Cox-Ingersoll-Ross model

Chapter 31 Cox-Ingersoll-Ross model In the Hull& Whitemodel, rt is a Gaussianprocess. Since, for each t , rt is normallydistributed, there is a positiveprobabilitythat rt  0 . The Cox-Ingersoll-Rossmodel is the simplest one which avoids negative interest rates. We begin with a d -dimensional Brownian motion W 1 ;W 2 ;::: ;W d  .Let 0 and 0 be constants. For j =1;::: ;d ,let X j 0 2 IR be given so that X 2 1 0 + X 2 2 0 + :::+ X 2 d 0  0; and let X j be the solution to the stochastic differential equation dX j t=, 1 2 X j t dt + 1 2 dW j t: X j is called the Orstein-Uhlenbeck process. It always has a drift toward the origin. The solution to this stochastic differential equation is X j t=e , 1 2 t  X j 0 + 1 2  Z t 0 e 1 2 u dW j u  : This solution is a Gaussian process with mean function m j t=e , 1 2 t X j 0 and covariance function s; t= 1 4  2 e , 1 2 s+t Z s^t 0 e u du: Define rt 4 = X 2 1 t+X 2 2 t+:::+ X 2 d t: If d =1 ,wehave rt= X 2 1 t and for each t , IP frt  0g =1 , but (see Fig. 31.1) IP  There are infinitely many values of t0 for which rt= 0  =1 303 304 t r(t) = X (t) x x 1 2 ( X (t), X (t) ) 1 1 2 2 Figure 31.1: rt can be zero. If d  2 , (see Fig. 31.1) IP f There is at least one value of t0 for which rt= 0g =0: Let f x 1 ;x 2 ;::: ;x d =x 2 1 +x 2 2 +:::+ x 2 d .Then f x i =2x i ; f x i x j =  2 if i = j; 0 if i 6= j: Itˆo’s formula implies drt= d X i=1 f x i dX i + 1 2 d X i=1 f x i x i dX i dX i = d X i=1 2X i  , 1 2 X i dt + 1 2 dW i t  + d X i=1 1 4  2 dW i dW i = ,rt dt +  d X i=1 X i dW i + d 2 4 dt =  d 2 4 , rt ! dt +  q rt d X i=1 X i t p rt dW i t: Define W t= d X i=1 Z t 0 X i u p ru dW i u: CHAPTER 31. Cox-Ingersoll-Ross model 305 Then W is a martingale, dW = d X i=1 X i p r dW i ; dW dW = d X i=1 X 2 i r dt = dt; so W is a Brownian motion. We have drt=  d 2 4 , rt ! dt +  q rt dW t: The Cox-Ingersoll-Ross (CIR) process is given by drt= ,rt dt +  q rt dW t; We define d = 4  2  0: If d happens to be an integer, then we have the representation rt= d X i=1 X 2 i t; but we do not require d to be an integer. If d2 (i.e.,  1 2  2 ), then IP f There are infinitely many values of t0 for which rt=0g =1: This is not a good parameter choice. If d  2 (i.e.,   1 2  2 ), then IP f There is at least one value of t0 for which rt= 0g =0: With the CIR process, one can derive formulas under the assumption that d = 4  2 is a positive integer, and they are still correct even when d is not an integer. For example, here is the distribution of rt for fixed t0 .Let r0  0 be given. Take X 1 0 = 0;X 2 0 = 0; ::: ; X d,1 0 = 0;X d 0 = q r0: For i =1;2;::: ;d, 1 , X i t is normal with mean zero and variance t; t=  2 4 1 , e ,t : 306 X d t is normal with mean m d t=e , 1 2 t q r0 and variance t; t .Then rt= t; t d,1 X i=1  X i t p t; t ! 2 | z  Chi-square with d , 1= 4, 2  2 degrees of freedom + X 2 d t | z  Normal squaredand independent of the other term (0.1) Thus rt has a non-central chi-square distribution. 31.1 Equilibrium distribution of r t As t!1 , m d t!0 .Wehave rt= t; t d X i=1  X i t p t; t ! 2 : As t!1 ,wehave t; t=  2 4 , and so the limiting distribution of rt is  2 4 times a chi-square with d = 4  2 degrees of freedom. The chi-square density with 4  2 degrees of freedom is f y = 1 2 2= 2 ,  2  2  y 2, 2  2 e ,y=2 : We make the change of variable r =  2 4 y . The limiting density for rt is pr= 4  2 : 1 2 2= 2 ,  2  2   4  2 r  2, 2  2 e , 2  2 r =  2  2  2  2 1 ,  2  2  r 2, 2  2 e , 2  2 r : We computed the mean and variance of rt in Section 15.7. 31.2 Kolmogorov forward equation Consider a Markov process governed by the stochastic differential equation dX t=bXt dt + X t dW t: CHAPTER 31. Cox-Ingersoll-Ross model 307 - h 0 y Figure 31.2: The function hy  Because we are going to apply the following analysis to the case X t= rt , we assume that X t  0 for all t . We start at X 0 = x  0 at time 0. Then X t is random with density p0;t;x;y (in the y variable). Since 0 and x will not change during the following, we omit them and write pt; y  rather than p0;t;x;y .Wehave IEhXt = Z 1 0 hy pt; y  dy for any function h . The Kolmogorov forward equation(KFE) is a partial differentialequationin the “forward” variables t and y . We derive it below. Let hy  be a smooth function of y  0 which vanishes near y =0 and for all large values of y (see Fig. 31.2). Itˆo’s formula implies dhX t = h h 0 X tbX t + 1 2 h 00 X t 2 X t i dt + h 0 X tX t dW t; so hX t = hX 0 + Z t 0 h h 0 X sbX s + 1 2 h 00 X s 2 X s i ds + Z t 0 h 0 X sX s dW s; IEhXt = hX 0 + IE Z t 0 h h 0 X sbX s dt + 1 2 h 00 X s 2 X s i ds; 308 or equivalently, Z 1 0 hy pt; y  dy = hX 0 + Z t 0 Z 1 0 h 0 y by ps; y  dy ds + 1 2 Z t 0 Z 1 0 h 00 y  2 y ps; y  dy ds: Differentiate with respect to t to get Z 1 0 hy p t t; y  dy = Z 1 0 h 0 y by pt; y  dy + 1 2 Z 1 0 h 00 y  2 y pt; y  dy : Integration by parts yields Z 1 0 h 0 y by pt; y  dy = hy by pt; y      y=1 y=0 | z  =0 , Z 1 0 hy  @ @y bypt; y  dy ; Z 1 0 h 00 y  2 y pt; y  dy = h 0 y  2 y pt; y      y=1 y=0 | z  =0 , Z 1 0 h 0 y  @ @y   2 ypt; y   dy = ,hy  @ @y   2 ypt; y       y=1 y=0 | z  =0 + Z 1 0 hy  @ 2 @y 2   2 ypt; y   dy : Therefore, Z 1 0 hy p t t; y  dy = , Z 1 0 hy  @ @y bypt; y  dy + 1 2 Z 1 0 hy  @ 2 @y 2   2 ypt; y   dy ; or equivalently, Z 1 0 hy  " p t t; y + @ @y bypt; y  , 1 2 @ 2 @y 2   2 ypt; y    dy =0: This last equation holds for every function h of the form in Figure 31.2. It implies that p t t; y + @ @y by pt; y  , 1 2 @ 2 @y 2   2 ypt; y   =0: (KFE) If there were a place where (KFE) did not hold, then we could take hy   0 at that and nearby points, but take h to be zero elsewhere, and we would obtain Z 1 0 h " p t + @ @y bp , 1 2 @ 2 @y 2  2 p  dy 6=0: CHAPTER 31. Cox-Ingersoll-Ross model 309 If the process X t has an equilibrium density, it will be py  = lim t!1 pt; y : In order for this limit to exist, we must have 0 = lim t!1 p t t; y : Letting t!1 in (KFE), we obtain the equilibrium Kolmogorov forward equation @ @y bypy , 1 2 @ 2 @y 2   2 ypy  =0: When an equilibrium density exists, it is the unique solution to this equation satisfying py   0 8y  0; Z 1 0 py  dy =1: 31.3 Cox-Ingersoll-Ross equilibrium density We computed this to be pr= Cr 2, 2  2 e , 2  2 r ; where C =  2  2  2  2 1 ,  2  2  : We compute p 0 r= 2, 2  2 : pr r , 2  2 pr = 2  2 r  , 1 2  2 ,r  pr; p 00 r= , 2  2 r 2  , 1 2  2 ,r  pr+ 2  2 r ,pr+ 2  2 r  , 1 2  2 , r  p 0 r = 2  2 r  , 1 r  , 1 2  2 , r ,  + 2  2 r  , 1 2  2 , r 2  pr We want to verify the equilibrium Kolmogorov forward equation for the CIR process: @ @r  , rpr , 1 2 @ 2 @r 2  2 rpr = 0: (EKFE) 310 Now @ @r  , rpr = ,pr+, rp 0 r; @ 2 @r 2  2 rpr = @ @r  2 pr+  2 rp 0 r =2 2 p 0 r+ 2 rp 00 r: The LHS of (EKFE) becomes ,pr+,rp 0 r ,  2 p 0 r , 1 2  2 rp 00 r = pr  , +,r ,  2  2  2 r  , 1 2  2 , r + 1 r  , 1 2  2 , r+ , 2  2 r , 1 2  2 ,r 2  = pr   , 1 2  2 , r 2  2 r  , 1 2  2 , r , 1 2  2 2  2 r  , 1 2  2 , r + 1 r  , 1 2  2 , r , 2  2 r  , 1 2  2 , r 2  =0; as expected. 31.4 Bond prices in the CIR model The interest rate process rt is given by drt= ,rt dt +  q rt dW t; where r0 is given. The bond price process is B t; T =IE " exp  , Z T t ru du      F t  : Because exp  , Z t 0 ru du  B t; T =IE " exp  , Z T 0 ru du      F t  ; the tower property implies that this is a martingale. The Markov property implies that B t; T  is random only through a dependence on rt . Thus, there is a function B r;t;T of the three dummy variables r;t;T such that the process B t; T  is the function B r;t;T evaluated at r t;t;T , i.e., B t; T =Brt;t;T: CHAPTER 31. Cox-Ingersoll-Ross model 311 Because exp n , R t 0 ru du o B rt;t;T is a martingale, its differential has no dt term. We com- pute d  exp  , Z t 0 ru du  B rt;t;T  = exp  , Z t 0 ru du  ,rtBrt;t;T dt + B r rt;t;T drt+ 1 2 B rr rt;t;T drt drt+ B t rt;t;T dt  : The expression in ::: equals = ,rB dt + B r  , r dt + B r  p rdW + 1 2 B rr  2 rdt+B t dt: Setting the dt term to zero, we obtain the partial differential equation , rBr;t;T+ B t r;t;T+  , rB r r;t;T+ 1 2  2 rB rr r;t;T= 0; 0  tT; r0: (4.1) The terminal condition is B r;T;T= 1; r  0: Surprisingly, this equation has a closed form solution. Using the Hull & White model as a guide, we look for a solution of the form B r;t;T= e ,rCt;T ,At;T  ; where C T; T = 0;AT; T = 0 .Thenwehave B t =,rC t , A t B; B r = ,CB; B rr = C 2 B; and the partial differential equation becomes 0=,rB +,rC t , A t B ,  , rCB + 1 2  2 rC 2 B = rB,1 , C t + C + 1 2  2 C 2  , BA t + C  We first solve the ordinary differential equation ,1 , C t t; T +C t; T + 1 2  2 C 2 t; T = 0; CT; T = 0; andthenset At; T = Z T t Cu; T  du; 312 so AT; T = 0 and A t t; T = ,C t; T : It is tedious but straightforward to check that the solutions are given by C t; T = sinh T , t  cosh T , t + 1 2  sinh T , t ; At; T =, 2  2 log 2 4 e 1 2 T ,t  cosh T , t + 1 2  sinh T , t 3 5 ; where  = 1 2 q  2 +2 2 ; sinh u = e u , e ,u 2 ; cosh u = e u + e ,u 2 : Thus in the CIR model, we have IE " exp  , Z T t ru du      F t  = B rt;t;T; where B r;t;T = exp f,rC t; T  , At; T g ; 0  tT; r0; and C t; T  and At; T  are given by the formulas above. Because the coefficients in drt= ,rt dt +  q rt dW t do not depend on t , the function B r;t;T depends on t and T only through their difference  = T , t . Similarly, C t; T  and At; T  are functions of  = T , t . We write B r; instead of B r;t;T , and we have B r; = exp f,rC   , A g ;   0;r0; where C  = sinh    cosh + 1 2  sinh  ; A =, 2  2 log 2 4 e 1 2   cosh + 1 2  sinh  3 5 ;  = 1 2 q  2 +2 2 : We have B r 0;T= IEexp  , Z T 0 ru du  : Now ru  0 for each u , almost surely, so B r0;T is strictly decreasing in T . Moreover, B r0; 0 = 1; [...]... in T Moreover, Br0; 0 = 1; CHAPTER 31 Cox-Ingersoll-Ross model 313 lim B r0; T  = IE exp , T !1 But also, Z1 0 ru du = 0: Br0; T  = exp f,r0C T  , AT g ; so r0C 0 + A0 = 0; lim r0C T  + AT  = 1; T !1 and r0C T  + AT  is strictly inreasing in T 31.5 Option on a bond The value at time t of an option on a bond in the CIR model is "  ZT   + F t ; vt; rt =...  ; cosh T  + 1 sinh T  2 2 3 1 T 2 AT  = , 2 2 log 4 cosh T  e 1 sinh T  5 ; +2 C T  = = 1 2 q 2 + 2 2: (cal) = 0, CHAPTER 31 Cox-Ingersoll-Ross model 317 ^r ^ 6, log B ^0; 0; T  Goes to 1 Strictly increasing -T ^ Figure 31.4: Bond price in CIR model 6 r0C T  + AT  ^ ^r ^ , log B ^0; 0; T  ^ 'T  -T Figure 31.5: Calibration The function... dt = '0t dt in the first integral to get  Z T^  0t dt ; ^ Br0; 0; T  = IE exp , r't' ^ ^ 0 ^ and this will be B ^0; 0; T  if we set r ^ ^ rt = r't '0t: ^^ CHAPTER 31 Cox-Ingersoll-Ross model 315 31.7 Calibration ^r^ t ^ B ^t; ^; T  = B  ^ rt ^ ^ '0 t  ! ^ ^ ; 't; 'T   ^ '^ ^ ^ = exp ,rt C 't;^ T  , A't; 'T  ^^ '0 t n o = exp ,rtC t; T  , At;... log B r0; 0; T2; ^ so 'T1 ^ 'T2 Thus ' is a strictly increasing time-change-function with the right properties 316 31.8 Tracking down '0 0 in the time change of the CIR model Result for general term structure models: @ , @T log B0; T  Justification: T =0 = r0:  ZT  B0; T  = IE exp , ru du : 0  ZT  , log B0; T  = , log IE exp , ru du 0  R T ru du  , IE rT e @ , @T log... a function B r; t; T  of the three dummy variables r; t; T such that the process B t; T  is the function B r; t; T  evaluated at rt; t; T , i.e., Bt; T  = Brt; t; T : CHAPTER 31 Cox-Ingersoll-Ross model Because exp pute 311 n Rt o , 0 ru du Brt; t; T  is a martingale, its differential has no dt term We com d exp , = exp , Zt Z0t ru du B rt; t; T   ru du ,rtBrt; t;... European derivative securities on the bond are priced using the same partial differential equation with the terminal condition appropriate for the particular security 31.6 Deterministic time change of CIR model Process time scale: In this time scale, the interest rate rt is given by the constant coefficient CIR equation q drt =  , rt dt + rt dW t: Real time scale: In this time scale, the interest...  +  , rBr r; t; T  + 2 2rBrr r; t; T  = 0; 0  t T; r  0: The terminal condition is (4.1) Br; T; T  = 1; r  0: Surprisingly, this equation has a closed form solution Using the Hull & White model as a guide, we look for a solution of the form B r; t; T  = e,rC t;T ,At;T ; where C T; T  = 0; AT; T  = 0 Then we have Bt = ,rCt , At B; Br = ,CB; Brr = C 2B; and the partial differential... T  , At; T  ; ^^^ ^ ^ ^^ ^ where ^ ^ ' ^^^ C t; T  = C 't;^ T  '0t ^^^ ^ ^ At; T  = A't; 'T  ^ ^ ^ do not depend on ^ and T only through T , t, since, in the real time scale, the model coefficients t are time dependent ^r ^ ^ Suppose we know r0 and B ^0; 0; T  for all T 2 ^ n ^ 0; T  We calibrate by writing the equation o ^r ^ B ^0; 0; T  = exp ,r0C 0; T  , A0;... t + 1 sinh T , t 2 2 3 1 2 log 4 e 2 T ,t 5; At; T  = , 2 1 cosh T , t + 2 sinh T , t C t; T  = where = 1 2 q u ,u u ,u sinh u = e , e ; cosh u = e + e : 2 2 2 + 2 2; Thus in the CIR model, we have "  ZT   IE exp , ru du F t = Brt; t; T ; t where Br; t; T  = exp f,rC t; T  , At; T g ; 0  t T; r  0; and C t; T  and At; T  are given by the formulas above Because... , r + , 2  , 1 2 , r2 + 2 2 2r  r = pr  , 1 2 , r 2  , 1 2 , r 2 2  , 1 2 , r 2 2r  1 1 + r  , 1 2 , r , 2 r  , 2 2 , r2 2 2 ,1 2 = 0; 2 2r 2 as expected 31.4 Bond prices in the CIR model The interest rate process rt is given by drt =  , rt dt + q rt dW t; where r0 is given The bond price process is "  ZT   B t; T  = IE exp , ru du F t : t Because exp , Zt 0 . Chapter 31 Cox-Ingersoll-Ross model In the Hull& Whitemodel, rt is a Gaussianprocess. Since, for each t. CHAPTER 31. Cox-Ingersoll-Ross model 317 - 6 Goes to 1 Strictly increasing ^ T , log ^ B ^r0; 0; ^ T  Figure 31.4: Bond price in CIR model - 6 ^r0C