional vector valuedfunction of time which gives the volatility of the forward LIBOR rate  L i ,  ν  i  ·  W  n +1 is thescalar product of   ν  i  and  W  n +1 ,  L i, 0  is a real number that represents the value of the forwardLIBOR rate  L i  at time  T  0 ,  i  = 1 , 2 , ... ,n . Models analogous to (2) hold when  T  i  < t ≤ T  i +1 , i  = 1 , 2 ,... ,n − 1 . These models can be studied with the methods suggested here and areomitted for simplicity. In this work we assume  p  = 1 , that is we consider the one-factor LMM.The LMM has been widely used to price interest rate derivatives, mostly European andBermudanswaptions. Inparticular, inordertopriceEuropeanswaptions, approximateformulaehave been developed, for instance, in [3] and [5]. Bermudan swaptions have been priced in theLMM model using ad-hoc Monte Carlo simulations (see, for instance, [1], [7], [9]).In this work we propose a new approach to evaluate European derivatives based on the onefactor LMM model, that makes use of the transition probability density function associated tothe LMM model. In particular we have developed a closed-form approximation of the transi-tion probability density function of the forward LIBOR rates  L i ( t ) , given  L i ( T  0 ) ,  T  0  < t ≤ T  1 , i  = 1 , 2 ,... ,n . This formula is derived using a suitable power series expansion of the solu-tion of the Fokker-Planck equation associated to the stochastic differential equations (2). Theapproximation of the transition probability density function obtained is employed to price Eu-ropean interest rate derivatives using the method of discounted expectations. Due to the factthat only one factor has been considered in the LMM, the integral that gives the derivative priceis a  1 − dimensional integral, that can be easily evaluated using a numerical quadrature rule.Let us describe the technique used to approximate the transition probability density functionassociated to the LMM model (2)-(3). We consider the change of variables: Z  i ( t ) =  log  L i ( t ) L i, 0   + 12    tT  0 ν  2 i  ( s ) ds, T  0  < t ≤ T  1 , i  = 1 , 2 ,... ,n.  (4)The set of stochastic differential equations and initial conditions (2)-(3) can be rewritten as: dZ  i ( t ) = − n   j = i +1 δ   j ν  i ( t ) ν   j ( t ) L  j, 0 e Z  j ( t ) − 12    tT  0 ν  2 j ( s ) ds 1 +  δ   j L  j, 0 e Z  j ( t ) − 12    tT  0 ν  2 j ( s ) ds dt  +  ν  i ( t ) dW  n +1 ( t ) ,T  0  < t ≤ T  1 , i  = 1 , 2 ,... ,n,  (5) Z  i ( T  0 ) = 0 , i  = 1 , 2 ,... ,n,  (6)Note that since we have chosen  p  = 1  the scalar term  ν  i ( t ) dW  n +1 ( t )  appearing in (5) is thenatural counterpart of the term  ν  i ( t ) · dW  n +1 ( t )  appearing in (2).Let  z   = ( z  1 ,z  2 ,... ,z  n ) ,  τ   =  t − T  0 , and let  p ( T  0 , 0 ,t,z  ) =  p ( T  0 ,τ,z  ) ,  0  < τ < T  1 − T  0 , z   ∈  R n , denote the transition probability density function of having  Z  i  =  z  i  at time  t ,  i  =1 , 2 ,... ,n , given  Z  i  = 0  at time  T  0 ,  i  = 1 , 2 ,... ,n .The transition probability density function  p ( τ,z  )  satisﬁes the Fokker-Planck equation: ∂p∂τ   = n  i =1 n   j = i +1 δ   j  ν  i ( τ  )  ν   j ( τ  ) L  j, 0 e z j − 12    τ T  0  ν  2 j ( s ) ds 1 +  δ   j L  j, 0 e z j − 12    τ T  0  ν  2 j ( s ) ds ∂p∂z  i + 12 n  i =1 n  k =1  ν  i ( τ  )  ν  k ( τ  )  ∂  2  p∂z  i ∂z  k , z   ∈ R n ,  0  < τ   ≤ T  1 − T  0 ,  (7)2  where   ν  i ( τ  ) =  ν  i ( τ   +  T  0 ) ,  i  = 1 , 2 ,... ,n , with initial condition:  p (0 ,z  ) =  δ  ( z  1 ) δ  ( z  2 ) ...δ  ( z  n ) , z   ∈ R n ,  (8)where  δ  ( · )  is the Dirac delta function.Let  ν   j  =  1 T  1 − T  0   T  1 T  0 ν   j ( t ) dt ,  j  = 1 , 2 ,... ,n , we consider the problem (7)-(8) when we sub-stitute to   ν   j ( τ  )  the function  ν   j + ε  j ( ν   j ( τ  ) − ν   j ), where  ε  j  is a real parameter,  j  = 1 , 2 ,... ,n . Wenote that when  ε  j  = 0 ,  j  = 1 , 2 ,... ,n  we have an approximate problem with time independentvolatilities and that when  ε  j  = 1 ,  j  = 1 , 2 ,... ,n  we have the srcinal problem (7)-(8).We approximate the transition probability density function  p ( τ,z  )  with its perturbation se-ries in powers of the variables  δ  i  and  ε  j ,  i  = 2 , 3 ,... ,n ,  j  = 1 , 2 ,... ,n , with base point in δ  i  = 0 ,  ε  j  = 0 ,  i  = 2 , 3 ,... ,n ,  j  = 1 , 2 ,... ,n , and this perturbation series is truncated atﬁrst-order. That is, we determine the coefﬁcients of the following asymptotic expansion:  p ( T  0 ,τ,z  ) =  p 0 ( T  0 ,τ,z  ) + n  l =2 δ  l  p δ, 1 l  ( T  0 ,τ,z  ) + n  l =1 ε l  p ε, 1 l  ( T  0 ,τ,z  )+ O ( n  l =2 δ  2 l  + n  l =1 ε 2 l ) , δ  i ,ε  j  → 0 , i  = 2 , 3 ,... ,n, j  = 1 , 2 ,... ,n, z   ∈ R n .  (9)Note that  p ( T  0 ,τ,z  )  is independent of   δ  1 . Substituting the expansion (9) in (7), (8), equatingthe terms of the same order, we get, for the zeroth and for the ﬁrst-order coefﬁcients of theexpansion (9), initial value problems for suitable partial diffrential equations. These initialvalue problems have been solved analytically.The resulting approximation of   p ( T  0 ,τ,z  )  has been used to price interest rate derivativesof European type using the method of discounted expectations. In particular we have usedthe transition probability density function to price European payer swaptions and interest ratespread options. For the sake of brevity, only the case of european swaptions will be presentedhere.We recall that a European swaption is a contract that gives the holder the right to enter, at aﬁxed time, let say at time  T  1 , in a payer swap. According to a payer swap the holder pays, attime  T  i +1 , the amount due on the basis of a ﬁxed interest rate  E   over the periods  [ T  i ,T  i +1 ] , andreceives the ﬂoating interest rate  L i ( T  i ) ,  i  = 1 , 2 ,... ,n .At time  T  1  a payer swaption has the following payoff function: Payoff  sw ( L 1 ( T  1 ) ,L 2 ( T  1 ) ,... ,L n ( T  1 )) =  Max ( n  i =1 δ   j ( L i ( T  1 ) − E  ) P  i +1 ( T  1 )) , 0) .  (10)Note that the functions  P  i +1 ( T  1 ) ,  i  = 1 , 2 ,... ,n , can be expressed in terms of the interestrates  L i ( T  1 ) ,  i  = 1 , 2 ,... ,n , using the relations given in (1) and the fact that  P  i +1 ( T  i +1 ) = 1 , i  = 1 , 2 ,... ,n .The value of the swaption  V  sw ( T  0 )  at time  T  0  is evaluated using the discounted expectationformula, that is: V  sw ( T  0 ) =  P  n +1 ( T  0 )    + ∞−∞ Payoff  sw ( L 1 ( T  1 ) ,L 2 ( T  1 ) ,... ,L n ( T  1 )) P  n +1 ( T  1 )  p ( T  0 ,T  1 − T  0 ,z  ) dz  1 dz  2 ,... ,dz  n ,  (11)3  where, for  i  = 1 , 2 ,... ,n , the value assumed by the LIBOR interest rate random variable L i ( T  1 )  that we continue to denote with  L i ( T  1 )  can be expressed in terms of the variables  z  i inverting the relations (4), that is: L i ( T  1 ) =  L i, 0 e z i − 12    T  1 T  0 ν  2 i  ( s ) ds ,i  = 1 , 2 ,... ,n,  (12)and  P  n +1 ( T  1 )  can be expressed in terms of the forward LIBOR interest rates  L i ( T  1 )  and henceof the variables  z  i  using relations (1), (12), and the fact that  P  i +1 ( T  i +1 ) = 1 ,  i  = 1 , 2 ,... ,n .In (11)  p ( T  0 ,T  1 − T  0 ,z  )  is approximated using (9). In this approximation the  n -dimensionalintegral (11) can be reduced to a one-dimensional integral, and therefore can be easily evaluatedusing a numerical qaudrature rule.We have applied the method described above to compute the price at time  T  0  = 0  of aEuropean swaption, where  n  = 10 ,  T  1  = 0 . 5 year ,  δ  i  = 0 . 5 year ,  E   = 0 . 05 year − 1 ,  L i, 0  =0 . 05 year − 1 ,  i  = 1 , 2 ,... , 10 . The volatility functions  ν  i ( t ) ,  i  = 1 , 2 ,... ,n  have been chosenas done in [4]. We found that the price of the swaption contract can be computed, using formula(11), where the probability density function  p ( T  0 ,T  1 − T  0 ,z  )  is approximated by the asymptoticexpansion (9), with a relative error equal to  2 . 5 e − 4 . The swaption price is obtained in lessthan  0 . 3  seconds using a computer with a AMD Athlon Processor 1000 MHZ 256 MB Ram.This time is about  500  times smaller than the time necessary to obtain the option price withapproximatively the same accuracy using a Monte Carlo simulation. References [1] L. Andersen, A Simple Approach to the Pricing of Bermudan Swaptions in the Multi-Factor LIBOR Market Model,  Journal of Computational Finance  3  (1999) 5-32.[2] L. V. Ballestra, G. Pacelli, F. Zirilli, A Numerical Method to Price European DerivativesBased on the One Factor LIBOR Market Model of Interest Rates, in preparation.[3] A. Brace, M. Gatarek, M. Musiela, The Market Model of Interest Rate Dynamics,  Mathe-matical Finance  7  (1997) 127-155.[4] D. Brigo, F. Mercurio, M. Morini, The LIBOR Model Dynamics: Approximations, Cali-bration and Diagnostics,  European Journal of Operational Research  163  (2005) 30-51.[5] J. Hull, A. White, Forward Rate Volatilities, Swap Rates Volatilities and the Implementa-tion of the LIBOR Market Model,  Journal of Fixed Income  10  (2000) 46-62.[6] F. Jamshidian, LIBOR and Swap Market Models and Measures,  Finance and Stochastics 1  (1997) 293-330.[7] M. S. Jensen, M. Svenstrup, Efﬁcient Control Variates and Strategies for Bermudan Swap-tions in a LIBOR Market Model,  Journal of Derivatives  12  (2005) 20-33.[8] K. Miltersen, K. Sandmann, D. Sondermann, Closed Form Solutions for Term StructureDerivatives with Lognormal Interest Rates,  Journal of Finance  52  (1997) 409-430.[9] R. Pietersz, A. Pelsser, M. van Regenmortel, Fast Drift-Approximated Pricing in the BGMmodel,  Journal of Computational Finance  8  (2004) 93-124.4

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