A two-factor model (Duffie & Kan)

Chapter 32 A two-factor model (Duffie & Kan) Let us define: X 1 t= Interest rate at time t X 2 t= Yield at time t on a bond maturing at time t +  0 Let X 1 0  0 , X 2 0  0 be given, and let X 1 t and X 2 t be given by the coupled stochastic differential equations dX 1 t=a 11 X 1 t+a 12 X 2 t+b 1  dt +  1 q  1 X 1 t+ 2 X 2 t+dW 1 t; (SDE1) dX 2 t=a 21 X 1 t+a 22 X 2 t+b 2  dt +  2 q  1 X 1 t+ 2 X 2 t+ dW 1 t+ q 1, 2 dW 2 t; (SDE2) where W 1 and W 2 are independent Brownian motions. To simplify notation, we define Y t 4 =  1 X 1 t+  2 X 2 t+; W 3 t 4 = W 1 t+ q 1, 2 W 2 t: Then W 3 is a Brownian motion with dW 1 t dW 3 t= dt; and dX 1 dX 1 =  2 1 Y dt; dX 2 dX 2 =  2 2 Y dt; dX 1 dX 2 =  1  2 Y dt: 319 320 32.1 Non-negativity of Y dY =  1 dX 1 +  2 dX 2 = 1 a 11 X 1 +  1 a 12 X 2 +  1 b 1  dt + 2 a 21 X 1 +  2 a 22 X 2 +  2 b 2  dt + p Y  1  1 dW 1 +  2  2 dW 1 +  2 q 1 ,  2  2 dW 2  = 1 a 11 +  2 a 21 X 1 + 1 a 12 +  2 a 22 X 2  dt + 1 b 1 + 2 b 2  dt + 2 1  2 1 +2 1  2  1  2 +  2 2  2 2  1 2 q Y t dW 4 t where W 4 t=  1  1 + 2  2 W 1 t+ 2 p 1, 2  2 W 2 t q  2 1  2 1 +2 1  2  1  2 +  2 2  2 2 is a Brownian motion. We shall choose the parameters so that: Assumption 1: For some  ,  1 a 11 +  2 a 21 =  1 ;  1 a 12 +  2 a 22 =  2 : Then dY = 1 X 1 +  2 X 2 +   dt + 1 b 1 + 2 b 2 ,  dt + 2 1  2 1 +2 1  2  1  2 +  2 2  2 2  1 2 p YdW 4 =Y dt + 1 b 1 + 2 b 2 ,  dt + 2 1  2 1 +2 1  2  1  2 +  2 2  2 2  1 2 p YdW 4 : From our discussion of the CIR process, we recall that Y will stay strictly positive provided that: Assumption 2: Y 0 =  1 X 1 0 +  2 X 2 0 + 0; and Assumption 3:  1 b 1 +  2 b 2 ,   1 2  2 1  2 1 +2 1  2  1  2 +  2 2  2 2 : Under Assumptions 1,2, and 3, Y t  0; 0  t1; almost surely, and (SDE1) and (SDE2) make sense. These can be rewritten as dX 1 t=a 11 X 1 t+a 12 X 2 t+b 1  dt +  1 q Y t dW 1 t; (SDE1’) dX 2 t=a 21 X 1 t+a 22 X 2 t+b 2  dt +  2 q Y t dW 3 t: (SDE2’) CHAPTER 32. A two-factor model (Duffie & Kan) 321 32.2 Zero-coupon bond prices The value at time t  T of a zero-coupon bond paying $1 at time T is B t; T =IE " exp  , Z T t X 1 u du      F t  : Since the pair X 1 ;X 2  of processes is Markov, this is random only through a dependence on X 1 t;X 2 t . Since the coefficients in (SDE1) and (SDE2) do not depend on time, the bond price depends on t and T only through their difference  = T , t . Thus, there is a function B x 1 ;x 2 ; of the dummy variables x 1 ;x 2 and  ,sothat B X 1 t;X 2 t;T , t=IE " exp  , Z T t X 1 u du      F t  : The usual tower property argument shows that exp  , Z t 0 X 1 u du  B X 1 t;X 2 t;T , t is a martingale. We compute its stochastic differential and set the dt term equal to zero. d  exp  , Z t 0 X 1 u du  B X 1 t;X 2 t;T , t  = exp  , Z t 0 X 1 u du  ,X 1 Bdt+B x 1 dX 1 + B x 2 dX 2 , B  dt + 1 2 B x 1 x 1 dX 1 dX 1 + B x 1 x 2 dX 1 dX 2 + 1 2 B x 2 x 2 dX 2 dX 2  = exp  , Z t 0 X 1 u du  ,X 1 B +a 11 X 1 + a 12 X 2 + b 1 B x 1 +a 21 X 1 + a 22 X 2 + b 2 B x 2 , B  + 1 2  2 1 YB x 1 x 1 + 1  2 YB x 1 x 2 + 1 2  2 2 YB x 2 x 2  dt +  1 p YB x 1 dW 1 +  2 p YB x 2 dW 3  The partial differential equation for B x 1 ;x 2 ; is , x 1 B , B  +a 11 x 1 + a 12 x 2 + b 1 B x 1 +a 21 x 1 + a 22 x 2 + b 2 B x 2 + 1 2  2 1  1 x 1 +  2 x 2 + B x 1 x 1 +  1  2  1 x 1 +  2 x 2 + B x 1 x 2 + 1 2  2 2  1 x 1 +  2 x 2 + B x 2 x 2 =0: (PDE) We seek a solution of the form B x 1 ;x 2 ; = exp f,x 1 C 1   , x 2 C 2   , A g ; valid for all   0 and all x 1 ;x 2 satisfying  1 x 1 +  2 x 2 + 0: (*) 322 We must have B x 1 ;x 2 ;0 = 1; 8x 1 ;x 2 satisfying (*) ; because  =0 corresponds to t = T . This implies the initial conditions C 1 0 = C 2 0 = A0 = 0: (IC) We want to find C 1  ;C 2 ;A for 0 .Wehave B  x 1 ;x 2 ;=  ,x 1 C 0 1 ,x 2 C 0 2 ,A 0   Bx 1 ;x 2 ;; B x 1 x 1 ;x 2 ;=,C 1 Bx 1 ;x 2 ;; B x 2 x 1 ;x 2 ;=,C 2 Bx 1 ;x 2 ;; B x 1 x 1 x 1 ;x 2 ;=C 2 1 Bx 1 ;x 2 ;; B x 1 x 2 x 1 ;x 2 ;=C 1 C 2 Bx 1 ;x 2 ;; B x 2 x 2 x 1 ;x 2 ;=C 2 2 Bx 1 ;x 2 ;: (PDE) becomes 0=Bx 1 ;x 2 ;  ,x 1 + x 1 C 0 1 +x 2 C 0 2 +A 0 ,a 11 x 1 + a 12 x 2 + b 1 C 1   , a 21 x 1 + a 22 x 2 + b 2 C 2   + 1 2  2 1  1 x 1 +  2 x 2 + C 2 1  + 1  2  1 x 1 +  2 x 2 + C 1  C 2   + 1 2  2 2  1 x 1 +  2 x 2 + C 2 2    = x 1 B x 1 ;x 2 ;  , 1+C 0 1 ,a 11 C 1   , a 21 C 2   + 1 2  2 1  1 C 2 1  + 1  2  1 C 1  C 2  + 1 2  2 2  1 C 2 2   + x 2 Bx 1 ;x 2 ;  C 0 2 , a 12 C 1   , a 22 C 2   + 1 2  2 1  2 C 2 1  + 1  2  2 C 1  C 2  + 1 2  2 2  2 C 2 2   + Bx 1 ;x 2 ;  A 0  , b 1 C 1 , b 2 C 2  + 1 2  2 1 C 2 1  + 1  2 C 1  C 2  + 1 2  2 2 C 2 2    We get three equations: C 0 1  =1+a 11 C 1  + a 21 C 2   , 1 2  2 1  1 C 2 1   ,  1  2  1 C 1  C 2   , 1 2  2 2  1 C 2 2  ; (1) C 1 0 = 0; C 0 2  =a 12 C 1  +a 22 C 2   , 1 2  2 1  2 C 2 1   ,  1  2  2 C 1  C 2   , 1 2  2 2  2 C 2 2  ; (2) C 2 0 = 0; A 0  =b 1 C 1 +b 2 C 2 , 1 2  2 1 C 2 1   ,  1  2 C 1  C 2   , 1 2  2 2 C 2 2  ; (3) A0 = 0; CHAPTER 32. A two-factor model (Duffie & Kan) 323 We first solve (1) and (2) simultaneously numerically, and then integrate (3) to obtain the function A  . 32.3 Calibration Let  0  0 be given. The value at time t of a bond maturing at time t +  0 is B X 1 t;X 2 t; 0  = expf,X 1 tC 1  0  , X 2 tC 2  0  , A 0 g and the yield is , 1  0 log B X 1 t;X 2 t; 0 = 1  0 X 1 tC 1  0 + X 2 tC 2  0 +A 0  : But we have set up the model so that X 2 t is the yield at time t of a bond maturing at time t +  0 . Thus X 2 t= 1  0 X 1 tC 1  0 + X 2 tC 2  0 +A 0  : This equation must hold for every value of X 1 t and X 2 t , which implies that C 1  0 =0;C 2  0 = 0 ;A=0: We must choose the parameters a 11 ;a 12 ;b 1 ; a 21 ;a 22 ;b 2 ;  1 ; 2 ;;  1 ;; 2 ; so that these three equations are satisfied. 324 . (3) A 0 = 0; CHAPTER 32. A two-factor model (Duffie & Kan) 323 We first solve (1) and (2) simultaneously numerically, and then integrate (3) to obtain. Chapter 32 A two-factor model (Duffie & Kan) Let us define: X 1 t= Interest rate at time t X 2 t= Yield at time t on a bond maturing at time