The Cox-Ingersoll-Ross Model

Transcription

1 The Cox-Ingersoll-Ross Model Mahias Thul, Ally Quan Zhang June 2, 2010 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 1

2 References Cox, John C.; Ingersoll, Jonahan E.; Ross, Sephen A. An Ineremporal General Equilibrium Model of Asse Prices Economerica, Vol. 53, No. 2 (March 1985), pp Cox, John C.; Ingersoll, Jonahan E.; Ross, Sephen A. A Theory of he Term Srucure of Ineres Raes Economerica, Vol. 53, No. 2 (March 1985), pp The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 2

3 Moivaion develop a general framework o model he erm srucure of ineres raes, price bonds and derivaive producs general decision beween he (i) arbirage approach wih exogenously given ineres rae dynamics and he (ii) equilibrium approach ha deermines hem endogenously Cox-Ingersoll-Ross (CIR) adop an equilibrium approach o endogenously deermine he risk-free rae The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 3

4 Ouline I) ouline he CIR general producion economy framework II) inroduce a one-facor represenaion of he model economy III) deermine he opimal consumpion sraegy in he one-facor model IV) derive he equilibrium risk-free rae V) develop he dynamics of he risk-free rae VI) price coningen claims in he one-facor model VII) compare he equilibrium and he arbirage approach The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 4

5 I) The general CIR Producion Economy - Model Assumpions 1) consumpion good : single consumpion good ha canno be sored and has o be eiher consumed or invesed serves as boh, he inpu and he oupu of he producion process all values are measure in unis of his good The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 5

6 2) producion opporuniies : n differen risky echnologies ransformaion process of producion opporuniy i is dη i (x, ) η i (x, ) = µ i(x, )d + σ i (x, )dz i i = 1,..., n where µ i and σ i are he exogenous insananeous drif and diffusion, x is he vecor of sae variables and dz i is he incremen of a Wiener process he single good is boh he inpu and he oupu of he producion process assume ha µ i and σ i fulfil condiions s.. he above SDE is well-defined and has a unique soluion consan reurns o scale : he yield is independen of he invesed volume due o he lineariy of he SDE covariance (σ i dz i ) (σ j dz j ) = σ i,j d here are no limiaions regarding he amoun ha can be invesed ino he producion The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 6

7 3) sae variables : k facors represening he sae of he echnology he sae variable i follows he process dx i = a i (x, )d + b i (x, )dζ i i = 1,..., k where a i and b i are he local drif and diffusion funcions covariance (b i dζ i ) (b j dζ j ) = b i,j d, (σ i dz i ) (b j dζ j ) = φ i,j d he sae vecor x has o represen all necessary informaion in aggregae form 4) marke : coninuous rading in fricionless marke a equilibrium prices marke paricipans are price-akers The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 7

8 5) individuals : idenical individuals w.r.. iniial capial endowmen, preferences and expecaions he sochasic dynamics and curren sae of he economy are public knowledge a each poin in ime, choose heir insananeous consumpion C and porfolio n weighs ω i, i = 1,..., n for he n producion opporuniies s.. i=1 ω i = 1 and ω i 0 maximise expeced lifeime uiliy subjec o he budge consrain { [ ]} T max E U(C s, s)ds Cs,ω i,s, s,i F where T is heir planning horizon The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 8

9 6) uiliy funcions : sricly increasing, concave, wice differeniable von Neumann-Morgensern uiliy funcion wih consan relaive risk-aversion (RRA) [ C γ U(C, ) = e ρ 1 ] γ where γ is one minus he coefficien of RRA 1 γ := C U CC U C > 0 U(C, ) is an isoelasic uiliy funcion ime separabiliy : ime preferences ener he uiliy only via he pre-facor e ρ where ρ > 0 is he ime preference facor he relaive proporion of risky invesmen from oal wealh W is consan The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 9

10 7) risk-free rae : he insananeous risk-free rae r is endogenously deermined in equilibrium applies o all individuals and for boh, borrowing and lending shadow riskless rae a which individuals are indifferen beween borrowing and lending and choose zero ransacion volume zero ne supply in he whole economy 8) coningen claims : here exiss a marke for derivaive insrumens ha have payoffs in unis of he consumpion good (e.g. bonds, fuures, opions) derivaives are in zero ne supply he value P (W, x, ) could be a funcion of he aggregae wealh, he sae vecor and ime equilibrium prices are independen of aggregae wealh due o he assumpion of a consan RRA uiliy funcion (excep if he payoff is a funcion of W ) The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 10

11 Represenaive Individual he homogeneiy assumpions allow o apply Rubinsein s aggregaion heorem equilibrium prices can be deermined assuming a represenaive individual (RI) who maximises expeced uiliy under his budge consrain he RI invess he fracion of wealh ha he does no consume ino producion due o zero ne supply of he risk-free insrumen and he coningen claims he dynamic budge equaion is dw = W n ω i µ i d C d + W i n ω i σ i dz i i The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 11

12 II) A One-Facor Model Economy one producion opporuniy (n = 1) and one sae variable (k = 1) where (σdz ) (bdζ ) = φd dη(x, ) η(x, ) = µ(x, )d + σ(x, )dz dx = a(x, )d + b(x, )dζ he RI s opimizaion problem becomes { [ ]} T max E U(C s, s)ds Cs, s F subjec o he dynamic budge equaion dw = (W µ(x, ) C )d + W σ(x, )dz The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 12

13 III) Opimal Consumpion Sraegy in he One-Facor Model define he indirec uiliy funcion J(W, x, ) by J(W, x, ) := max Cs, s { E [ T ]} U(C s, s)ds F according o Bellman s Principle of Opimaliy, opimal sraegies are ime consisen J(W, x, ) = max Cs, s + max Cs, s { = max Cs, s { [ +d E U(C s, s)ds { [ ]} ]} T E U(C s, s)ds +d F +d F [ ]} +d E U(C s, s)ds + J(W +d, x +d, + d) F The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 13

14 assuming ha we can apply Iô s lemma, J(W +d, x +d, + d) can compued as J(W +d, x +d, + d) J(W, x, ) + J W dw + J x dx + J d J W 2 (dw ) J x 2 (dx ) J W x dw dx = J(W, x, ) + J W (W µ C )d + J W W σdz + J x ad + J x bdζ + J J b 2 d + 2 x 2 d J W x W φd 2 J W 2 W 2 σ2 d noe ha he Wiener processes z and ζ are maringales and hus heir changes have zero expeced value The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 14

15 plugging his resul back ino he definiion of he indirec uiliy funcion resuls he Hamilon-Jacobi-Bellman (HJB) equaion 0 = max C { U(C, ) + J (W µ C ) + J a + J W x } 2 J W 2 W 2 σ J b x 2 where J(W, x, ) and d were cancelled ou 2 J W x W φ using he Dynkin operaor L o simplify he noaion yields 0 = max C {U(C, ) + LJ(W, x, )} The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 15

16 he opimizaion problem can be solved in hree seps: 1) deermine he opimal consumpion level C (J) depending on he indirec uiliy funcion J 2) recover J by solving he PDE ha is obained by subsiuing he opimal consumpion C (J) ino he Bellman equaion 3) solve for he opimal consumpion level C by subsiuing he indirec uiliy funcion ino C (J) The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 16

17 Sep 1: Deermine C (J) le 0 = max C {Ψ(C )} s.. C 0, hen Ψ(C ) = U(C, ) + J (W µ C ) + J a + J W x J W 2 W 2 σ J b x 2 he Kuhn-Tucker firs order condiions (FOC) for C are Ψ = U J 0 C C W C Ψ C = C U C C J W = 0 2 J W x W φ The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 17

18 excep for he rivial soluion C = 0, he opimal consumpion is chosen such ha marginal uiliy of consumpion equals o he marginal uiliy of fuure wealh U = J C W he FOC are no only necessary bu also sufficien due o he sric concaviy of he uiliy funcion U given he concree uiliy funcion, he opimal consumpion rae C (J) can be deermined [ C γ U(C, ) = e ρ 1 ] γ U C = e ρ ( C ) γ 1 C (J) = ( e ρ J W ) 1 γ 1 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 18

19 Sep 2: Solve for J subsiuing back ino he HJB equaion and grouping similar erms yields a non linear PDE for J ha can in general no be solved explicily for isolasic uiliy funcions, he indirec uiliy funcion akes he form J(W, x, ) = f(x, )U(W, ) + g(x, ) logarihmic uiliy is a limiing case of he isoelasic uiliy when γ 0 or equivalenly RRA 1 [ C γ lim γ 0 e ρ 1 ] = e ρ ln(c ) γ in his special case, f(x, ) is independen of wealh W a funcion of ime only f(x, ) = f() = 1 e ρ(t ) ρ The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 19

20 Sep 3: Solve for C differeniaing J(W, x, ) w.r.. W yields [f()u(w, ) + g(x, )] = f() U = 1 e ρ(t ) W W ρw using he FOC allows o solve for C U C = J C W = W ρ 1 e ρ(t ) under logarihmic uiliy C depends on he curren wealh W, he ime preference facor ρ and he planning horizon (T ) only he opimal consumpion C is independen of he funcion g(x, ) and hus of he producion opporuniies in he economy The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 20

21 IV) Equilibrium Risk-free Rae in he One-Facor Model since he risk-free insrumen is in zero ne supply, i is no being held by he represenaive invesor deermine he risk-free rae endogenously such ha he invesor is no beer off by rading in he money marke, i.e. he is indifferen beween an invesmen in he producion opporuniy and he risk-free insrumen denoe by ω η 0 he proporion of wealh ha is invesed in he producion opporuniy, hen (1 ω η )W is he amoun invesed in he risk-free insrumen he dynamic budge equaion is dw = (ω η W µ(x, ) + (1 ω η )W r C )d + ω η W σ(x, )dz = (ω η W (µ(x, ) r ) + W r C )d + ω η W σ(x, )dz The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 21

22 he new opimizaion problem is 0 = max C,ωη {Ψ(C, ω η )} s.. C 0, ω η 0 wih Ψ(C, ω η ) = U(C, ) + J (ω η W (µ r ) + W r C ) + J a + J W x J W 2 ω 2 η W 2 σ J b x 2 he FOC for C do no change and he FOC for ω η are Ψ = J W (µ r ) + 2 J ω η W W 2 Ψ ω η = J ω η W (µ r ) + 2 J ω η W W 2 again, he FOC are boh necessary and sufficien ω η W 2 σ2 + 2 J W x ω η W φ ω 2 η W 2 σ2 + 2 J W x W φ 0 2 J W x ω η W φ = 0 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 22

23 we wan o find he opimum condiional on he invesor only consuming or invesing in he producion opporuniy bu wihou holding he risk-free asse solving he second FOC for r and seing ω η = 1 yields r = µ + 2 J/ W 2 W σ J/ W x φ J/ W J/ W he equilibrium ineres rae r depends on (i) he insananeous mean reurn µ(x, ) of he opimally invesed wealh (ii) a erm reflecing he uncerainy abou he reurns of he producion opporuniy (iii) a erm reflecing he uncerainy abou he sae of he echnology The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 23

24 noe ha by definiion and hus 2 J/ W 2 J/ W W = γ 1 r = µ + (γ 1)σ J/ W x φ J/ W for he special case of logarihmic uiliy (γ = 0), he indirec uiliy funcion becomes J(W, x, ) = f() ln(w ) + g(x, ) 2 J W x = 0 and we ge r = µ(x, ) σ 2 (x, ) under logarihmic uiliy, he ineres rae only depends on he sochasic dynamics of he producion opporuniy r is independen of he fuure producion risk arising from he sae variable x (i.e. a(x, ) and b(x, )) The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 24

25 V) Equilibrium Dynamics of he Risk-free Rae - Model Assumpions 1) facor dynamics : he drif and diffusion coefficiens or he sae variable are a(x, ) = a 0 + a 1 x and b(x, ) = b 0 x, i.e. dx = (a 0 + a 1 x )d + b 0 x dζ where a 0, a 1 and b 0 are consans, a 0 0 and (dz ) (dζ ) = ρd for a 0 > 0 and a 1 < 0, x is a non-negaive mean-revering random variable noe ha (dx ) 2 = b 2 0 x d The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 25

26 2) producion dynamics : he means and variances of he raes of reurn of he producion process are proporional o x, i.e. µ(x, ) = ˆµx and σ(x, ) = ˆσ x dη η = ˆµx d + ˆσ x dz given a fixed x = x his yield a geomeric Brownian moion for η and hus normally disribued reurns ln ( ηt η 0 ) N ([ˆµ 12 ˆσ2 ] xt, ˆσ 2 xt ) echnological progress increases boh, he mean-reurn and he variance of he producion process The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 26

27 Ineres Rae Dynamics using he assumpion abou he facor and producion dynamics allows he equilibrium ineres rae r becomes r = µ(x, ) σ 2 (x, ) = ˆµx ˆσ 2 x applying Iô s lemma yields he diffusion process of he risk-free rae dr = = ( ˆµ ˆσ 2) dx (a 0 (ˆµ ˆσ 2) ) + a 1 r d + b 0 ˆµ ˆσ 2 r dζ = κ ( r r ) d + σ r dζ where κ = a 1, r = ( a 0 (ˆµ ˆσ 2 )) a 1 1 and σ = b 0 ˆµ ˆσ 2 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 27

28 Sochasic Properies of r (i) r follows a square roo diffusion ha is almos surely non-negaive (ii) he ineres rae is elasically pulled owards a long erm value r > 0 (iii) κ > 0 deermines he speed of mean-reversion (iv) Feller condiion : if 2κ r σ 2 and r > 0, hen he process does no reach zero almos surely (v) r follows a non-cenral chi-square disribuion (vi) r for s, he condiional mean and variance are E [r F s ] = r s e κ( s) + r (1 e κ( s)) Var [r F s ] = σ2 κ r s (e κ( s) e 2κ( s)) + r σ2 2κ (1 e κ( s)) 2 his direcly follows from he disribuional properies or can be derived using Iô s lemma (see appendix A) The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 28

29 Empirical Properies of Ineres Raes ha are refleced by r (i) r shows mean-reversion (ii) ineres raes can no become negaive r is a suiable model for he nominal ineres rae (iii) even if he Feller condiion is no fulfilled and r = 0, hen his value is no absorbing (iv) he absolue variance of he ineres rae increases when r increases The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 29

30 Sample Pahs of he CIR Process shor rae [% p.a.] ime [years] Figure 1: Sample pahs for he CIR process wih r 0 = 0.10, κ = 1.0, r = 0.10, σ = The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 30

31 VI) Equilibrium Pricing of Coningen Claims jus like he risk-free rae, all coningen claims are in zero ne supply and hus no held by he represenaive invesor in equilibrium denoe by P (W, x, ) he price of a coningen claim by Iô s lemma, is differenial akes he form dp = P d + P W dw + P 2 P x 2 (dx ) 2 + x dx P W x dw dx 2 P W 2 (dw ) 2 collecing all drif erms, we se α(w, x, ) o be he insananeous reurn of he derivaive securiy o obain dp = α(w, x, )P d + P W W σdz + P x bdζ The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 31

32 accordingly, we choose β η (W, x, )P = P W W σ, β x (W, x, )P = P x b o ge he following expression for he insananeous yield of he coningen claim dp P = α(w, x, )d + β η (W, x, )dz + β x (W, x, )dζ le ω η be he proporion of wealh ha is invesed in he producion opporuniy and ω P be he proporion invesed in he derivaive insrumen (1 ω η ω P ) W is he amoun invesed in he risk-free insrumen he dynamic budge equaion is dw = (ω η W (µ r ) + ω P W (α r ) + W r C ) d +W (ω η σ + ω P β η ) dz + W ω P β x dζ The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 32

33 he new opimizaion problem is 0 = max C,ωη,ω P {Ψ(C, ω η, ω P )} s.. C 0, ω η 0 wih Ψ(C, ω η, ω P ) = U(C, ) + J (ω η W (µ r ) + ω P W (α r ) + W r C ) + J a + J W x ] J W 2 W J b x 2 [ (ω η σ + ω P β η ) 2 + (ω P β x ) (ω η σ + ω P β η ) ω P β x ρ 2 J W x W [(ω η σ + ω P β η ) ρ + ω P β x ] b The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 33

34 he new FOC for ω P is Ψ = J W (α r ) ω P W + 2 J W 2 W 2 ( ) [ω P β 2 η + β2 x + 2β ηβ x ρ + 2 J W x W [β η ρ + β x ] b = 0 ] + ω η σβ η + ω η σβ x ρ since here is zero ne supply for he coningen claim, we se ω η = 1, ω P = 0 and solve for he equilibrium excess reurn α r o ge α r = [ ] 2 J/ W 2 W σ 2 J/ W x bρ J/ W J/ W [ ] + 2 J/ W 2 W σρ 2 J/ W x b J/ W J/ W β η β x The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 34

35 Marke Prices of Facor Risks he equaion for he equilibrium excess reurn α r allows us o define he marke prices of facor risks as θ η = 2 J/ W 2 W σ 2 J/ W x bρ J/ W J/ W θ x = 2 J/ W 2 W σρ 2 J/ W x b J/ W J/ W once again, noe ha under iso-elasic uiliy 2 J/ W 2 W = 1 γ, J/ W 2 J/ W x J/ W = 0 α r is he excess reurn ha makes he represenaive invesor indifferen beween buying and selling he coningen claim The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 35

36 θ η and θ x relae he excess reurn o he amoun of risk aken in β η and β x jus like he equilibrium risk-free rae, he marke prices of facors risks are independen of he aggregae wealh in he economy equilibrium prices of derivaives whose payoffs are independen of wealh are hus independen of W hemselves as shown earlier, for he special case of logarihmic uiliy (γ = 0), he RRA is consan a one and he marke prices of risk simplify o θ η = σ(x, ), θ x = σ(x, )ρ he marke prices of risk do no only depend on he invesor s uiliy funcion bu also on he dynamics of he producion opporuniy and he sae variable as well as heir correlaion The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 36

37 Fundamenal Valuaion Equaion by Iô s lemma, he differenial of a coningen claim P (x, ) wih wealh-independen payoff akes he form dp = P = P d + dx P x 2 x 2 (dx ) 2 ( P + P a(x, ) P b 2 (x, ) x 2 x 2 ) d + P x b(x, )dζ using he previous, he insananeous drif α(x, ) and he insananeous diffusion β x (x, ) are α(x, ) = 1 ( P P + P a(x, ) + 1 x 2 β x (x, ) = 1 P ( P x b(x, ) ) ) 2 P b 2 (x, ) x 2 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 37

38 he coningen claim s dynamics can hen be expressed as dp = α(x, )P d + β x (x, )P dζ plugging hese expressions for α(x, ) and β(x, ) ino he equaion for he equilibrium excess reurn yields he fundamenal valuaion equaion α(x, ) r = θ x β x (x, ) 1 ( P P + P a(x, ) + 1 ) 2 P b 2 (x, ) x 2 x 2 r = σ(x, ) P ( ) P b(x, ) x P + P [a(x, ) σ(x, )b(x, )ρ] P b 2 (x, ) = rp x 2 x 2 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 38

39 Valuaion of Zero-Bonds le P (x,, T ) be he ime price of a defaul-free zero bond ha pays one uni of he consumpion good in T noe ha he payoff of he zero bond is independen of wealh is equilibrium dynamics can be described by he fundamenal valuaion equaion when uiliy is logarihmic P + P [a(x, ) σ(x, )b(x, )ρ] P b 2 (x x 2 x 2, ) = r P we replace he general drif and diffusion erms by he concree choice ha has been made o derive he CIR process (a(x, ) = a 0 + a 1 x, b(x, ) = b 0 x σ(x, ) = ˆσ x ) o ge P + P (a 0 + a 1 x ˆσb 0 ρx ) P b 2 x 2 x 2 0 x = r P and The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 39

40 since he equilibrium ineres rae is r = (ˆµ ˆσ 2) x, we can apply a change of variable from P (x,, T ) o P (r,, T ) where P = P ( ˆµ ˆσ 2), x r 2 P x 2 = 2 P r 2 (ˆµ ˆσ 2) 2 and obain P + P r ( a 0 ( ˆµ ˆσ 2) + a 1 r ˆσb 0 ρr ) P x 2 b 2 0 ( ˆµ ˆσ 2) r = r P finally, using he noaion of he ineres rae dynamics and seing ψ = ˆσb 0 ρ yields P + P (κ ( r r ) ψr ) P r 2 x 2 σr = r P ψr is he covariance of ineres rae changes wih he proporional change in opimally invesed wealh (he ineres rae s bea ) The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 40

41 he boundary condiion for he zero bond is P (r, T, T ) = 1 he CIR process falls ino he general class of affine erm srucure models where he drif and he squared diffusion erm are affine funcions of r here exiss a closed form soluion for zero bond prices under affine erm srucure models and for he case of he CIR model i is P (r,, T ) = A(τ)e B(τ)r, τ = T where A(τ) = B(τ) = θ = [ 2θe (θ+κ+ψ)τ 2 (θ + κ + ψ) (e θτ 1) + 2θ 2 ( e θτ 1 ) (θ + κ + ψ) (e θτ 1) + 2θ (κ + ψ) σ 2 ] 2κ r/ σ 2 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 41

42 Comparaive Saics parameer sensiiviy of P (r,, T ) r τ θ κ, r > θ κ, r < θ ψ σ 2 (convex) (convex) (concave) (convex) (concave) (concave) The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 42

43 Term Srucure of Ineres Raes for differen r yield [% p.a.] mauriy [years] Figure 2: Term srucures of ineres raes for r 0 {0.025, 0.050,..., 0.175}, κ = 1.0, r = 0.10, σ = The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 43

44 Term Srucure of Ineres Raes for differen κ yield [% p.a.] mauriy [years] Figure 3: Term srucures of ineres raes for r 0 = 0.05, κ {0.5, 1.0, 1.5, 2.0}, r = 0.10, σ = The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 44

45 VII) In Conras: he Arbirage Approach he dynamics of he sae variable x and he risk-free rae are given exogenously derivaive insrumens are priced relaive o given marke prices he marke price of risk (MPR) relaes he excess reurn o he amoun of risk aken and is deermined exogenously α r = θ x (x, )β x if no exogenous prices are available ha can be used o deermine he MPR, one has o impose assumpions abou is funcional form closing he model by choosing a cerain funcion form of θ x migh resul in inernal inconsisencies and a model ha is no free of arbirage The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 45

46 he equilibrium approach deermines he MPR endogenously wihin he specific model of he economy no all funcional forms of θ x can be obained wihin an equilibrium model bu all endogenously deermined MPR yield a pricing model ha is free of arbirage The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 46

47 Thank you! - Quesions? The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 47

48 Appendix A - Derivaion of he firs wo Momens of he CIR Process we noe ha here is negaive geomeric drif proporional o κ and hus se X = f(, x) = e κ x o obain df(, x) = f (, r )d + f x (, r )dr f xx(, r ) (dr ) 2 = κe κ r d + e κ κ ( r r ) d + e κ σ r dζ = e κ κ rd + e κ σ r dζ The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 48

49 inegraing boh sides yields e κ r = e κs r s + κ r = e κs r s + r s e κτ dτ + σ ( e κ e κs) + σ since an Iô inegral has zero expeced value, we ge s s rτ dζ rτ dζ E [r F s ] = r s e κ( s) + r (1 e κ( s)) we ge he following limiing relaionships for he mean-reversion facor κ lim E [r F s ] = r, κ lim κ 0 E [r F s ] = r s noe ha dx as compued above can be expressed as dx = e κ κ rd + e κ 2 σ X dζ The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 49

50 applying he Iô formula o compue d (X ) 2 yields d (X ) 2 = 2X dx + (dx ) 2 = 2e κ κ rx d + 2e κ 2 σx X dζ + e κ σ 2 X d inegraing boh sides gives us X 2 = X 2 s + ( 2κ r + σ 2) s e κτ X τ dτ + 2 σ when aking he expeced value, he Iô inegrals drops ou again s e κτ 2 X τ Xτ dζ τ E [ ] X 2 F s = X 2 s + ( 2κ r + σ 2) = X 2 s + ( 2κ r + σ 2) s s e κτ E [X τ F s ] dτ e κτ [r s e κs + r (e κτ e κs )] dτ The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 50

51 solving he inegral gives a closed form soluion for he expeced value of X 2 E [ ] X 2 F s back subsiuion of E [ r 2 E [ ] r 2 F s = X 2 + σ2 s +2κ r e κs (r s + r ) κ ] Fs = e 2κ E [ X 2 = e 2κ( s) r s + 2κ r + σ2 κ (e κ e κs) 2κ r + σ2 + 2κ Fs ] yields 2κ r + σ2 + r (1 e 2κ( s)) 2κ he variance of r can hen be compues as Var [r F s ] = E [ r 2 Var [r F s ] = σ2 κ r s (e κ( s) e 2κ( s)) + r σ2 2κ r (e 2κ e 2κs) (r s + r ) (e κ( s) e 2κ( s)) Fs ] (E [r F s ]) 2 (1 e κ( s)) 2 The Cox-Ingersoll-Ross Model - Mahias Thul, Ally Quan Zhang 51