关于CIR模型中即期利率的条件密度及贴现债券定价_英文_

关于CIR模型中即期利率的条件密度及贴现债券定价_英文_ Ξ On the Conditional Density Function of CIR Model and an Analytical Sol ution f or Bond Price Via Varia ble Separation Method ) () ()(L IAO Chan2ggao 廖长高,L I Xian2ping 李贤平,XUP ing 徐萍 ( )Department of Statistics , Fudan University , S hanghai 200433 , China Abstract :This paper deals with the conditional density function of spot interest rate of CIR model and gives the explicit expression of bond price as a function of the spot interest rate . As you can see ,our method is differ2 ent from the expectation method under the equivalent martingale measure . Key words :Conditional density function ;CIR model ; Spot interest rate ; Kolmogorov backwark e2 quation ;Affine term structure CL C Number :F224 . 4 ;O211 . 9 AMS( 2000) Subject Cla ssif ication :62P20 ; 90A09 ; 90 H12 () Article ID :100129847 2002Supplement20081204 Document code :A Pricing theory is very crucial in finance . Many models have been built up to study the dynamics of ( ) spot interest rate r t . Under some assumptions ,i . e . ,no arbitrage opportunity in the market ,we can ob2 tain the basic pricing formulae for default2free bonds. In this paper ,our aim is to derive the conditional ( ) density function of spot interest rate in CIR model through Kolmogorov backward equationPDEand ob2 tain bond prices by solving two ordinary differential equations which differs from that applied in CIR mod2 el using expectation method under the unique martingale measure . Generally speaking ,one can apply the same procedure to obtain the conditional density function for other types of spot interest rate processes by finding the corresponding orthogonal polynomials and then simplifying a summation of series. We are also interested in the conditional moments of bond prices ,so ,firstly ,we need to derive the explicit expression of bond prices. We assume the logarithm of bond price is proportional to the spot interest rate ,that is ( ( ) ) ( ) ( ) )( ln p r , t , T= A t , T- B t , T?r ,1 ( ) ( ) ( ) where r = r t, A = A t , Tand B = B t , Tare two deterministic functions of time t and maturity date T. As you can see ,this formula applies for all affine term structure models which was proposed and () κ(θdeeply researched by Duffie and Kan 1996. They assume the interest rate process r follows d r = - 2 2 ) σσκθσ σ ( ) ( ) r+ + rdw t, where ,,andare some positive constants , w tis a standard Brownian 0 0 1 Ξ Received date :May 11 ,2002 motion . It’s very easy to handle it through variable transformation and then attain a similar model as CIR. We can then restrict our model as follows κ(θ ) σ ( ) ()dr = - rdt + rdW t ,2 1 2 σκ(θ ( ) ) λ( )rp + - r tp + p = rp + rp ,3 rr r t r 2 ( ) where p = p r , t , T, pand pare the first and second order derivatives with respect to r , respectively. r rr () θFrom 2,we conclude that the spot interest rate will converge to the long2term constant mean ,, mean2 while this process also guarantees the following two features : First , negative interest rate is not permissi2 ( ) ble ;Second ,a higher interest rate means a higher volatility. 3is the fundamental pricing equation for discount bond p , and we can derive it by the common use of arbitrage2free pricing argument . ( ) To begin with our work ,we firstly let f y , s | x , t denote the conditional density function and prove the following proposition. () Proposition The diffusion process 2has the following conditional density function q/ 2 v 1/ 2 - u - v ( ) ( ) ()2 uv, we 4 f y , s | x , t =I q u κθκ 22( ) - k s - tw ?( ) where , u ?wxe , v ?wy , q ? - 1 , and I ?is the modified Bessel q 2 2 ( ) - k s - t σσ( )1 - e function of the first kind of order q . c 2σκθκα Proof Denote by a = / 2 , b = - , c = ,= - 1 , the conditional density function of a () process 2has the following backward Kolgomorov equation 2 5 f 5 f 5f ( ) )(= c + by+ a .5 2 5 s 5 y 5 y ( )() Let h = - by/ a , so 4can be turned into the density function of variable h , denoted by g ? 2 5 g 5 g 5g bc ()6 -= - + bh bh . 5 s 2 a 5 h 5 h φ( ) ( ) ( ) ( ) λ() Assume f = hd t, and let d′t/ d t= - , combined with equation 5we can get 2 φφbc d d φλ()+ bhbh = - .7 - - 2a dh dh λIf we choose = - bn , appealing to the knowledge of higher transcendental functions ,i . e . see Erdléyi n (α) ( ) () ( ) 1953,one should remember the solution of equation 7is Laguerre polynomials L hwith parameter n c (α) α () = - 1 , usually normalized by the conditions L 0> 0 and n a (α) (α) (α) n h{ L ( ) ) (α ) ( ( ) ) ( ) h+ + 1 - hL h′+ nL h= 0 ; n n n ? Γ( α) n + 1 + (α) 2 ( ) ( ) ()[ L x] m xd x8 = ,n ?Γ( ) Γ(α ) n + 1+ 10 α - x Γ(α ) ( ) ( ) where m x= x? e / + 1for x > 0 ; m x= 0 for x < 0. The Laguerre system (α) L ( )( ) y comprises the unique orthogonal polynomials with respect to the density function m x.n n ?0 ( ) So the relevant spectral representation for g ?is ?(α) (α) ( ) bn s - t( ) ( ) ( ) π ( ))(g h, s | h, t= eL h L h m h ,9 1 0 n 1 n 0 1 n ? n = 0 Γ( ) Γ(α ) n + 1+ 1 b b πNote that h= - y , h= - x , we can revert to get the expres2. where = n 1 0 Γ( α ) n + + 1a a ) ( sion for f ?as ? b b b (α) b (α)( ) bn s - t )( ) (π f y , s | x , t= eL L 10 m - y ? - .- y- xn n n ? a a a a n = 0 () Appealing to a classical result of series sum formula ,we simplify the left2hand side of 10to the following abbreviate form ( )αααb s - t +1 by/ ( )( ) - b/ ayee b x + y/ a ( )=×expf y , s | x , t ( )b s - t ( ) b s - t )1 - (ee1 - ()11 α2 - / 2 ( )b s - t b ( )α b s - t 2 b xye xye×I . 2 ( ) b s - t a)( a 1 - e () Now compared with 4,we have justified our propositon. () () We guess bond price has the form1;then from 3,we obtain 1 2 2 σκ(θ ) ψ( ) rpB - B [- r- r ] p + A′- Br′p - rp = 0.2 Rearrange the above equationa and note that r is a random variable ,so the coefficient of r and the constant term equal zero ,respectively ,that is 1 2 2 σ(κ ψ) ()B + + B - B′- 1 = 0 ,12 2 θκ()- B + A′= 0. 13 2 σ dB 22 2 2 γ(ψ κ) ση γ κ ψ() Let = + + 2,= + + , from 12we have an = d t . Take 2 2 2σ(κ ψ) γ[B + + ]- indefinite integral and rearrange ,we get 2 σκ ψ γ)(- B + + - γ t= e c , For all 0 ? t ? T. 2 σκ ψ γB + + + ( ) ( ) ( ) One should also notice p r , T , T= 1 , so we get A T , T= B T , T= 0 , therefore c = (κ ψ γ) - + - and it also follows that κ ψ γ + + γ( )T - t γ(κ ψ γ) (κ ψ γ) (κ ψ γ) (κ ψ γ) 2+ + - + - [ - + - + e ?+ + ] = B γ( ) T - t2(κ ψ γ) (κ ψ γ) σ [ - + - + e+ + ]()14 γ( )T - t 2 e - 1 = . γ( )T - t γ η e- 1 2+ () () Now ,according as 13and 14,we can solve A explicitly γ( )T - t 1 - e θκθκA =B d t = 2d t γ( )T - t?? η γ - e1 + 2 θκθκ - 2 2s - s - s γ (η( ) γ) (η(γ) ln e - 1+ 2- ln 1 - e + 2e/ + c= γ ηη 2- γ θκ2ηθκ2 = +ln s + c . s 2 2 η( ) γ σe- 1+ 2σγ γ( ) ( ) Wherein s = T - tNote the terminal condition A T , T= 0 , so c = 0 and therefore . γ θκ2ηθκ2 ( ) ()T - t. 15 A = +ln γ( ) T - t2 2 η( ) γσσe- 1+ 2 Based upon the expression of bond price ,we can discuss some properties ,i . e . monotone increasing or de2 θσκcreasing interval and convexity about the maturity date T , the revertiong mean , the volatilityand . () () () () From 1, 4, 14and 15,we have the following one order conditonal moment of bond price A ( )( ) s , T - B s , Ty ( ) Ep y , s | x , t= eEe t t ? k - u ? B u e A k + q- y - y c = eye d y?Γ( ) ? k !k + q + 10 k = 0 ? ()k 16 1 u - u + A e = q +1 ? 1 + B / c ( ) 1 + B / ck ! k = 0 - B u 1 + A c + B = e × q +1 () 1 + B / c. Similarly ,we can also derive the second order conditional moment as 2 2( ( ) ) ( ) ( ) V arp y , s | x , t= Epy , s | x , t- EP y , s | x , t] t t t () - 2 B u - 2 B u 17 11 2 A c +2 B c + B e - e = e. q +12 q +2 () () 1 + 2 B / c 1 + B / cReference : 1 J C Cox ,J E Ingersoll and S A Ross. A theory of the term structure of interest ratesJ . Econometrica ,1985 ,53 :385 ～ 407. John H Cochrane . Asset pricingM . Princeton : Princeton university press ,2001. 2 3 Samuel Karlin , Howard M Taylor. A second course in stochastic processM . New York :Academic press ,inc . ,1981. 4 Erdé1yi . Higher Transcendental Functions ,Vol . IIM . New 2York :Mc Graw2Hill ,1953. 5 武宝亭 ,李庆士. 随机过程与随机微分方程M . 成都 :电子科技大学出版社 ,1994. 关于 CIR 模型中即期利率的条件密度及贴现债券定价 廖长高 ,李贤平 ,徐萍 () 复旦大学统计运筹系 ,上海 200433 摘要 :本文对 CIR 模型运用 Kolgomorov 向后方程给出即期利率的条件密度所满足的偏微 分方程 ,并用谱方法将该密度表达成一个 Laguerre 型的正交级数之和 ,借助一个经验级数和公 式 ,我们得出了该和式的第一类修正贝塞尔函数表达式. 同时 ,我们猜测贴现债券价格关于即 期利率的特殊表达式 ,并通过求解两个一阶常微分方程验证了我们的猜测. 在文章的结尾 ,我 们给出了贴现债券价格的条件期望和方差以反映必要的矩信息. 关键词 :条件密度函数 ;CIR 模型 ;即期利率 ;柯尔摩哥洛夫向后方程 ;仿射期限结构