Quantitative Finance and Investments Advanced Formula Sheet. Fall 2016/Spring 2017

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1 Quanave Fnance and Invesmens Advanced Formula Shee Fall 2016/Sprng 2017 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. Werushayouwllfndhenclusonofheformulapackageobeavaluablesudyade 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 racce, Brgo and Mercuro Chaper 3 Table 3.1 Summary of nsananeous shor rae models Model Dynamcs r > 0 r AB AO V dr = k[θ r ]d + σdw N N Y Y CIR dr = k[θ 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 = k[θ r ]d + σdw N N Y Y BK dr = r [η ³ a ln r ]d + σr dw Y LN N N MM dr = r hη λ ln r d + σr dw Y LN N N γ 1+γ CIR++ r = x + ϕ, dx = k[θ 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() =k[θ r()]d + σdw(), r(0) = r 0 (3.6) r() =r(s)e k( s) + θ 1 e k( s) + σ R s e k( u) dw (u) (3.7) E {r() F s } = r(s)e k( s) + θ 1 e k( s) Var{r() F s } = σ2 1 e 2k( s) 2k (3.8) B(,T )r() (, T )=A(, T )e (3.9) dr() =[kθ B(, T )σ 2 kr()]d + σdw T () (3.11) dr() =[kθ (k + λσ)r()]d + σdw 0 (), r(0) = r 0 (3.12) dr() =[b ar()]d + σdw 0 () (3.13) r() =r(s)e a( s) + b 1 e a( s) + σ R s a 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 ] n(1 ˆα) (3.16) V c 2 = 1 h n r ˆαr 1 n ˆβ(1 2 ˆα) (3.19) E {r() F s } = r(s)e a( s) and Var{r() F s } = r 2 (s)e ³e 2a( s) σ2 ( s) 1 (3.20) (, T )= rp R sn(2 r snh y) R f(z)sn(yz)dzdy + 2 π Γ(2p) rp K 2p (2 r) (3.21) dr() =k(θ r())d + σ p r()dw (), r(0) = r 0 (3.22) dr() =[kθ (k + λσ)r()]d + σ p r()dw 0 (), r(0) = r 0 2

3 (3.23) E {r() F s } = r(s)e k( s) + θ 1 e k( s) Var{r() F s } = r(s) σ2 e k( s) e 2k( s) + θ σ2 1 e k( s) 2 k 2k (3.24) B(,T )r() (, T )=A(, T )e (3.25) 2kθ/σ 2 2h exp {(k + h)(t )/2} A(, T )= 2h +(k + h)(exp {(T )h} 1) B(, T )= 2(exp{(T )h} 1) 2h +(k + h)(exp {(T )h} 1), h = k 2 +2σ 2 (3.27) dr() =[kθ (k + B(, T )σ 2 )r()]d + σ p r()dw T () (3.28) p T r() r(s) (x) =p χ 2 (υ,δ(,s))/q(,s)(x) =q(, s)p χ 2 (υ,δ(,s))(q(, s)x) q(, s) =2[ρ( s)+ψ + B(, T )] and δ(, s) = 4ρ( s)2 r(s)e h( s) q(, s) age 68 R(, T )=α(, T )+β(, T )r(), B(,T )r() (, T )=A(, T )e (3.29) σ f (, T )= B(, T ) σ(, r()) T age 69 dr() = b(, r())d + σ(, r())dw () b(, x) =λ()x + η(), σ 2 (, x) =γ()x + δ() B(, T )+λ()b(, T ) 1 2 γ()b(, T )2 +1=0, B(T,T)=0 [ln A(, T )] η()b(, T )+1 2 δ()b(, T )2 =0, A(T,T)=1 age 69/70 Vascek λ() = k, η() =kθ, γ() =0, δ() =σ 2 age 70 CIR λ() = k, η() =kθ, γ() =σ 2, δ() =0 b(x) =λx + η, σ 2 (x) =γx + δ µ θ age 71 lm E{r() F s } =exp a + σ2 4a µ µ 2θ (3.31) lm Var{r() F s } =exp a + σ2 σ 2 exp 1 2a 2a (3.32) dr() =[ϑ() a()r()]d + σ()dw () (3.33) dr() =[ϑ() ar()]d + σdw() (3.34) ϑ() = fm (0,) + af M (0,)+ σ2 T 2a (1 e 2a ) (3.35) r() =r(s)e a( s) + R s e a( u) ϑ(u)du + σ R s e a( u) dw (u) = r(s)e a( s) + α() α(s)e a( s) + σ R 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 } = r(s)e a( s) + α() α(s)e a( s) Var{r() F s } = σ2 1 e 2a( s) 2a (3.38) dx() = ax()d + σdw(), x(0) = 0 age 74 x() =x(s)e a( s) + σ R s e a( u) dw (u) (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 r 3 (3.48) x = V 1 3=σ 2a [1 e 2a 1 µ M,j (3.49) k =round x +1 (3.50) 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 = μ(x α ; α)d + σ(x α ; α)dw x (3.65) x (, T )=Π x (, T, x α ; α) (3.66) r = x + ϕ(; α), 0 h (3.67) (, T )=exp R T ϕ(s; α)ds Π x (, T, r ϕ(; α); α) (3.68) ϕ(; α) =ϕ (; α) :=f M (o, ) f x (0,; α) (3.69) h exp R T ϕ(s; α)ds = Φ (, T, x 0 ; α) := M (0,T) Π x (0,,x 0 ; α) Π x (0,T,x 0 ; α) M (0,) (3.70) Π(, T, r ; α) =Φ (, T, x 0 ; α)π (, 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 age 100 ϕ VAS (; α) =f M (0,)+(e k 1) k2 θ σ 2 /2 k 2 η j,k σ2 2k 2 e k (1 e k ) x 0 e k age 101 (, T )= M (0,T)A(0,)exp{ B(0,)x 0 } M (0,)A(0,T)exp{ B(0,T)x 0 } A(, T )exp{ B(, T )[r ϕ VAS (; α)]} (3.76) dx() =k(θ x())d + σ p x()dw (), x(0) = x 0, r() =x()+ϕ() (3.77) ϕ CIR (; α) =f M (0,) f CIR (0,; α) f CIR 2kθ(exp{h} 1) (0,; α) = 2h +(k + h)(exp{h} 1) + x 4h 2 exp{h} 0 [2h +(k + h)(exp{h} 1)] 2 h = k 2 +2σ 2 4

5 Chaper 4 (4.4) r = x()+y()+ϕ(), r(0) = r 0 (4.5) dx() = ax()d + σdw 1 (), x(0) = 0 dy() = by()d + ηdw 2 (), y(0) = 0 (4.6) E{r() F s } = x(s)e a( s) + y(s)e 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 R 0 e a( u) dw 1 (u)+η R 0 e b( u) dw 2 (u)+ϕ() (4.8) dx() = ax()d + σdfw 1 () dy() = by()d + ηρdfw 1 ()+η p 1 ρ 2 dfw 2 () where dw 1 () =ddfw 1 () and dw 2 () =ρdfw 1 ()+ p 1 ρ 2 dfw 2 () ) ) 1 e a(t 1 e b(t (4.9) M(, T )= x()+ y() a b (4.10) V (, T )= σ2 T + 2 a 2 a e a(t ) 1 2a e 2a(T ) 3 2a + η2 T + 2 b 2 b e b(t ) 1 2b e 2b(T ) 3 2b +2ρ ση ab (4.11) (, T )=exp T + e a(t ) 1 a ½ R T + e b(t ) 1 b ϕ(u)du 1 e a(t ) x() a e (a+b)(t ) 1 a + b ) 1 e b(t y()+ 1 ¾ b 2 V (, T ) (4.12) ϕ() =f M (0,T)+ σ2 1 e at 2 + η2 1 e bt 2 + ρ ση 2a 2 2b 2 ab (1 e at )(1 e bt ) n (4.13) exp R o T ϕ(u)du = M (0,T) ½ M (0,) exp 12 ¾ [V (0,T) V (0,)] (4.14) (, T )= M (0,T) exp {A(, T )} M (0,) A(, T ):= 1 ) ) 1 e a(t 1 e b(t [V (, T ) V (0,T)+V(0,)] x() y() 2 a b (4.15) (, T )=A(, T )exp{ B(a,, T )x() B(b,, T )y()} (4.16) σ f (, T )= p σ 2 e 2a(T ) + η 2 e 2b(T ) +2ρσηe (a+b)(t ) 5

6 age 152 Cov(df (, 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 a(t 1+T 2 2) + η 2 e b(t 1+T 2 2) +ρση e at 1 bt 2 +(a+b) + e at 2 bt 1 +(a+b) Corr(df (, T 1 ),df(, T 2 )) = σ2 e a(t 1+T 2 2) + η 2 e b(t 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 ) age 153 f(, T 1 T 2 )= ln (, T 1) ln (, T 2 ) T 2 T 1 df (, T 1,T 2 )=...d + B(a,, T 2) B(a,, T 1 ) σdw 1 () T 2 T 1 + B(b,, T 2) B(b,, T 1 ) ηdw 2 () T 2 T 1 σ f (, T 1,T 2 )= p σ 2 β(a,, T 1,T 2 ) 2 + η 2 β(b,, T 1,T 2 ) 2 +2ρσηβ(a,, T 1,T 2 )β(b,, T 1,T 2 ) where β(z,, T 1,T 2 )= B(z,, T 2) B(z,,T 1 ) T 2 T 1 Cov(df (, T 1,T 2 ),df(, T 3,T 4 )) d σ 2 B(a,, T 2) B(a,, T 1 ) B(a,, T 4 ) B(a,, T 3 ) T 2 T 1 T 4 T 3 +η 2 B(b,, T 2) B(b,, T 1 ) B(b,, T 4 ) B(b,, T 3 ) T 2 T 1 T 4 T 3 B(a,, T2 ) B(a,, T 1 ) B(b,, T 4 ) B(b,, T 3 ) +ρση T 2 T 1 T 4 T 3 + B(a,, T 4) B(a,, T 3 ) B(b,, T 2 ) B(b,, T 1 ) T 4 T 3 T 2 T 1 s age 160 σ 3 = dz 3 () = σ σ2 2 (ā b) 2 +2 ρ σ 1σ 2 b ā σ 1 dz 1 () σ 2 ā b dz 2(), σ 4 = σ 2 σ 3 ā b age 161 a =ā, b = b, σ = σ 3, η = σ 4, ρ = σ 1 ρ σ 4 σ 3 6

7 ϕ() =r 0 e ā + R 0 θ(v)e ā( v) dv ā = a, b = b, σ1 = p σ 2 + η 2 +2ρση, σ 2 = η(a b) ρ = σρ + η p σ2 + η 2 +2ρση, θ() =dϕ() + aϕ() d Managng Cred Rsk: The Grea Challenge for Global Fnancal Markes, Caouee, e. al. Chaper 20 (20.2) R p = N X EAR (20.3) V p = N j=1 (20.5) UAL p = N age 403 N X X j σ σ j ρ j j=1 N X X j σ σ j ρ j 1 CV ar(cl)=ead LGD µ µ ρφ 1 (CL)+Φ 1 (D) Φ D 1 ρ 1+(M 2.5) b(d) 1 1.5b(D) Bond-CDS Bass Handbook: Measurng, Tradng and Analysng Bass Trades, Elzalde, Docor, and Saluk age 13, Equaon 1 S = D (1 R) age 15, Equaon 2 FR = U AI RA + FC age 18, Equaon 3 V[c + p] B SS = RF A age 25, Equaon 4 BT1 =CN (100 R U C F C)+BN (R+CR B FC) age 25, Equaon 5 BT2 =BN (100 + CR B FC) CN (U + C + FC) age 43, Equaon 7 CN = B R 100 R U BN 7

8 A Survey of Behavoral Fnance, Barbers and Thaler (1) (x, p : y,q) =π(p)v(x) +π(q)v(y) (2) π v(x ) where v = xα f x 0 λ( x) α f x<0 and π = w( ) w( ), w( )= γ ( γ +(1 ) γ ) 1/γ (3) D +1 D = e g D+σ D ε +1 (4) (5) C +1 = e g C+σ C η +1 C µ µµ µ ε 0 1 w N, η 0 w 1 (6) E 0 ρ C1 γ =0 1 γ " µc+1 γ (7) 1 = ρe R +1# C,..d.over me (8) R +1 = D = 1+ +1/D +1 D +1 /D D (9) r +1 = d +1 +cons. d +1 d +cons. (10) E π v[(1 w)r f,+1 + wr +1 1] (11) E 0 ρ C1 γ 1 γ + b 0C γ ˆv(X +1 ) =0 (13) R +1 = +1 + D +1 (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 ṽ(x +1,z ) ρ r +1+j + E lm ρ j (p +j d +j )+cons. j (16) r r f = β.1 (F 1 r f )+...+ β,k (F K r f ) (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 +0.5(σD 2 +γ2 σc 2 2γσ Cσ D w) f (20) R +1 = D = 1+ +1/D +1 D +1 = 1+f /D D f e g D+σ D ε +1 8

9 CAIA Level II: Advanced Core Topcs n Alernave Invesmens, Black, Chambers, Kazem Chaper 16 (16.1) repored (16.2) repored (16.3) rue (16.4) rue = α + β 0 rue = α rue =(1/α) repored = repored 1 + β 1 rue 1 + β 2 rue α(1 α) rue 1 + α(1 α) 2 rue 2 + [(1 α)/α] repored 1 +[(1/α) ( repored repored 1 )] (16.5) R,repored β 0 R,rue + β 1 R 1,rue + β 2 R 2,rue + (16.6) repored (16.7) repored =(1 ρ) rue =(1 ρ) rue + ρ repored 1 + ρ repored 1 (16.8) R,repored (1 ρ)r,rue + ρr 1,repored (16.9) R,rue =(R,repored ρr 1,repored )/(1 ρ) (16.10) ˆρ = corr(r,repored R 1,repored ) (16.11) ρ,j = σ j /(σ σ j ) (16.12) R repored Chaper 21 age 262 = α + β 1 R repored 1 + β 2 R repored β k R repored k + ε Y = S I E H where Y = yeld, S = oal solar radaon over he area per perod, I = fracon of solar radaon capured by he crop canopy, E = phoosynhec effcency of he crop (oal plan dry maer per un of solar radaon), H = harves ndex (fracon of oal dry maer ha s harvesable) Managng Invesmen orfolo: A Dynamc rocess, Magnn, Tule, no, McLeavey Chaper 8 age 523 TRCI = CR + RR + SR age 553 RR n, =(R + R 1 + R R n )/n age 554 r n DD = r, 0)] 2 n 1 age 555 ARR rf Sharpe Rao = SD age 556 ARR rf Sorno Rao = DD 9

10 The Secular and Cyclc Deermnans of Capalzaon Raes: The Role of ropery Fundamenals, Macroeconc Facors, and "Srucural Changes," Chervachdze, Cosello, Wheaon (1) Log(C j, )=a 0 + a 1 log(c j, 1 )+a 2 log(c j, 4 )+a 3 log(rri j, )+a 4 RTB + a 7 Q2 (1.1) RRI j, s = RR j, /M RR j +a 8 Q3 + a 9 Q4 + a 10 D j (2) Log(C j, )=a 0 + a 1 log(c j, 1 )+a 2 log(c j, 4 )+a 3 log(rri j, s )+a 4 RTB (2.1) DEBTFLOW = TNBL /GD +a 5 SREAD + a 6 DEBTFLOW + a 7 Q2 + a 8 Q3 + a 9 Q4 + a 10 D j (3) Log(C j, )=a 0 + a 1 log(c j, 1 )+a 2 log(c j, 4 )+a 3 log(rri j, s )+a 4 RTB +a 5 SREAD + a 6 DEBTFLOW + a 7 Q2 + a 8 Q3 + a 9 Q4 (4) Log(C j, )=a 0 + a 1 yearq + a 2 log(c j, 1 )+a 3 log(c j, 4 )+a 4 log(rri j, s )+a 5 RT B +a 6 SREAD + a 7 DEBTFLOW + a 7 Q2 + a 8 Q3 + a 9 Q4 + a 10 D j Analyss of Fnancal Tme Seres, Tsay Chaper 9 (9.1) r = α + β 1 f β m f m +, =1,...,T,,...,k (9.2) r = α + βf +, =1,...,T (9.3) R = α 1 T + Fβ 0 + E (9.4) R = Gξ 0 + E (9.5) r = α + β r m +, =1,...,k =1,...,T (9.11) Var(y )=wσ 0 r w, =1,...,k (9.12) Cov(y,y j )=wσ 0 r w j,, j =1,...,k (9.13) k Var(r )=r(σ r )= k λ = k Var(y ) (9.14) ˆΣ r [ˆσ j,r ]= 1 T 1 (9.15) ˆρ r = Ŝ 1 ˆΣ r Ŝ 1 T =1(r r)(r r) 0, r = 1 T T r =1 (9.16) r μ = βf + (9.17) Σr = Cov(r )=E[(r μ)(r μ) 0 ]=E[(βf + )(βf + ) 0 ]=ββ 0 + D (9.18) Cov(r, f )=E[(r μ)f]=βe(f 0 f)+e( 0 f)=β 0 (9.19) ˆβ [ ˆβ j ]= hpˆλ1 ê 1 pˆλ2 ê 2 pˆλm ê m (9.20) LR(m) = T 1 16 (2k +5) 23 m ³ ln ˆΣ r ln ˆβ ˆβ 0 + ˆD 10

11 Handbook of Fxed Income Secures, Fabozz Chaper 69 (69 4) Asse Allocaon s (w s w B s ) R B s (69 5) Secury Selecon s w s (R s R B s ) (69 12) α k f k αb k f B k = s α k,s f k,s s α B k,s f B k,s Chaper 70 (70 1) Asse Allocaon w µ w s s w wb s (TR B w B s TR B ) (70 2) Secor Managemen ws (TRs TRs B ) s (70 3) Top-Level Exposure (w w B ) TR B (70 4) Asse Allocaon w µ w s s w wb s (ER B w B s ER B ) (70 5) Secor Managemen ws (ERs ERs B ) s (70 6) Top-Level Exposure (w w B ) ER B (70 7) Ouperformance from average carry yavg yavg B (70 8) Key rae conrbuons ω j yj yavg ω B j yj yavg B j (70 9) Ouperformance from avg. parallel shfs OAD OAD B y avg (70 10) Ouperformance from reshapng KRD j KRDj B ( yj y avg ) j (70 11) Asse Allocaon OASD µ w s OASDs wb s OASDs B OASD OASD B s (70 12) Secury Selecon s OAS s B OAS B w s OASD s OAS s OAS B s (70 13) Spread Duraon Msmach (OASD OASD B ) OAS B w s OASDs ws B OASDs B OAS B s (70 14) Asse Allocaon s (70 15) Secury Selecon s w s OASD s OAS s OAS B s 11

12 Inroducon o Cred Rsk Modelng, 2nd ed., Bluhm, Overbeck, Wagner Chaper 6 age 237 M n = M1 n Guaranees and Targe Volaly Funds, Morrson and Tadrowsk age 4 w equy ˆσ equy =mn µ σarge ˆσ equy 2 = λ ˆσ equy, 100% 2 +(1 λ) 1 µ µ 2 S ln S roxy Funcons for he rojecon of Varable Annuy Greeks, Clayon, Morrson, Turnbull, and Vysnasuskas age 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) age 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 V proxy (S,R,σ )) 2 σ sress1 σ sress2 σ sress3 σ base σ base σ base ˆ ˆρ ˆV = ˆV sress1 ˆV sress2 ˆV sress3 ˆV base ˆV base base ˆV 12

13 age 6 µ S S w h S µ S w h S, R R h R, S S h S, σ σ ³ ˆV base h σ V proxy, R R h R 2 (S,R,σ ), σ σ ³ 2 ˆV base h σ V proxy (, S,R,σ ) Recen Advances n Cred Rsk Modelng, Capuano, Chan-Lau, Gasha, Mederos, Sanos, and Souo (II.1) E =max(0,v D) ln V µ D + μ 1 2 σ2 T (II.2) DD T = σ T (II.3) x = a M + p 1 a 2 Z (II.4) rob{x < x M} = q ( M) =Φ Ã! x a M p 1 a 2 (II.5) p K+1 (0, M) =p K (0, M)(1 q K+1 ( M)) (II.6) p K+1 (l, M) =p K (l, M)(1 q K+1 ( M)) + p K (l 1, M)q K+1 ( M), l =1,...,K (II.7) p K+1 (K +1, M) =p K (K, M)q K+1 ( M) (II.8) p(l, ) = R pn (l, M)φ(M)dM (III.1) τ =nf{ 0 V K} 13

14 Marke Models: A Gude o Fnancal Daa Analyss, Chaper 6, Aledander (6.1) = XW (6.2) X = w w w k k (6.3) σ K σ AT M = b(k S) (6.4) (σ K σ AT M )=w K1 1 + w K2 2 + w K3 3 (6.5) = γ S + ε (6.6) σ AT M = α + β S + ε (6.7) β K, = β + Σw K γ (6.8) y = a + b + e (6.9) y = a + X b + e (6.10) y = c + Xd + e (6.11) rcac = (1.45) (14.71) rparbas (17.21) rsocgen (20.55) rdan (6.12) Sochasc Modelng, Theory and Realy from an Acuaral erspecve (I.B-1) ds = μsd + σsdz (I.B-2) ln S T N(ln S 0 +(r σ 2 /2)T,σ T ) (I.B-3) μ =lns 0 +(r σ 2 /2)T, σ = σ T (I.B-4) ĉ = 1 N c N (I.B-5) c = S 0 N(d 1 ) Ke rt N(d 2 ) (I.B-6) d 1 = ln(s 0/K)+(r + σ 2 /2)T σ, d 2 = d 1 σ T T (I.B-7) MC samplng error = 1 Sdev(c ) N (I.B-8) 1 f = 2 (f(u 1)+f(u 2 )) (I.B-9) 1 Sdev( f) N (I.B-10) (I.B-11) f (u) g (u)+g(u) 1 Sdev(f(u) g(u)) N 14

15 (I.B-12) h = 1 n N (I.B-13) (I.B-14) j=1 f(v (j) ) k ˆf = (x +1 x )h k (x +1 x ) Sdev(h(j) lm of he sum should be k) 1 N f(z ) (I.B-15) N g(z ) µ N f(z ) (I.B-16) Sdev g(z ) ) n (I.B-17) S 0 = e r [ps 0 u +(1 p)s 0 d] (I.B-18) (I.B-19) p = er d u d u = e σ and d = u 1 (here s an error n he book formula, he upper (I.B-20) C 0 = e r [pc u +(1 p)c 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, s n ha goes from 0 o m) (I.B-22) S m = S 0 (1 η)u n d m n, n =0, 1,...,m (same error) r (I.B-23) p = µr 12 12σ 2 σ , p 0 = 2 3, p + = p + 2 6, u = e σ 3, d = u 1 µ S (I.B-24) log N(μ( r),σ 2 ( r)) (I.B-25) log S r µ S+1 S ρ() N(μ ρ(),σ 2 ρ() ) (I.B-26) p j =r(ρ( +1)=j ρ() =), =1, 2,...,K, j =1, 2,...,K (here s an error n he book formula, y +1should be +1) (I.B-27) L(Θ) =f(y 1 Θ)f(y 2 Θ,y 1 )f(y 3 Θ,y 1,y 2 ) f(y n Θ,y 1,y 2,...,y n 1 ) (I.B-28) f(ρ(),ρ( 1),y Θ,y 1,y 2,...,y 1 ) for ρ() =1, 2 and ρ( 1) = 1, 2 (I.B-29) π 1 p(ρ( 1) = Θ,y 1,y 2,...,y 1 ) (I.B-30) p j = p(ρ() =j ρ( 1) =, Θ) µ y μ j (I.B-31) g j, = f(y ρ() =j, Θ) =φ = σ j " 1 exp 1 σ j 2π 2 µ # 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) f(y 1 Θ) =f(ρ(0) = 1,y 1 Θ)+f(ρ(0) = 2,y 1 Θ) µ µ y1 μ 1 y1 μ 2 = π 1,0 φ + π 2,0 φ σ 1 (II.A-1) S(0, 1) = ln{1/[1 + C(0, 1)]} S(0, 2) = (1/2) ln{[1 C(0, 2) exp( S(0, 1))]/[1 + C(0, 2)]} S(0, 3) = (1/3) ln{[1 C(0, 3) exp( S(0, 1) C(0, 3) exp( 2S(0, 2))]/[1 + C(0, 3)]} (II.A-2) r = σ(r/ r) γ φ (II.A-3) F ( + T )=F 1 ( + T )+ Λ q,t+1 φ q, q (II.A-4) F ( + T )=F 1 ( + T )+ µ Λ g,t+1 φ q, + Λ q,t+1 Λ q,t+1 /2+ T +1 Λ q, q (II.A-5) E[exp( 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)) (II.A-6) V = E[exp( F 0 (0) F 1 (1) F 2 (2) F N (N))CF N ] (II.A-7) F ( + T )=F 1 ( + T )+[ Λ q,t+1 φ q, ] g +[δ arge (1/ arge )(F ( arge + T ) F 0 ( arge + T ))] +[(1 δ arge )(F ( arge + T ) F ( arge + T +1)] µ (II.A-8) 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[ a(j 1)], Λ >1,j =0 (II.A-10) Facor1=Facor1 (0) + ρ 12 Facor2 (0) + ρ 13 Facor3 (0) Facor2=(1 ρ 2 12) 1/2 Facor2 (0) + ρ 13 Facor3 (0) q σ 2 Facor3=(1 ρ 2 13 ρ 2 13) 1/2 Facor3 (0) (II.A-11) S = S 1 exp[f 1 ( 1) + σ 1 φ (e) 1 σ 1/2] 2 (II.A-12) X( +1) =X() exp(rff RFD) (II.A-13) X( + ) =X() exp((rf 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 /2)d + σ dw } (II.A-16) S = S 1 exp{(μ σ 2 /2)d + σ φ } 16

17 (II.A-17) S = S 1 exp{(f ( + d) σ 2 /2)d + σ φ } (II.A-18) S = S 1 exp{(f ( + d) q σ 3 /2)d + σ φ } (II.A-19) ds = μ S d + σ(s,)dw (II.A-20) ds = μ S d + σ()s α dw (II.A-21) ds = μ S d + V S dw, dv = κ(θ V )d + v V dz, d[s, 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) E(Value of Equy) =A(Value of Asses) N(d 1 ) F (Face Value of Deb) e T N(d 2 ) d 1 = log(a/f )+(r + σ2 A/2) T σ A T d 2 = d 1 σ A T (II.A-26) (II.A-27) (II.A-28) (II.A-29) Spread = (II.A-30) σ E = σ A N(d 1 ) A E Rsk Neural robably of Defaul = N( d 2 ) Recovery Rae = A N( d 1) N( d 2 ) Threshold = Φ 1 (D) h q() =h() exp R h(τ)dτ 0 π() 1 R q(τ)dτ τ=0 R T =0 R T [1 R A() r]q()ν()d =0 q() {u()+e()]d + π(t ) u(t ) d ln(h )=α(β ln(h ))d + γdz 17