Quantitative Finance and Investments Advanced Formula Sheet. Fall 2013/Spring 2014

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1 Quaniaive Finance and Invesmens Advanced Formula Shee Fall 013/Spring 014 ThisishesamesheeusedforFall013.Theonlychangeisonhiscoverpage. Morning and afernoon exam bookles will include a formula package idenical o he one aached o his sudy noe. The exam commiee believes ha by providing many key formulas, candidaes will be able o focus more of heir exam preparaion ime on he applicaion of he formulas and conceps o demonsrae heir undersanding of he syllabus maerial and less ime on he memorizaion of he formulas. The formula shee was developed sequenially by reviewing he syllabus maerial for each major syllabus opic. Candidaes should be able o follow he flow of he formula package easily. We recommend ha candidaes use he formula package concurrenly wih he syllabus maerial. No every formula in he syllabus is in he formula package. Candidaes are responsible for all formulas on he syllabus, including hose no on he formula shee. Candidaes should carefully observe he someimes suble differences in formulas and heir applicaion o slighly differen siuaions. Candidaes will be expeced o recognize he correc formula o apply in a specific siuaion of an exam quesion. Candidaes will noe ha he formula package does no generally provide names or definiions of he formula or symbols used in he formula. Wih he wide variey of references and auhors of he syllabus, candidaes should recognize ha he leer convenions and use of symbols may vary from one par of he syllabus o anoher and hus from one formula o anoher. Werushayouwillfindheinclusionofheformulapackageobeavaluablesudyaide ha will allow for more of your preparaion ime o be spen on masering he learning objecives and learning oucomes. 1

2 Ineres Rae Models - Theory and racice, Brigo and Mercurio Chaper 3 Table 3.1 Summary of insananeous shor rae models Model Dynamics 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 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 + i 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 Y Y EEV r = x + ϕ, dx = x [η a ln x ]d + σx dw Y* SLN N N *raes are posiive under suiable condiions for he deerminisic funcion 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 } = σ 1 e k( s) k (3.8) B(,T )r() (, T )=A(, T )e (3.9) dr() =[kθ B(, T )σ kr()]d + σdw T () (3.11) dr() =[kθ (k + λσ)r()]d + σdw 0 (), r(0) = r 0 (3.1) 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 ir i 1 n r n i r i 1 n n r i 1 ( n r i 1) n (3.15) ˆβ = [r i ˆαr i 1 ] n(1 ˆα) (3.16) V c = 1 h n r i ˆαr i 1 n ˆβ(1 i ˆα) (3.19) E {r() F s } = r(s)e a( s) and Var{r() F s } = r (s)e ³e a( s) σ ( s) 1 (3.0) (, T )= rp R sin( r sinh y) R f(z)sin(yz)dzdy + π 0 0 Γ(p) rp K p ( r) (3.1) dr() =k(θ r())d + σ p r()dw (), r(0) = r 0 (3.) dr() =[kθ (k + λσ)r()]d + σ p r()dw 0 (), r(0) = r 0

3 (3.3) E {r() F s } = r(s)e k( s) + θ 1 e k( s) Var{r() F s } = r(s) σ e k( s) e k( s) + θ σ 1 e k( s) k k (3.4) B(,T )r() (, T )=A(, T )e (3.5) kθ/σ h exp {(k + h)(t )/} A(, T )= h +(k + h)(exp {(T )h} 1) B(, T )= (exp{(t )h} 1) h +(k + h)(exp {(T )h} 1), h = k +σ (3.7) dr() =[kθ (k + B(, T )σ )r()]d + σ p r()dw T () (3.8) p T r() r(s) (x) =p χ (υ,δ(,s))/q(,s)(x) =q(, s)p χ (υ,δ(,s))(q(, s)x) q(, s) =[ρ( s)+ψ + B(, T )] and δ(, s) = 4ρ( s) r(s)e h( s) q(, s) age 68 R(, T )=α(, T )+β(, T )r(), B(,T )r() (, T )=A(, T )e (3.9) σ f (, T )= B(, T ) σ(, r()) T age 69 dr() = b(, r())d + σ(, r())dw () b(, x) =λ()x + η(), σ (, x) =γ()x + δ() B(, T )+λ()b(, T ) 1 γ()b(, T ) +1=0, B(T,T)=0 [ln A(, T )] η()b(, T )+1 δ()b(, T ) =0, A(T,T)=1 age 69/70 Vasicek λ() = k, η() =kθ, γ() =0, δ() =σ age 70 CIR λ() = k, η() =kθ, γ() =σ, δ() =0 b(x) =λx + η, σ (x) =γx + δ µ θ age 71 lim E{r() F s } =exp a + σ 4a µ µ θ (3.31) lim Var{r() F s } =exp a + σ σ exp 1 a a (3.3) dr() =[ϑ() a()r()]d + σ()dw () (3.33) dr() =[ϑ() ar()]d + σdw() (3.34) ϑ() = fm (0,) + af M (0,)+ σ T a (1 e a ) (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,)+ σ a (1 e a ) 3

4 (3.37) E{r() F s } = r(s)e a( s) + α() α(s)e a( s) Var{r() F s } = σ 1 e a( s) a (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( i+1 ) x( i )=x i,j } = x i,j e a i =: M i,j Var{x( i+1 ) x( i )=x i,j } = σ 1 e a i =: V a i r 3 (3.48) x i = V i 1 3=σ a [1 e a i 1 µ Mi,j (3.49) k =round x i+1 (3.50) p u = η j,k + η j,k 6Vi,p m = 3V i 3 η j,k,p 3Vi d = η j,k 6Vi 3V i (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 i 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 i 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) k θ σ / k η j,k σ k 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 kθ(exp{h} 1) (0,; α) = h +(k + h)(exp{h} 1) + x 4h exp{h} 0 [h +(k + h)(exp{h} 1)] h = k +σ 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 (), y(0) = 0 (4.6) E{r() F s } = x(s)e a( s) + y(s)e b( s) + ϕ() Var{r() F s } = σ 1 e a( s) + η 1 e b( s) +ρ ση 1 e (a+b)( s) a b a + b (4.7) r() =σ R 0 e a( u) dw 1 (u)+η R 0 e b( u) dw (u)+ϕ() (4.8) dx() = ax()d + σdfw 1 () dy() = by()d + ηρdfw 1 ()+η p 1 ρ dfw () where dw 1 () =ddfw 1 () and dw () =ρdfw 1 ()+ p 1 ρ dfw () ) ) 1 e a(t 1 e b(t (4.9) M(, T )= x()+ y() a b (4.10) V (, T )= σ T + a a e a(t ) 1 a e a(t ) 3 a + η T + b b e b(t ) 1 b e b(t ) 3 b +ρ 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 V (, T ) (4.1) ϕ() =f M (0,T)+ σ 1 e at + η 1 e bt + ρ ση a b ab (1 e at )(1 e bt ) n (4.13) exp R o T ϕ(u)du = M (0,T) ½ M (0,) exp 1 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() a b (4.15) (, T )=A(, T )exp{ B(a,, T )x() B(b,, T )y()} (4.16) σ f (, T )= p σ e a(t ) + η e b(t ) +ρσηe (a+b)(t ) 5

6 age 15 Cov(df (, T 1 ),df(, T )) d = σ B T (a,, T 1) B T (a,, T )+η B T (b,, T 1) B T (b,, T ) B +ρση T (a,, T 1) B T (b,, T )+ B T (a,, T ) B T (b,, T 1) = σ e a(t 1+T ) + η e b(t 1+T ) +ρση e at 1 bt +(a+b) + e at bt 1 +(a+b) Corr(df (, T 1 ),df(, T )) = σ e a(t 1+T ) + η e b(t 1+T ) σ f (, T 1 )σ f (, T ) + ρση e at 1 bt +(a+b) + e at bt 1 +(a+b) σ f (, T 1 )σ f (, T ) age 153 f(, T 1 T )= ln (, T 1) ln (, T ) T T 1 df (, T 1,T )=...d + B(a,, T ) B(a,, T 1 ) σdw 1 () T T 1 + B(b,, T ) B(b,, T 1 ) ηdw () T T 1 σ f (, T 1,T )= p σ β(a,, T 1,T ) + η β(b,, T 1,T ) +ρσηβ(a,, T 1,T )β(b,, T 1,T ) where β(z,, T 1,T )= B(z,, T ) B(z,,T 1 ) T T 1 Cov(df (, T 1,T ),df(, T 3,T 4 )) d σ B(a,, T ) B(a,, T 1 ) B(a,, T 4 ) B(a,, T 3 ) T T 1 T 4 T 3 +η B(b,, T ) B(b,, T 1 ) B(b,, T 4 ) B(b,, T 3 ) T T 1 T 4 T 3 B(a,, T ) B(a,, T 1 ) B(b,, T 4 ) B(b,, T 3 ) +ρση T T 1 T 4 T 3 + B(a,, T 4) B(a,, T 3 ) B(b,, T ) B(b,, T 1 ) T 4 T 3 T T 1 s age 160 σ 3 = dz 3 () = σ 1 + σ (ā b) + ρ σ 1σ b ā σ 1 dz 1 () σ ā b dz (), σ 4 = σ 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 + η +ρση, σ = η(a b) ρ = σρ + η p σ + η +ρση, θ() =dϕ() + aϕ() d Managing Credi Risk: The Grea Challenge for Global Financial Markes, Caouee, e. al. Chaper 0 (0.) R p = N X i EAR (0.3) V p = N j=1 (0.5) UAL p = N age 403 N X i X j σ i σ j ρ ij j=1 N X i X j σ i σ j ρ ij 1 CV ar(cl)=ead LGD µ µ ρφ 1 (CL)+Φ 1 (D) Φ D 1 ρ 1+(M.5) b(d) 1 1.5b(D) Liquidiy Risk Measuremen and Managemen: Guide o Global Bes racices, Maz and Neu A racioner s Chaper age 33 age 33 log V () =α + β + σε log V q () =α + β σφ 1 (q) Bond-CDS Basis Handbook: Measuring, Trading and Analysing Basis Trades, Elizalde, Docor, and Saluk age 13, Equaion 1 S = D (1 R) age 15, Equaion FR = U AI RA + FC age 18, Equaion 3 V[c + p] B SS = RF A age 5, Equaion 4 BT1 =CN (100 R U C F C)+BN (R+CR B FC) age 5, Equaion 5 BT =BN (100 + CR B FC) CN (U + C + FC) age 43, Equaion 7 CN = B R 100 R U BN 7

8 A Survey of Behavioral Finance, Barberis and Thaler (1) (x, p : y,q) =π(p)v(x) +π(q)v(y) () i π i v(x i ) where v = xα if x 0 λ( x) α if x<0 and π i = w( i ) w(i ), 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, i.i.d.over ime (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 lim ρ j (p +j d +j )+cons. j (16) r i r f = β i.1 (F 1 r f )+...+ β i,k (F K r f ) (17) r i, r f, = α i + β i,1 (F 1, r f, )+...+ β i,k (F K, r f, )+ε i, (18) R f = 1 ρ eγg C+0.5γ σ C (19) 1 = ρ 1+f e g D γg C +0.5(σD +γ σc γσ Cσ D w) f (0) 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 Topics in Alernaive Invesmens, Black, Chambers, Kazemi Chaper 16 (16.1) repored (16.) repored (16.3) rue (16.4) rue = α + β 0 rue = α rue =(1/α) repored = repored 1 + β 1 rue 1 + β rue + + α(1 α) rue 1 + α(1 α) rue + [(1 α)/α] repored 1 +[(1/α) ( repored repored 1 )] (16.5) R,repored β 0 R,rue + β 1 R 1,rue + β R,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 i,j = σ ij /(σ i σ j ) (16.1) R repored Chaper 1 age 6 = α + β 1 R repored 1 + β R repored + + β k R repored k + ε Y = S I E H where Y = yield, S = oal solar radiaion over he area per period, I = fracion of solar radiaion capured by he crop canopy, E = phoosynheic efficiency of he crop (oal plan dry maer per uni of solar radiaion), H = harves index (fracion of oal dry maer ha is harvesable) Managing Invesmen orfolio: A Dynamic rocess, Maginn, Tule, ino, McLeavey Chaper 8 age 53 TRCI = CR + RR + SR age 553 RR n, =(R + R 1 + R R n )/n age 554 r n DD = i r, 0)] n 1 age 555 ARR rf SR = SD age 556 ARR rf SR = DD 9

10 The Secular and Cyclic Deerminans of Capializaion Raes: The Role of ropery Fundamenals, Macroeconic Facors, and "Srucural Changes," Chervachidze, Cosello, Wheaon (1) Log(C j, )=a 0 + a 1 log(c j, 1 )+a log(c j, 4 )+a 3 log(rri j, )+a 4 RTB + a 7 Q (1.1) RRI j, s = RR j, /M RR j +a 8 Q3 + a 9 Q4 + a 10 D j () Log(C j, )=a 0 + a 1 log(c j, 1 )+a log(c j, 4 )+a 3 log(rri j, s )+a 4 RTB (.1) DEBTFLOW = TNBL /GD +a 5 SREAD + a 6 DEBTFLOW + a 7 Q + a 8 Q3 + a 9 Q4 + a 10 D j (3) Log(C j, )=a 0 + a 1 log(c j, 1 )+a log(c j, 4 )+a 3 log(rri j, s )+a 4 RTB +a 5 SREAD + a 6 DEBTFLOW + a 7 Q + a 8 Q3 + a 9 Q4 (4) Log(C j, )=a 0 + a 1 yearq + a 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 Q + a 8 Q3 + a 9 Q4 + a 10 D j Analysis of Financial Time Series, Tsay Chaper 3 µ (v+1)/ Γ[(v +1)/] (3.7) f( v) = Γ(v/) p 1+, v > (v )π v µ (3.8) (a m+1,...,a T α, A m )= T v +1 a ln 1+ =m+1 (v )σ ξ + 1 f[ξ( + ω) v] if < ω/ ξ (3.9) g( ξ,v) = ξ + 1 f[( + ω)/ξ v] if ω/ ξ (3.10) f(x) = v exp 1 x/λ v, <x<, 0 <v λ (1+1/v) Γ(1/v) (3.14) GARCH(m, s): a = σ, σ = α 0 + m α i a i + s β j σ j j=1 + 1 ln(σ ) (3.3) GARCH(1, 1)-M: r = μ + cσ + a, a = σ, σ = α 0 + α 1 a 1 + β 1 σ 1 (3.8) EGARCH(m, s): ln(σ )=α 0 + s a i + γ i a i α i + m β j ln(σ σ j) i Chaper 8 (8.1) μ = E(r ), Γ 0 = E[(r μ)(r μ) 0 ] (8.) Γ [Γ ij ( )] = E[(r μ)(r μ) 0 ] (8.3) ρ [ρ ij ( )] = D 1 Γ D 1 j=1 10

11 (8.4) Γ ij ( ) ρ ij ( ) = p Γii (0)Γ jj (0) = Cov(r i,r j, ) sd(r i )sd(r j ) (8.5) ˆΓ = 1 T (r r)(r r) 0, 0 T = +1 (8.6) ˆρ = ˆD 1ˆΓ ˆD 1, 0 (8.7) Q k (m) =T m 1 =1 T r(ˆγ 0 ˆΓ 1ˆΓ 0 ˆΓ 1 0 ) (8.11) r = a + Φa 1 + Φ a + Φ 3 a 3 + (8.13) r = φ 0 + Φr Φ p r p + a, p > 0 (8.3) r = θ 0 + a Θ 1 a 1 Θ q a q or r = θ 0 + Θ(B)a (8.33) x = αβ 0 x 1 + p 1 (8.34) Φ j = p i=j+1 Φ i x i + a q Θ j a j j=1 Φ i, j =1,...,p 1, αβ 0 = Φ p + Φ p Φ 1 I = Φ(1) (8.35) x = μ + Φ 1 x Φ p x p + a (8.39) x = μd + αβ 0 x 1 + Φ 1 x Φ p 1 x p+1 + a Chaper 9 (9.1) r i = α i + β i1 f β im f m + i, =1,...,T,,...,k (9.) r = α + βf +, =1,...,T (9.3) R i = α i 1 T + Fβ 0 i + E i (9.4) R = Gξ 0 + E (9.5) r i = α i + β i r m + i, i =1,...,k =1,...,T (9.11) Var(y i )=wiσ 0 r w i, i =1,...,k (9.1) Cov(y i,y j )=wiσ 0 r w j, i, j =1,...,k (9.13) k Var(r i )=r(σ r )= k λ i = k Var(y i ) (9.14) ˆΣ r [ˆσ ij,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 i ij ]= hpˆλ1 ê 1 pˆλ ê pˆλm ê m 11

12 (9.0) LR(m) = T 1 16 (k +5) 3 m ³ ln ˆΣ r ln ˆβ ˆβ 0 + ˆD Handbook of Fixed Income Securiies, Fabozzi Chaper 70 (70 1) Asse Allocaion w µ w s s w wb s (TR B w B s TR B ) (70 ) Secor Managemen ws (TRs TRs B ) s (70 3) Top-Level Exposure (w w B ) TR B (70 4) Asse Allocaion 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 1