Basic Economic cenario Generaor: echnical pecicaions Jean-Charles CROIX IFA - Universié Lyon 1 January 1, 13
Conens Inroducion 1 1 Risk facors models 3 1.1 Convenions............................................ 3 1. hor rae............................................. 3 1..1 Vasicek exended model................................. 3 1.. Discreizaion....................................... 4 1..3 Forward rae exracion................................. 4 1.3 ock and Real-esae...................................... 4 1.3.1 Black & choles model under sochasic shor rae.................. 5 1.3. Discreizaion....................................... 5 1.4 Credi and liquidiy risks.................................... 5 1.4.1 Discreizaion....................................... 5 Asses valuaion 6.1 Basic asses............................................ 6.1.1 Defaul-free zero-coupon bond............................. 6.1. Defaul-free ordinary bond............................... 6. Defaul-able asses........................................ 7..1 Corporae bond...................................... 7.. Credi Defaul wap................................... 8.3 Derivaives............................................ 8.3.1 hor-rae European derivaive............................. 9.3. ock European derivaive................................ 9.3.3 Converible bond..................................... 9 Bibliography 1 A echnical annex 11 A.1 uppor funcions......................................... 11 A. Heah, Jarrow and Moron HJM approach.......................... 11 A.3 European call on zero-coupon bonds.............................. 1 A.3.1 No free-lunch valuaion................................. 1 A.3. P, s law under -forward measure......................... 13 A.3.3 European pu on zero-coupon bonds.......................... 14 A.3.4 Vasicek exended model applicaion.......................... 14 A.4 European call on sock...................................... 15 A.4.1 No-free lunch principle.................................. 15 A.4. European Pu on sock................................. 16 A.4.3 Vasicek exended model applicaion.......................... 17 1
Inroducion his documen presens all models and asses included in he basic economic scenario generaor. Each chaper conains closed formulas and coding specicaions associaed discreizaion mehods and conaining le.
Chaper 1 Risk facors models In his rs chaper, dieren risk facors and heir respecive models are presened. Each model is heoreically deailed wih specic coding mehods when i's needed. In his economic scenario generaor EG, all risk facors are modeled under he risk-neural measure. he shor rae is sochasic and is diusion process follows he Vasicek Exended model also known as Hull-Whie model. ocks and real-esae asses are driven hrough Black-choles wih sochasic shor rae model. Defaul and liquidiy spreads are modeled under he Longsa Mihal and Neis LMN model which consider defaul as Poisson process incremen. No correlaion are considered excep a linear relaion beween socks and zero-coupon bonds. 1.1 Convenions In order o simplify his documen's reading, a few convenions are assumed: f, is he forward rae beginning in and nishing in, r is he shor rae in, P, is he zero-coupon bond price in giving a uni in, ɛ is a sandard gaussian random variable. 1. hor rae Under olvency, economic scenario generaors have o ake he ineres rae erm srucure as an inpu for shor rae calibraion. his consrain is respeced in he Vasicek exended model, which is a direc applicaion of Heah, Jarrow and Moron HJM framework see he annex of his documen for more informaion. his simple model is convenien as i provides closed formula for bonds see asses valuaion chaper. In order o clarify formulas, hese addiional convenions are added: K = 1 e k, k L = σ k 1 e k. 1.1 1..1 Vasicek exended model Vasicek exended model is derived from he HJM approach cf. echnical annex wih a diusion process for he forward rae as follows: df, = µ f, d + σ f, dw 1 1. 3
wih W 1 a sandard brownian moion under he risk-neural measure. In his paricular model, he volailiy srucure is specied in order o exponenially decrease wih mauriy: σ f, = σe k 1.3 From his expression, one can derive he forward rae's rend cf echnical annex: µ f, = σ f, σ f, udu = σ e k e k 1.4 k Now ha he rend and he volailiy srucure are known, inegraion of equaion 1. beween and provides: f, = f, + f, = f, + µ f s, ds + σ k σ f s, dŵ 1 s, e k s e k s ds + In which one can immediaely derive he shor rae: σe K s dŵ 1 s. 1.5 r = f, 1.6 1.. Discreizaion Numerical implemenaion requires discree equaions. A discree version of equaion 1.6 is obained hereby: r +δ e kδ r = f, + δ e kδ f, + σ K + δ e kδ K +δ + σe kδ e k s dŵ s 1 1.7 where δ is he discreizaion sep. Finally, he shor rae is coded in he le Mehods_PahsGeneraion.R as follows: r +δ = e kδ r + f, + δ e kδ f, + σ 1..3 Forward rae exracion K + δ e kδ K + Lδɛ 1.8 As previously menioned, Vasicek exend model requires he forward rae srucure f, as an inpu. he approach used in his EG is based on he equaion 1.9: R, + δ + δ R, f, = lim δ δ which is compued as follow in Funcions_ForwardExracion.R: 1.9 f, = R, + 1 1 + 1 1 R, 1 1.1 1 he iniial erm srucure is exraced from he le ZCraes.csv obained from he French Insiue of Acuaries, cf User's guide for more informaion conaining zero-coupon raes a a monhly frequency for he nex 4 years. 1.3 ock and Real-esae ock and real-esae asses are modeled under an exended version of Black & choles under sochasic shor rae [1]. 4
1.3.1 Black & choles model under sochasic shor rae In his EG, sock and real-esae follow he same diusion process: d = r d + ρσ dw, d = r d + ρσ dŵ 1 + 1 ρ σ dŵ 1.11 where Ŵ 1 and Ŵ are independen brownian moions obained by Cholesky decomposiion. Iô's lemma drives o he soluion: = exp r u du σ + ρσ Ŵ 1 + 1 ρ σ Ŵ 1.1 1.3. Discreizaion In order o numerically generae pahs for, equaion 1.1 needs o be discreized and is coded in Mehods_PahsGeneraion.R as follows: +δ = exp r +δ σ δ + ρσ δɛ1 + 1 ρ σ δɛ Noe ha his discreizaion doesn' creae any bias. 1.4 Credi and liquidiy risks 1.13 In his EG, credi and liquidiy risks are considered in order o include defaul-able bonds and credi defaul swaps CD. he LMN model considers credi and liquidiy even as he resul of a Poisson process wih sochasic inensiies [4]. he credi inensiy λ is driven hrough he CIR model Cox, Ingersoll e Ross [] wich impose posiiviy, mean aracion and heeroscedasiciy: dλ = α βλd + σ λ λdŵ 4 1.14 where α,β and σ λ are posiive real numbers. Liquidiy risk inensiy is modeled as a simple whie noise: dγ = ηdŵ 3 1.15 where η is a posiive real number. Each noaion grade AAA, AA, ec. is hen modeled by dieren coecien values. 1.4.1 Discreizaion he wo inensiy processes need o be discreized in order o be numerically generaed in he EG and are coded in Mehods_PahsGeneraion.R. Discreizaion is direc and exac for liquidiy: γ +δ = γ + η δɛ 1.16 Whereas defaul inensiy requires he use of he Milsein scheme which is a second order Iô-aylor developmen: λ +δ = λ + αβ λ δ + σ λ λ δɛ + σ λ 4 δɛ 1 1.17 5
Chaper Asses valuaion In his chaper, muliple producs are inroduced and priced according o models presened in chaper 1. he discouning facor is dene hereby:.1 Basic asses δ = exp r u du In his secion, defaul-free zero-coupon bonds and defaul-free ordinary bonds are presened..1.1 Defaul-free zero-coupon bond Zero-coupon prices are direcly derived from he expression of he forward rae as follows: δ P, = E = exp f, udu δ As he forward rae dynamic is governed by Vasicek exended model, he zero-coupon price is compued in Mehods_PriceDisribuion_ZCBond hereby: P, = P, P, exp K L + K f, r his las formula requires he prices P, and P,. hese wo values are obained direcly from he zero-coupon rae R, in ZCraes.csv as follows:.1..3 P, = exp R,.4 In he le ZCraes.csv, zero-coupon raes are given monhly. In order o exend he possible mauriies in he EG, a linear exrapolaion is realized for zero-coupon bond prices: wih δ [, 1 1 ]. P, + δ = P, + δ P, + 1 1 P, 1 1.5.1. Defaul-free ordinary bond Defaul-free bonds can be considered as a sum of zero-coupon bonds wih dieren mauriies yearly coupons in his example: P, = P, + c 6 i=+1 P, i.6
where c is he coupon rae. In he EG, mauriies beween coupon daes are modeled and i's necessary o add coupon's ime value o he clean price o ge he bond diry price: Diry price = Clean price + Coupon's ime value.7 he coupon's ime value is he nex coupon a a proraa emporis value. Considering c he bond coupon rae, n c he number of coupons each year, he mauriy and he valuaion dae, he clean price is compued as follows: β 1 C p c, n c,, = P, + c P, α i i= β = n c + 1 α i = i n c where α i is he β i-h coupon's dae and β he number of remaining coupons. In his EG, mauriy is considered o be equal o he las coupon dae o avoid addiional echnical diculies. Coupon's ime value is compued hereby: C c c, n c,, = c α β + 3.9 36 3 where 36 represens he marke 3 days of delay. Finally, he diry price is coded as follows in Mehods_PriceDisribuion_Bond.R:. Defaul-able asses.8 Oc, n c,, = C p c, n c,, + C c c, n c,,.1 In his secion, all defaul-able asses are valuaed under chaper 1 diusions...1 Corporae bond In he LMN model [4], corporae bonds are valuaed hereby: CBc, ω, = c Au expbuλ CuP, u exp γ udu + A expb λ C P, exp γ + 1 ω expbuλ CuP, ugu + λ Hu exp γ udu.11 where funcions A,B,C,G and H are deailed in annex and ω is he par sill paid in case of defaul. Considering discree coupons and changing he dae of valuaion, clean price formula is derived from equaion.11 hereby: β 1 CB p c, n c,,, ω = c Aα i expbα i λ Cα i P, α i exp γ α i i= + A expb λ C P, exp γ + 1 ω expbuλ CuP, ugu + λ Hu exp γ udu.1 ame noaions and behavior are used for he coupon's ime value nor defaul nor liquidiy spread considered. Finally, he marke value for a corporae bond is given by he following formula: 7
β 1 CB p c, n c,,, ω = c Aα i expbα i λ Cα i P, α i exp γ α i Discreizaion i= + A expb λ C P, exp γ + 1 ω expbuλ CuP, ugu + λ Hu exp γ udu + C c ρ, n c,,.13 In he EG, corporae bonds are coded in he le Mehods_PriceDisribuion_CorporaeBond.R. Discreizaion is required for inegral calculus wih n seps and δ = n. Considering ha he coupons are discree and no coninuous, he valuaion formula becomes: β 1 CBc, n c,,, ω = c Aα i expbα i λ Cα i P, α i exp γ α i i= + A expb λ C P, exp γ n + 1 ω δ expbiδλ CiδP, + iδgiδ + λ Hiδ exp γ iδ i=1 + C c c, n c,,.14.. Credi Defaul wap Credi defaul swap is valuaed hrough he LMN model [4]. he premium s is supposed coninuous and is he fair value beween he insurer and he proecion buyer. he valuaion formula a iniial dae is presened hereby: sω,, = ω expbλ P, G + Hλ d A expbλ P, d In he EG, CD don' necessarily sar a dae, hus he previous formula is adaped as follows:.15 sω,, = ω expbuλ P, ugu + Huλ du Au expbuλ P, udu.16 Discreizaion he CD coninuous premium is compued in he le Mehods_PriceDisribuion_CDPremium.R. Formula.16 requires discreizaion for he inegrals calculus wih n seps δ = n : sω,, = ω n i=1 δ expbiδλp,+iδgiδ+hiδλ n i=1 δaiδ expbiδλp,+iδ.17.3 Derivaives In his secion, vanilla derivaives producs are valuaed. 8
.3.1 hor-rae European derivaive In his secion, European calls and pus on zero-coupon bonds are valuaed. he price of a call on a zero-coupon bond saring a wih mauriy s a dae is valuaed hereby see annex for deails: C,, s, K = P, snd 1 KP, Nd, 1 P, s d 1 = H, s ln H, s +, P, K d = d 1 H, s, σ H, s = e k 3 ks 1 e ks e k.18 wih < s. his produc is coded in he le Mehods_PriceDisribuion_EuroCallPu_ZC.R. Pu prices are derived hrough he Call-Pu relaion see annex for deails..3. ock European derivaive In his secion, European calls and pus on sock are valuaed. he call price a dae wih mauriy and srike K is compued hereby: C,, K = Nd 1 KP, Nd, ln P, K + 1 τ d 1 =, τ d = d 1 τ, σ τ = k + ρσσ σ + σ + k k ρσσ K L k K..19 his formula is deailed in annex and can be found in [3]..3.3 Converible bond A converible bond is composed of a bond non-defaul-able in his EG and a call on sock wih same mauriy and srike equal o he bond principal. he valuaion is compued from his equaion: where α is an adjusmen parameer. OCc,, = coρ,, + C,, 1. 9
Bibliography [1] choles M. Black F. he pricing of opions and corporae liabiliies. Journal of Poliical Economy, 813:637654, 1973. [] Ross. Cox J., Ingersoll J. A heory of he erm srucure of ineres raes. Economerica, 53:38548, 1985. [3] Quiard-Pinon F. Mahémaiques nancières. EM,. [4] Neis E. Longsa F., Mihal. Corporae yield spreads: Defaul risk or liquidiy? new evidence from he credi-defaul swap marke. NBER Working paper, 1418, April 4. 1
Appendix A echnical annex his echnical annex presens deailed demonsraion for risk facor models and asses valuaion. A.1 uppor funcions his secion deails funcions used for CD and corporae bonds valuaion: φ = σ + β κ = β + φ β φ αβ + φ 1 κ A = exp σ 1 κe φ B = β φ φ σ + σ 1 κe φ η 3 C = exp 6 e φ 1 exp αβ + φ α σ 1 κ 1 κe φ G = α φ σ αβ + φ + φσ 1 κ H = exp σ 1 κe φ α σ + α σ +1 A.1 A. Heah, Jarrow and Moron HJM approach In heir aricle, Heah, Jarrow and Moron esablish a mehod o model ineres raes erm srucure hrough he no-free lunch hypohesis. o do so, hey suppose he following diusion process for he forward rae: df, = µ f, d + σ f, dŵ 1 wih W a sandard brownian moion under he risk-neural probabiliy. In his framework, auhors suppose a Black & choles ype diusion for he zero-coupon bond: dp, P, = r d σ P, dŵ 1 his relaion leads o a link beween he rend and he volailiy srucure of he forward rae. Indeed, a zero-coupon bond is valuaed as follows: A. A.3 11
hen, hrough he Leibniz rule: P, = exp lnp, = f, udu. f, udu, A.4 d lnp, = f, d df, udu, = r d µ f, ud + σ f, udŵ 1 du, = r µ f, udu d σ f, udu dŵ 1. A.5 he Iô's lemma gives us: 1 d lnp, = P, dp, 1 d < P, >, P, = r 1 σ P, d σ P, dŵ 1 A.6 which leads o he following equaliies by idenicaion: hus: µ f, udu = 1 σ P,, σ f, udu = σ P, A.7 Finally, µ f, udu = 1 σ f, udu A.8 A.3 European call on zero-coupon bonds µ f, = σ f, σ f, udu A.9 In his secion, he valuaion of European zero-coupon bonds is deailed ino he Vasicek exended model framework. A.3.1 No free-lunch valuaion Under his principle, wih s >, he call price becomes: C,, K = E Q P, s K + δ Wih he -forward probabiliy associaed o he following Radon-Nykodin densiy: A.1 1
I comes wih a = {P, s K}: dq dq = δ P, A.11 C,, K = E Q P, s K + δ = P, E Q P, s K + = P, E Q P, s1 a KQ a A.1 A.3. P, s law under -forward measure Using Iô's lemma in A.5, i comes: P, s = P, s exp r u du σ P u, sdŵ u 1 1 σp u, sdu A.13 o bring back his equaion under he -forward measure, we use he following Radon-Nykodin measure: dq dq = P, P, δ = exp r u du σ P u, sdŵ u 1 1 σp u, sdu δ = exp σ P u, sdŵ 1 u 1 σ P u, sdu A.14 hen, Girsanov heorem idenies a -brownian moion W : Observing hese wo relaions: d W u = dŵu + σ P u, sdu A.15 P, s = P, s exp r u du σ P u, sdŵ u 1 1 σp u, sdu P, = P, exp r u du σ P u, dŵ u 1 1 A.16 σp u, du As P, = 1, he rs line can be rewrien using he second one: P, s = P, s = P, s = Wih: P, s P, exp P, s P, exp P, s P, exp [σ P u, s σ P u, ]dŵ 1 u 1 [σ P u, s σ P u, ]d W u 1 σ P u, sdu 1 [σ P u, s σ P u, ]d W u 1 1 [σ P u, s σ P u, ]du [σ P u, s σ P u, ] du [σ P u, s σ P u, ]du A.17 13
W = H = [σ P u, s σ P u, ]dz u 1 [σ P u, s σ P u, ] du [σ P s, σ P, ] du A.18 One can observe ha W hen: is disribued according o N H, H under he -forward measure. P, s = a can be rewrien as {expw K P, P,s }, so: P, s P, expw C,, K = P, snd 1 KP, Nd d 1 = 1 P, s H ln P, K + H d = d 1 H A.19 A. A.3.3 European pu on zero-coupon bonds Call-Pu pariy applies o european opions: he pu expression derives direcly from A.: + P u = Call + KP, A.1 P u,, K = Call,, K P, s + KP, = P, snd 1 1 + KP, 1 Nd = KP, N d P, sn d 1 A. A.3.4 Vasicek exended model applicaion In his model, he volailiy srucure is he following: σ P, = H compuaion becomes: σe ku du = σ k 1 e k = σk A.3 H = = [σ P u, s σ P u, ] du σp u, sdu σ P u, sσ P u, du + σp u, du = A B + C where A,B and C are deailed hereby: A = σ k k e ks e ks + 1 B = σ k C = σ k k e ks e ks 1 k 1 e k 1 k e ks e ks + 1 k 1 e k + 1 k 1 e k k e ks e ks+ A.4 A.5 14
And nally: σ H = k e ks 1 e ks e k 3 A.6 A.4 European call on sock As a reminder, hese diusion are considered: dp P = r d σ P, dŵ 1 d = r d + ρσ dŵ 1 + 1 ρ σ dŵ dŵ 1 dŵ = A.7 A.4.1 No-free lunch principle he sandard valuaion mehod under no-free lunch hypohesis is used: C,, K = E Q K + δ = E Q δ 1 b E Q Kδ 1 b = Q b KP, Q b A.8 Wriing b = { K}, s erm becomes: Abou he second one: A = E Q δ, 1 b = E Q δ 1 b = E Q 1 b = Q b A.9 B = E Q δ 1 b = E Q P, δ P, P, 1 b = P, Q b A.3 Now, he objecive is o evaluae he probabiliy of even b under Q and Q measures. Iô's lemma gives: = exp r u du exp σ du + ρσ dŵ u 1 + 1 ρ σ dŵ u A.31 he Radon-Nykodin measure is: dq dq = δ = exp σ du + ρσ dŵ 1 u + 1 ρ σ dŵ u A.3 Muli-dimensional Girsanov's heorem shows wo brownian moions under Q :W 1 and W as follows: W 1 W = Ŵ 1 ρσ = Ŵ 1 ρ σ A.33 15
hus: [ ] P d = P d d [ ] P = dp [ 1 P + d P [ ] 1 + 1 dp + d < P, 1 > ] + ermes en d A.34 By Girsanov heorem, diusion coecien don' change and prices in he new numeraire are Q maringales: d [ ] P = [σ P, + ρσ ]d W u 1 σ 1 ρ d W u A.35 P hen, a change of ime uni gives: hus: τ = [σ P u, + σ ρσ P u, + σ ]du A.36 Which leads o: And nally: P, = P, exp Bτ 1 τ 1 = P, exp Bτ 1 τ 1 Q K = Q 1 K ln = Q ɛ + 1 τ τ P, K A.37 A.38 A.39 he compuaion is similar for B: = Nd 1 C,, K = Nd 1 KP, Nd, ln P, K + 1 τ d 1 =, τ A.4 d = d 1 τ A.4. European Pu on sock he Call-Pu pariy applies for european opions: he pu price is direcly derived: + P u = Call + KP, A.41 P u,, K = Call,, K + KP, = Nd 1 1 + KP, 1 Nd = KP, N d N d 1 A.4 16
A.4.3 Vasicek exended model applicaion Direc applicaion of previous formula gives: Which leads for he EG: τ = τ = σ K u + ρσ σk u + σ du σ k + ρσσ k + σ + σ k ρσσ k K L K A.43 A.44 17