Quantitative Finance and Investments Advanced Formula Sheet. Fall 2017/Spring 2018
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- Χαρά Αλαφούζος
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1 Quanave Fnance and Invesmens Advanced Formula Shee Fall 2017/Sprng 2018 Mornng and afernoon exam bookles wll nclude a formula package dencal o he one aached o hs sudy noe. The exam commee beleves ha by provdng many key formulas, canddaes wll be able o focus more of her exam preparaon me on he applcaon of he formulas and conceps o demonsrae her undersandng of he syllabus maeral and less me on he memorzaon of he formulas. The formula shee was developed sequenally by revewng he syllabus maeral for each major syllabus opc. Canddaes should be able o follow he flow of he formula package easly. We recommend ha canddaes use he formula package concurrenly wh he syllabus maeral. No every formula n he syllabus s n he formula package. Canddaes are responsble for all formulas on he syllabus, ncludng hose no on he formula shee. Canddaes should carefully observe he somemes suble dfferences n formulas and her applcaon o slghly dfferen suaons. Canddaes wll be expeced o recognze he correc formula o apply n a specfc suaon of an exam queson. Canddaes wll noe ha he formula package does no generally provde names or defnons of he formula or symbols used n he formula. Wh he wde varey of references and auhors of he syllabus, canddaes should recognze ha he leer convenons and use of symbols may vary from one par of he syllabus o anoher and hus from one formula o anoher. We rus ha you wll fnd he ncluson of he formula package o be a valuable sudy ade ha wll allow for more of your preparaon me o be spen on maserng he learnng objecves and learnng oucomes. 1
2 Ineres Rae Models - Theory and Pracce, Brgo and Mercuro Chaper 3 Table 3.1 Summary of nsananeous shor rae models Model Dynamcs r > 0 r AB AO V dr = kθ r d + σdw N N Y Y CIR dr = kθ r d + σ r dw Y NCχ 2 Y Y D dr = ar d + σr dw Y LN Y N EV dr = r η a ln r d + σr dw Y LN N N HW dr = kθ r d + σdw N N Y Y BK dr = r η a ln r d + σr dw Y LN N N MM dr = r η λ γ 1+γ ln r d + σr dw Y LN N N CIR++ r = x + ϕ, dx = kθ x d + σ x dw Y* SNCχ 2 Y Y EEV r = x + ϕ, dx = x η a ln x d + σx dw Y* SLN N N *raes are posve under suable condons for he deermnsc funcon ϕ. 3.5 dr = kθ rd + σdw, r0 = r r = rse k s + θ 1 e k s + σ s e k u dw u 3.7 E {r F s } = rse k s + θ 1 e k s Var{r F s } = σ2 1 e 2k s 2k B,T r 3.8 P, T = A, T e 3.9 dr = kθ B, T σ 2 krd + σdw T 3.11 dr = kθ k + λσrd + σdw 0, r0 = r dr = b ard + σdw r = rse a s + b 1 e a s + σ s a e a u dw 0 u 3.14 ˆα = n n r r 1 n r n r 1 n n r2 1 n r 1 2 n 3.15 ˆβ = r ˆαr 1 n1 ˆα 3.16 V 2 = 1 n r ˆαr 1 n ˆβ1 2 ˆα 3.19 E {r F s } = rse a s and Var{r F s } = r 2 se e 2a s σ2 s P, T = rp π 2 0 sn2 r snh y 0 fz snyzdzdy + 2 Γ2p rp K 2p 2 r 3.21 dr = kθ rd + σ rdw, r0 = r dr = kθ k + λσrd + σ rdw 0, r0 = r 0 2
3 3.23 E {r F s } = rse k s + θ 1 e k s Var{r F s } = rs σ2 e k s e 2k s + θ σ2 1 e k s 2 k 2k 3.24 B,T r P, T = A, T e kθ/σ 2 2h exp {k + ht /2} A, T = 2h + k + hexp {T h} 1 B, T = 2exp{T h} 1 2h + k + hexp {T h} 1, h = k 2 + 2σ dr = kθ k + B, T σ 2 rd + σ rdw T 3.28 p T r rs x = p χ 2 υ,δ,s/q,sx = q, sp χ 2 υ,δ,sq, sx q, s = 2ρ s + ψ + B, T and δ, s = 4ρ s2 rse h s q, s B,T r Page 68 R, T = α, T + β, T r, P, T = A, T e 3.29 σ f, T = B, T σ, r T Page 69 dr = b, rd + σ, rdw b, x = λx + η, σ 2, x = γx + δ B, T + λb, T 1 2 γb, T = 0, BT, T = 0 ln A, T ηb, T δb, T 2 = 0, AT, T = 1 Page 69/70 Vascek λ = k, η = kθ, γ = 0, δ = σ 2 Page 70 CIR λ = k, η = kθ, γ = σ 2, δ = 0 Page 71 bx = λx + η, σ 2 x = γx + δ θ lm E{r F s } = exp a + σ2 4a 2θ a + σ2 2a 3.31 lm Var{r F s } = exp exp 3.32 dr = ϑ ard + σdw 3.33 dr = ϑ ard + σdw σ 2 2a ϑ = f M 0, + af M 0, + σ2 T 2a 1 e 2a 3.35 r = rse a s + s e a u ϑudu + σ s e a u dw u = rse a s + α αse a s + σ s e a u dw u 3.36 where α = f M 0, + σ2 2a 2 1 e a 2 3
4 3.37 E{r F s } = rse a s + α αse a s Var{r F s } = σ2 1 e 2a s 2a 3.38 dx = axd + σdw, x0 = 0 Page 74 Page 74 Page 75 x = xse a s + σ s e a u dw u α Q{r < 0} = Φ σ 2 2α 1 e 2α T rudu F N B, T r α + ln P M 0, + 1V 0, T V 0,, V, T P M 0,T 2 where B, T = 1 a 1 e at and V, T = σ2 a T a e at 1 2a e 2aT 3 2a B,T r 3.39 P, T = A, T e where A, T = P M 0,T P M 0, exp { B, T f M 0, σ2 4a 1 e 2a B, T 2 } 3.40 ZBC, T, S, X = P, SΦh XP, T Φh σ p 1 e 2aT where σ p = σ BT, S and h = 1 2a σ p ln P,S + σp P,T X ZBP, T, S, X = XP, T Φ h + σ p P, SΦ h 3.42 Cap, T, N, X = N n 1 + Xτ 1 ZBP, 1,, 1+Xτ or Cap, T, N, X = N n P, 1Φ h + σp 1 + Xτ P, Φ h, where σp 1 e = σ 2a 1 B 2a 1, and h = 1 ln P, 1+Xτ σp P, 1 + σ p Flr, T, N, X = N n 1 + Xτ P, Φh P, 1 Φh σp 3.44 CBO, T, T, c, X = n c ZBO, T, T, X 3.45 PS, T, T, N, X = N n c ZBP, T,, X 3.46 RS, T, T, N, X = N n c ZBC, T,, X 3.47 E{x +1 x = x,j } = x,j e a =: M,j Var{x +1 x = x,j } = σ2 1 e 2a =: V 2 2a x = V 1 3 = σ 2a 1 e 2a 1 M,j 3.49 k =round x p u = η2 j,k + η j,k 6V 2 2, p m = 2 3V 3 η2 j,k, p 3V 2 d = η2 j,k 6V 2 2 3V 3.64 dx α = µx α ; αd + σx α ; αdw x η j,k 4
5 3.65 P x, T = Π x, T, x α ; α 3.66 r = x + ϕ; α, P, T = exp T ϕs; αds Π x, T, r ϕ; α; α 3.68 ϕ; α = ϕ ; α := f M o, f x 0, ; α 3.69 exp T ϕs; αds = Φ, T, x 0 ; α := P M 0, T Π x 0,, x 0 ; α Π x 0, T, x 0 ; α P M 0, 3.70 Π, T, r ; α = Φ, T, x 0 ; απ, T, r ϕ ; α; α 3.71 V x, T, τ, K = Ψ x, T, τ, K, x α ; α dϕ; α 3.74 dr = kθ + kϕ; α + kr d + σdw d Page 100 ϕ V AS ; α = f M 0, + e k 1 k2 θ σ 2 /2 k 2 Page 101 P, T = P M 0, T A0, exp{ B0, x 0 } P M 0, A0, T exp{ B0, T x 0 } A, T exp{ B, T r ϕ V AS ; α} σ2 2k 2 e k 1 e k x 0 e k 3.76 dx = kθ xd + σ xdw, x0 = x 0, r = x + ϕ 3.77 ϕ CIR ; α = f M 0, f CIR 0, ; α f CIR 0, ; α = h = k 2 + 2σ 2 Chaper r = x + y + ϕ, r0 = r dx = axd + σdw 1, x0 = 0 2kθexp{h} 1 2h + k + hexp{h} 1 + x 4h 2 exp{h} 0 2h + k + hexp{h} 1 2 dy = byd + ηdw 2, y0 = E{r F s } = xse a s + yse b s + ϕ Var{r F s } = σ2 1 e 2a s + η2 1 e 2b s + 2ρ ση 1 e a+b s 2a 2b a + b 4.7 r = σ 0 e a u dw 1 u + η 0 e b u dw 2 u + ϕ 4.8 dx = axd + σd W 1 dy = byd + ηρd W 1 + η 1 ρ 2 d W M, T = where dw 1 = d W 1 and dw 2 = ρd W ρ 2 d W 2 1 e at 1 e bt x + y a b 5
6 4.10 V, T = σ2 a 2 T + 2 a e at 1 2a e 2aT 3 2a + η2 T + 2 b 2 b e bt 1 2b e 2bT 3 2b +2ρ ση ab 4.11 P, T = exp T + e at 1 a { T + e bt 1 b e a+bt 1 a + b ϕudu 1 e at 1 e bt x y + 1 } a b 2 V, T 4.12 ϕt = f M 0, T + σ2 1 e at 2 η e bt 2 ση + ρ 2a 2 2b 2 ab 1 e at 1 e bt { 4.13 exp } T ϕudu = P M 0, T { P M 0, exp 12 } V 0, T V 0, 4.14 P, T = P M 0, T P M 0, exp {A, T } A, T := 1 1 e at 1 e bt V, T V 0, T + V 0, x y 2 a b 4.15 P, T = A, T exp{ Ba,, T x Bb,, T y} 4.16 σ f, T = σ 2 e 2aT + η 2 e 2bT + 2ρσηe a+bt Page 152 Covdf, T 1, df, T 2 d = σ 2 B T a,, T 1 B T a,, T 2 + η 2 B T b,, T 1 B T b,, T 2 B +ρση T a,, T 1 B T b,, T 2 + B T a,, T 2 B T b,, T 1 = σ 2 e at 1+T η 2 e bt 1+T 2 2 +ρση e at 1 bt 2 +a+b + e at 2 bt 1 +a+b Corrdf, T 1, df, T 2 = σ2 e at 1+T η 2 e bt 1+T 2 2 σ f, T 1 σ f, T 2 + ρση e at 1 bt 2 +a+b + e at 2 bt 1 +a+b σ f, T 1 σ f, T 2 6
7 Page 153 f, T 1 T 2 = ln P, T 1 ln P, T 2 T 2 T 1 df, T 1, T 2 =...d + Ba,, T 2 Ba,, T 1 σdw 1 T 2 T 1 + Bb,, T 2 Bb,, T 1 ηdw 2 T 2 T 1 σ f, T 1, T 2 = σ 2 βa,, T 1, T η 2 βb,, T 1, T ρσηβa,, T 1, T 2 βb,, T 1, T 2 where βz,, T 1, T 2 = Bz,, T 2 Bz,, T 1 T 2 T 1 Covdf, T 1, T 2, df, T 3, T 4 d σ 2 Ba,, T 2 Ba,, T 1 Ba,, T 4 Ba,, T 3 T 2 T 1 T 4 T 3 +η 2 Bb,, T 2 Bb,, T 1 Bb,, T 4 Bb,, T 3 T 2 T 1 T 4 T 3 Ba,, T2 Ba,, T 1 Bb,, T 4 Bb,, T 3 +ρση T 2 T 1 T 4 T 3 + Ba,, T 4 Ba,, T 3 Bb,, T 2 Bb,, T 1 T 4 T 3 T 2 T 1 Page 160 σ 3 = dz 3 = σ σ2 2 ā b ρ σ 1σ 2 b ā σ 1 dz 1 σ 2 ā b dz 2, σ 4 = σ 2 σ 3 ā b Page 161 a = ā, b = b, σ = σ 3, η = σ 4, ρ = σ 1 ρ σ 4 σ 3 ϕ = r 0 e ā + 0 θve ā v dv ā = a, b = b, σ1 = σ 2 + η 2 + 2ρση, σ 2 = ηa b ρ = σρ + η σ2 + η 2 + 2ρση, dϕ θ = + aϕ d Managng Cred Rsk: The Grea Challenge for Global Fnancal Markes, Caouee, e. al. Chaper R p = N X EAR 20.3 V p = N j=1 N X X j σ σ j ρ j 7
8 20.5 UAL p = N N X X j σ σ j ρ j 1 Page 403 j=1 CV arcl = EAD LGD ρφ 1 CL + Φ 1 P D Φ P D 1 ρ 1 + M 2.5 bp D 1 1.5bP D Bond-CDS Bass Handbook: Measurng, Tradng and Analysng Bass Trades, Elzalde, Docor, and Saluk Page 13, Equaon 1 S = P D 1 R Page 15, Equaon 2 F R = U AI RA + F C Page 18, Equaon 3 P V c + p BP SS = RF A Page 25, Equaon 4 BT P 1 = CN 100 R U CP F C+BN R+CR BP F C Page 25, Equaon 5 BT P 2 = BN CR BP F C CN U + CP + F C Page 43, Equaon 7 CN = BP R 100 R U BN A Survey of Behavoral Fnance, Barbers and Thaler 1 x, p : y, q = πpvx + πqvy 2 π vx where v = xα f x 0 λ x α f x < 0 and π = wp wp, wp = P γ P γ + 1 P γ 1/γ 3 D +1 D = e g D+σ D ε C +1 = e g C+σ C η +1 C ε 0 1 w N, 0 w 1 η 6 E 0 ρ C1 γ =0 1 γ C+1 γ 7 1 = ρe R +1 C,..d.over me 8 R +1 = D +1 + P +1 P = 1 + P +1/D +1 P /D D +1 D 9 r +1 = d +1 +cons. d +1 d +cons. 8
9 10 E π v1 wr f,+1 + wr E 0 ρ C1 γ 1 γ + b 0C γ ˆvX +1 =0 13 R +1 = P +1 + D +1 P 14 p d = E ρ d +1+j E 15 E 0 =0 j=0 j=0 ρ C1 γ 1 γ + b 0C γ ṽx +1, z 16 r r f = β.1 F 1 r f β,k F K r f ρ r +1+j + E lm ρ j p +j d +j +cons. j 17 r, r f, = α + β,1 F 1, r f, β,k F K, r f, + ε, 18 R f = 1 ρ eγg C+0.5γ 2 σ 2 C 19 1 = ρ 1 + f e g D γg C +0.5σD 2 +γ2 σc 2 2γσ Cσ D w f 20 R +1 = D +1 + P +1 P = 1 + P +1/D +1 P /D D +1 D = 1 + f e g D+σ D ε +1 f CAIA Level II: Advanced Core Topcs n Alernave Invesmens, Black, Chambers, Kazem Chaper P repored 16.2 P repored 16.3 P rue 16.4 P rue = α + β 0 P rue = αp rue = 1/α P repored = P repored 1 + β 1 P rue 1 + β 2 P rue α1 αp rue 1 + α1 α 2 P rue α/α P repored 1 + 1/α P repored P repored R,repored β 0 R,rue + β 1 R 1,rue + β 2 R 2,rue P repored 16.7 P repored = 1 ρp rue = 1 ρ P rue + ρp repored 1 + ρ P repored R,repored 1 ρr,rue + ρr 1,repored 16.9 R,rue = R,repored ρr 1,repored /1 ρ ˆρ = corrr,repored R 1,repored ρ,j = σ j /σ σ j R repored = α + β 1 R repored 1 + β 2 R repored β k R repored k + ε 9
10 Managng Invesmen Porfolo: A Dynamc Process, Magnn, Tule, Pno, McLeavey Chaper 8 Page 523 T RCI = CR + RR + SR Page 553 RR n, = R + R 1 + R R n /n Page 554 n DD = r, 0 2 n 1 Page 555 ARR rf Sharpe Rao = SD Page 556 ARR rf Sorno Rao = DD The Secular and Cyclc Deermnans of Capalzaon Raes: The Role of Propery Fundamenals, Macroeconc Facors, and Srucural Changes, Chervachdze, Cosello, Wheaon 1 LogC j, = a 0 + a 1 logc j, 1 + a 2 logc j, 4 + a 3 logrri j, + a 4 RT B + a 7 Q2 1.1 RRI j, s = RR j, /MRR j +a 8 Q3 + a 9 Q4 + a 10 D j 2 LogC j, = a 0 + a 1 logc j, 1 + a 2 logc j, 4 + a 3 logrri j, s + a 4 RT B 2.1 DEBT F LOW = T NBL /GDP +a 5 SP READ + a 6 DEBT F LOW + a 7 Q2 + a 8 Q3 + a 9 Q4 + a 10 D j 3 LogC j, = a 0 + a 1 logc j, 1 + a 2 logc j, 4 + a 3 logrri j, s + a 4 RT B +a 5 SP READ + a 6 DEBT F LOW + a 7 Q2 + a 8 Q3 + a 9 Q4 4 LogC j, = a 0 + a 1 yearq + a 2 logc j, 1 + a 3 logc j, 4 + a 4 logrri j, s + a 5 RT B +a 6 SP READ + a 7 DEBT F LOW + a 7 Q2 + a 8 Q3 + a 9 Q4 + a 10 D j Analyss of Fnancal Tme Seres, Tsay Chaper r = α + β 1 f β m f m + ɛ, = 1,..., T, = 1,..., k 9.2 r = α + βf + ɛ, = 1,..., T 9.3 R = α 1 T + Fβ + E 9.4 R = Gξ + E 9.5 r = α + β r m + ɛ, = 1,..., k = 1,..., T 9.11 Vary = w Σ r w, = 1,..., k 9.12 Covy, y j = w Σ r w j,, j = 1,..., k 10
11 9.13 k Varr = rσ r = k λ = k Vary 9.14 ˆΣ r ˆσ j,r = 1 T ˆρ r = Ŝ 1 ˆΣ r Ŝ 1 T =1r rr r, r = 1 T T r = r µ = βf + ɛ 9.17 Σ r = Covr = Er µr µ = Eβf + ɛ βf + ɛ = ββ + D 9.18 Covr, f = Er µf = βef f + Eɛ f = β 9.19 ˆβ ˆβj = ˆλ1 ê 1 ˆλ2 ê 2 ˆλm ê m 9.20 LRm = T k m ln ˆΣ r ln ˆβ ˆβ + ˆD Handbook of Fxed Income Secures, Fabozz Chaper Asse Allocaon s w P s w B s R B s 69 5 Secury Selecon s w P s R P s R B s α P k f P k αb k f B k = s α P k,s f P k,s s α B k,s f B k,s Chaper Asse Allocaon w P 70 2 Secor Managemen s s w P s w wb s T R B P w B s T R B w P s T R P s T R B s 70 3 Top-Level Exposure w P w B T R B 70 4 Asse Allocaon w P w P s w wb s ER B P w B s ER B 70 5 Secor Managemen s s w P s ER P s ER B s 70 6 Top-Level Exposure w P w B ER B 70 7 Ouperformance from average carry yavg P yavg B 70 8 Key rae conrbuons ω P j yj yavg P ω B j yj yavg B j 70 9 Ouperformance from avg. parallel shfs OAD P OAD B y avg Ouperformance from reshapng KRD P j KRDj B yj y avg j 11
12 70 11 Asse Allocaon OASD P w P s OASDs P wb s OASDs B OASD P OASD B s Secury Selecon s OAS s B OAS B w P s OASD P s OAS P s OAS B s Spread Duraon Msmach OASD P OASD B OAS B w P s OASDs P ws B OASDs B OAS B s Asse Allocaon s Secury Selecon s w P s OASD P s OAS P s OAS B s Inroducon o Cred Rsk Modelng, 2nd ed., Bluhm, Overbeck, Wagner Chaper 6 Page 237 M n = M n 1 Guaranees and Targe Volaly Funds, Morrson and Tadrowsk Page 4 w equy ˆσ equy = mn σarge 2 = λ ˆσ equy ˆσ equy, 100% S + 1 λ ln S Proxy Funcons for he Projecon of Varable Annuy Greeks, Clayon, Morrson, Turnbull, and Vysnasuskas Page 4 ˆV, V proxy S,, R, σ, 2 proxy S, R, σ = S V proxy S, R, σ ρ proxy S, R, σ = R V proxy S, R, σ V proxy S, R, σ = σ V proxy S, R, σ 12
13 Page 5 S sress1, S sress2, S sress3, S base, S base, S base, R sress1, R sress2, R sress3, R base, R base, R base, ˆ, proxy S,, R,, σ, 2 ˆρ, ρ proxy S,, R,, σ, 2 ˆV, V proxy S,, R,, σ, 2 ˆV base, Page 6 S, S w V proxy S,, R,, σ, 2 h S s w, h S σ sress1, σ sress2, σ sress3, σ base, σ base, σ base,, R, R, σ, σ ˆV base h R h σ, V proxy, S, S h S, R, R, σ, σ h R h σ ˆ, ˆρ, ˆV, S,, R,, σ, = 2 ˆV, sress1 ˆV, sress2 ˆV, sress3 2 ˆV base, V proxy, S,, R,, σ, base ˆV, base ˆV, base ˆV, Recen Advances n Cred Rsk Modelng, Capuano, Chan-Lau, Gasha, Mederos, Sanos, and Souo II.1 E = max0, V D ln V D + µ 1 2 σ2 T II.2 DD T = σ T II.3 II.4 II.5 II.6 II.7 x = a M + 1 a 2 Z Prob{x < x M} = q M = Φ x a M 1 a 2 p K+1 0, M = p K 0, M1 q K+1 M p K+1 l, M = p K l, M1 q K+1 M + p K l 1, Mq K+1 M, l = 1,..., K p K+1 K + 1, M = p K K, Mq K+1 M II.8 pl, = pn l, MφMdM III.1 τ = nf{ 0 V K} 13
14 Marke Models: A Gude o Fnancal Daa Analyss, Chaper 6, Aledander 6.1 P = XW 6.2 X = w 1 P 1 + w 2 P w k P k 6.3 σ K σ AT M = bk S 6.4 σ K σ AT M = w K1 P 1 + w K2 P 2 + w K3 P P, = γ, S + ε, 6.6 σ AT M = α + β S + ε 6.7 β K, = β + Σw K γ, 6.8 y = a + Pb + e 6.9 y = a + X b + e 6.10 y = c + Xd + e 6.11 rcac = rparbas rsocgen P P P P 4 Sochasc Modelng, Theory and Realy from an Acuaral Perspecve I.B-1 ds = µsd + σsdz I.B-2 ln S T Nln S 0 + r σ 2 /2T, σ T I.B-3 µ = ln S 0 + r σ 2 /2T, σ = σ T I.B-4 ĉ = 1 N c N I.B-5 c = S 0 Nd 1 Ke rt Nd 2 I.B-6 d 1 = lns 0/K + r + σ 2 /2T σ, d 2 = d 1 σ T T I.B-7 MC samplng error = 1 Sdevc N I.B-8 1 f = 2 fu 1 + fu 2 I.B-9 1 Sdev f N rdan I.B-10 I.B-11 f u g u + gu 1 Sdevfu gu N 14
15 I.B-12 h = 1 n N I.B-13 I.B-14 j=1 fv j k ˆf = x +1 x h k x +1 x Sdevhj lm of he sum should be k 1 N fz I.B-15 N gz N fz I.B-16 Sdev gz n I.B-17 S 0 = e r ps 0 u + 1 ps 0 d I.B-18 I.B-19 p = er d u d u = e σ and d = u 1 I.B-20 C 0 = e r pc u + 1 pc d I.B-21 S m = S 0 u n d m n, n = 0, 1,..., m here s an error n he book formula, he upper here s an error n he book formula, s n ha goes from 0 o m I.B-22 S m = S 0 1 ηu n d m n, n = 0, 1,..., m same error I.B-23 p = r 12 12σ 2 σ , p 0 = 2 3, p + = p + 2 6, I.B-24 I.B-25 u = e σ3, d = u 1 S log Nµ r, σ 2 r log S r S+1 S ρ Nµ ρ, σ 2 ρ I.B-26 p j = Prρ + 1 = j ρ =, = 1, 2,..., K, j = 1, 2,..., K here s an error n he book formula, y + 1 should be + 1 I.B-27 LΘ = fy 1 Θfy 2 Θ, y 1 fy 3 Θ, y 1, y 2 fy n Θ, y 1, y 2,..., y n 1 I.B-28 fρ, ρ 1, y Θ, y 1, y 2,..., y 1 for ρ = 1, 2 and ρ 1 = 1, 2 I.B-29 π, 1 pρ 1 = Θ, y 1, y 2,..., y 1 I.B-30 p j = pρ = j ρ 1 =, Θ y µ j I.B-31 g j, = fy ρ = j, Θ = φ = σ j 1 exp 1 σ j 2π 2 2 y µ j σ j 15
16 I.B-32 π, = 2 π k, 1 p k g, k=1 2 j=1 2 π, 1 p j g j, p 21 p 12 I.B-33 π 1,0 =, π 2,0 = p 12 + p 21 p 12 + p 21 I.B-34 fy 1 Θ = fρ0 = 1, y 1 Θ + fρ0 = 2, y 1 Θ y1 µ 1 y1 µ 2 = π 1,0 φ + π 2,0 φ II.A-1 S0, 1 = ln{1/1 + C0, 1} II.A-2 II.A-3 II.A-4 II.A-5 σ 1 S0, 2 = 1/2 ln{1 C0, 2 exp S0, 1/1 + C0, 2} σ 2 S0, 3 = 1/3 ln{1 C0, 3 exp S0, 1 C0, 3 exp 2S0, 2/1 + C0, 3} r = σr/ r γ φ F + T = F 1 + T + q F + T = F 1 + T + q Λ q,t +1 φ q, Eexp F 0 0 F 1 1 F 2 2 F N N = exp F 0 0 F 0 1 F 0 2 F 0 N Λ g,t +1 φ q, + Λ q,t +1 Λ q,t +1 /2 + T +1 Λ q, II.A-6 P V = Eexp F 0 0 F 1 1 F 2 2 F N NCF N II.A-7 II.A-8 F + T = F 1 + T + g Λ q,t +1 φ q, +δ arge 1/ arge F arge + T F 0 arge + T +1 δ arge F arge + T F arge + T + 1 F + T = F 1 + T + k T Λ q.t +1 φ q, + Λ q,t +1 Λ q,t +1 /2 + T +1 Λ q, II.A-9 Λ 1,j = Λ 1,1 exp aj 1, Λ >1,j = 0 II.A-10 F acor1 = F acor1 0 + ρ 12 F acor2 0 + ρ 13 F acor3 0 II.A-11 F acor2 = 1 ρ /2 F acor2 0 + ρ 13 F acor3 0 F acor3 = 1 ρ 2 13 ρ /2 F acor3 0 S = S 1 expf σ 1 φ e 1 σ 2 1/2 II.A-12 X + 1 = X exprf F RF D q II.A-13 X + = X exprf F RF D vol 2 /2 + sqr vol Z II.A-14 ds = µ S d + σ S dw II.A-15 ds = S exp{µ σ 2 /2d + σ dw } II.A-16 S = S 1 exp{µ σ 2 /2d + σ φ } 16
17 II.A-17 S = S 1 exp{f + d σ 2 /2d + σ φ } II.A-18 S = S 1 exp{f + d q σ 2 /2d + σ φ } II.A-19 II.A-20 ds = µ S d + σs, dw ds = µ S d + σs α dw II.A-21 ds = µ S d + V S dw, dv = κθ V d + v V dz, ds, V = ρd II.A-22 µ = F + d + rp σ S II.A-23 σ F = sqr 2 σ 1 S 1 1 II.A-24 mn n σ model w σ marke 2 σ model II.A-25 EValue of Equy = AValue of Asses Nd 1 II.A-26 II.A-27 II.A-28 F Face Value of Deb e rt Nd 2 d 1 = loga/f + r + σ2 A /2 T σ A T d 2 = d 1 σ A T II.A-29 Spread = II.A-30 σ E = σ A Nd 1 A E Rsk Neural Probably of Defaul = N d 2 Recovery Rae = A N d 1 N d 2 T hreshold = Φ 1 D q = h exp hτdτ 0 π 1 qτdτ τ=0 T =0 T 1 R A rqνd =0 q {u + ed + πt ut d lnh = αβ lnh d + γdz 17
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