Quantitative Finance and Investments Core Formula Sheet. Spring 2016

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1 Quaniaive Finance and Invesmens Core Formula Shee Spring 6 Morning and afernoon exam bookles will include a formula package idenical o he one aached o his sudy noe. The exam commiee believe 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. For example, here are several versions of he Black-Scholes-Meron opion pricing formula o differeniae beween insrumens paying dividends, ied o an index, ec. 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. We rus ha you will find he inclusion of he formula package o be a valuable sudy aide ha will allow for more of your preparaion ime o be spen on masering he learning objecives and learning oucomes. In sources where some equaions are numbered and ohers are no (nn) denoes ha here is no number assigned o ha paricular equaion. The only change from 5 is ha Chaper 9, Secion 3 from Wilmo has been removed from he syllabus.

2 An Inroducion o he Mahemaics of Financial Derivaives, 3rd Ediion, A. Hirsa and S. Nefci Chaper (.) S() = C() ( + r ) (+r ) S ( + ) S ( + ) C ( + ) C ( + ) ψ ψ h Q up (S + σ )+Q down (S σ ) (.66) S = +r (.67) C = ( + r) (.7) S = +d S u Q up + S d Q down +r (.7) C = +r (page 6) Chaper 3 (3.37) (3.49) E Q C+ C nx i + i f Chaper 4 Qup C up + + Q downc+ down C u Q up + C d Q down g(s)df (s) +r ( i i ) f(s)ds nx i + i g i (f( i ) f( i )) (4.4) df () =F s ds + F r dr + F d + F ssds + F rrdr + F sr ds dr Chaper 5 (5.) E[S I u ]= Z S f(s I u )ds, (5.8) P ( F () =+a )=p u < (5.9) P ( F () = a )= p (5.44) P (X +s x +s x,...,x )=P(X +s x +s x ) (5.45) r + r = E[(r + r ) I ]+σ(i,) W (5.48) dr = μ(r,)d + σ(r,)dw r+ α r (5.49) = + β R σ W + + R + α r + β R σ W+ Chaper 6 (6.3) E [S T ]=E[S T I ], < T (6.4) E S < (6.5) E [S T ]=S, for all <T

3 (6.9) E Q [e ru B +u ]=B, <u<t (6.) E Q [e ru S +u ]=S, <u (6.) M = N G N B (6.6) P ( N G =) λ G >P( N B =) λ B (6.7) E[ M ] λ G λ B > (6.44) X N(μ,σ ) (6.46) X +T = X + Z +T (6.5) Z = X μ (6.54) E [Z +T ]=X μ (6.55) E [Z +T ]=Z (6.56) S N(,σ ) (6.57) S = (6.58) Z = S dx u (6.6) E [Z +T σ (T + )] = Z σ (6.64) I I + I T I T (6.65) M = E P [Y T I ] (6.66) E P [M +s I ]=M (6.7) G T = f(s T ) (6.7) B T = e T r sds (6.7) M = E P GT B T I (6.6) M k = M + (6.8) E [M k ]=M (6.) C T = C + Chaper 7 kx H i [Z i Z i ] D s ds + g(c s )dm s (7.3) W k =[S k S k ] E k [S k S k ] (7.6) W k = kx W i (7.8) E k [W k ]=W k (7.9) V k = E [ Wk ] (7.3) " n # X V = E W k = k= nx k= V k 3

4 (7.3) V >A > (7.33) V <A < (7.34) V max =max k [Vk,k =,...,n] (7.35) V k >A 3, V max <A 3 < (7.36) E[ W k ] = σkh (7.56) S k S k = E k [S k S k ]+σ k W k Chaper 8 (8.7) ½ wih probabiliy λd dn = wih probabiliy λd (8.8) M = N λ (8.9) E[M ]= (8.5) E(e iux )=exp(φ(u)) (8.6) φ(u) =iγu Z + σ u + e iuy iuyi { y } dν(y) (8.8) X =(r q + ω) + Z (page 9) dν(y) =k(y)dy (8.9) k(y) = e λ py νy I y> + e λn y ν y θ (nn) λ p = σ 4 + θ σ ν σ (nn) λ n = θ σ 4 + σ ν (8.) σ k W k = + θ σ I y< ω wih probabiliy p ω wih probabiliy p... ω m (8.) E[σ k W k ] = σkh mx (8.3) p i ωi = σkh (8.8) ω i (h) = ω i h ri (8.9) p i (h) = p i h qi (8.33) q i +r i = (8.34) c i = ω i p i. wih probabiliy p m (8.58) J =(N λ) (8.6) ds = a(s,)d + σ (S,)dW + σ (S,)dJ (8.75) =< < < n = T (8.76) n = T 4

5 (8.77) S i = S i, i =,,...,n ½ ui S (8.78) S i+ = i d i S i wih probabiliy p i wih probabiliy p i (8.79) u i = e σ, for all i (8.8) d i = e σ, for all i (8.8) p i = h + μ i, for all i σ (8.89) S i+n S i (8.9) log S i+n S i (8.9) log S i+n S i = Z log u +(n Z)logd = Z log u d + n log d Chaper 9 (9.37) lim n E " n X k= Z # T σ(s k,k)[w k W k ] σ(s u,u)dw u = (9.38) S k S k = a(s k,k)h + σ(s k,k)[w k W k ], k =,,...,n " Z # T (9.39) E σ(s,) d < (9.4) (9.73) nx σ(s k,k)[w k W k ] k= (9.74) lim n E (9.76) If (9.77) (9.78) x dx = x T T " n # X x i+ T = i= (dx ) exiss, hen d = T (dx ) = d (9.79) (dw ) = d Z (9.85) E s σ u dw u = Z s lim E n σ u dw u, σ(s,)dw " n X x i+ i= <s< (dx ) # = 5

6 Chaper (page 7) ds = a d + σ dw, (.36) df = F ds + F S d + F σ S d F (.37) df = a + F S + F σ S (.64) Z F s ds u =[F (S,) F (S, )] d + F Z σ dw S F u + F ssσu du (.69) df = F d + F s ds + F s ds + Fs s ds + F s s ds +F s s ds ds nx (.79) Y () = N i ()P i () (.8) dy () = nx N i ()dp i ()+ nx nx dn i ()P i ()+ dn i ()dp i () (.8) ds = a d + σ dw + dj, (.8) E[ J ]= " Ã kx!# (.83) J = N λ h a i p i Ã kx! (.84) a = α + λ a i p i " # kx (.85) df (S,)= F + λ (F (S + a i,) F (S,))p i + F ssσ d + F s ds + dj F " kx # (.86) dj F =[F (S,) F (S,)] λ (F (S + a i,) F (S,))p i d (.87) S = lim s S s, s < Chaper (.4) ds = μs d + σs dw, [, ) (.3) S = S e {(a σ )+σw } (.34) ds = rs d + σs dw, [, ) (.38) S T = hs i e (r σ )T e σw T (.4) Z = e σw (.5) x = E[Z ]=e σ (.55) S = e r(t ) E [S T ] 6

7 (.7) ds = μs d + σ p S dw, [, ) (.74) ds = λ(μ S )d + σs dw (.78) ds = μs d + σdw (.79) ds = μd + σ dw (page 9) dσ = λ(σ σ )d + ασ dw ½ wih probabiliy λd (page 93) dn = wih probabiliy λd (.83) (.84) (page 94) ds =(μ λκ)d + σdw +(e J )dn S ds S =(μ λ κ )d + σd W +(e J )dn S = S e (r q+ω)+x(;σ,ν,θ) (page 94) f(x; σ, ν, θ) = Z σ πg exp (x θg) g /ν e g/ν σ g ν /ν Γ(/ν) dg Chaper (.3) P = θ F (S,)+θ S (.4) dp = θ df + θ ds (.5) ds = a(s,)d + σ(s,)dw, [, ) (.6) df = F d + F ssσ d + F s ds (.7) df = F s a + F ssσ + F d + F s σ dw (.) θ = (.) θ = F s (.) dp = F d + F ssσ d (.6) r(f (S,) F s S )=F + F ssσ (.7) rf + rf s S + F + F ssσ =, S, T (.3) P = F (S,) F s (S,)S (.4) dp = df (S,) F s ds S df s df s (S,)dS (.6) dp = df (S,) F s ds S F s + F ss μs + F sssσ S d + F ss σs dw F ss σ S d (.8) dp = df (S,) F s ds S [F ss (μ r)s d] F ss σs dw 7

8 (page ) dw =(σdw +(μ r)d) (.9) a F + a F s S + a F + a 3 F ss =, S, T (.3) F (S T,T)=G(S T,T) (.3) F + F s =, S, T (.33) F (S,)=αS α + β (.44) F (S,)=e αs α (.46).3F ss + F = (.47) F (S,)= α(s S ) +.3 α( ) + β( ) (.53) F (S,)= (S 4) 3( ),, S Chaper 3 (3.) a(s,)=μs (3.) σ(s,)=σs, [, ) (3.3) rf + rf s S + F + σ F ss S =, S, T (3.4) F (T )=max[s T K, ] (3.6) F (S,)=S N(d ) Ke r(t ) N(d ) (3.7) d = ln( S K )+(r + σ )(T ) σ T (3.8) d = d σ T (3.9) N(d )= Z d (3.) a(s,)=μs π e x dx (3.3) σ(s,)=σ(s,)s, [, ) (3.4) rf + rf s S + F + σ(s,) F ss S =, S, T (3.5) F (T )=max[s T K, ] (3.34) rf rf s S δ F F ssσ = (3.35) F (S T,S T,T)=max[, max(s T,S T ) K] (muli-asse opion) (3.36) F (S T,S T,T)=max[,(S T S T ) K] (spread call opion) (3.37) F (S T,S T,T)=max[,(θ S T + θ S T ) K] (porfolio call opion) (3.38) F (S T,S T,T)=max[,(S T K ), (S T K )] (dual srike call opion) (3.47) F + rs F S + σ S F S rf 8

9 F (3.48) F ij F i,j (3.49) rs F S rs F ij F i,j j S (3.5) rs F F i+,j F ij (3.5) S rs j S Fi+,j F ij S F S Chaper 4 F ij F i,j S S (4.6) dp( z) =P z dz <z < z + dz (4.7) Z (4.8) E[z ]= dp(z )= Z z dp(z ) Z (4.9) E [z E[z ]] = [z E[z ]] dp(z ) (4.9) E S + = S +R (4.3) E Q S + = S +r (4.4) z N(, ) (4.4) dp(z )= e (z) dz π (4.43) ξ(z )=e z μ μ (4.44) [dp(z )][ξ(z )] = π e (z )+μz μ dz (4.45) dq(z )= π e (z μ) dz (4.47) dq(z )=ξ(z )dp(z ) (4.48) ξ(z ) dq(z )=dp(z ) 9

10 (4.53) f(z,z )= π p Ω e σ (4.54) Ω = σ σ σ (4.55) Ω = σ σ σ (4.56) dp(z,z )=f(z,z )dz dz (4.57) ξ(z,z )=exp z μ, z μ Ω z μ z μ ½ z, z Ω (4.58) dq(z,z )=ξ(z,z )dp(z,z ) (4.59) dq(z,z )= (4.6) ξ(z )=e z Ω μ+ μ Ω μ (4.69) dq(z ) dp(z ) = ξ(z ) (4.74) dq(z )=ξ(z )dp(z ) (4.75) dp(z )=ξ(z ) dq(z ) π p Ω e z, z Ω z z X u du), (4.76) ξ = e ( X udw u [,T] (4.77) E he Xdui u <, [,T] Z Z u (4.83) E ξ s X s dw s I u = ξ s X s dw s (4.84) W = W Z X u du, (4.85) Q(A) =E P [ A ξ T ] (4.86) dw = dw X d (4.93) dq = ξ T dp (4.) A A A n = Ω (4.3) A + A +... An = Ω [,T] (4.7) E P [Z Ai ]=Q(A i ) Z (4.38) E P [g(x )] = g(x)f(x)dx (page 48) g(x )=Z h(x ) Z (4.4) E P [g(x )] = h(x) f(x)dx = E Q [h(x )] Ω Ω μ + μ, μ μ Ω μ μ dz dz ¾

11 Chaper 5 (5.) Y N(μ, σ ) (5.4) M(λ) =E[e Yλ ] (5.) M(λ) =e (λμ+ σ λ ) (5.5) S = S e Y, [, ) (5.5) E[S S u,u<]=s u e μ( s)+ σ ( s) (5.3) Z = e r S (5.3) E Q e r S S u,u< = e ru S u (5.3) E Q [Z Z u,u<]=z u i (5.38) E Q e r( u) S S u,u< = S u e r( u) e ρ( u)+ σ ( u) where under Q,Y N(ρ, σ ) (5.39) ρ = r σ (5.4) E Q e r S S u,u< = e ru S u (5.5) ds = rs d + σs dw h i (5.58) C = E Q e r(t ) max{s T K, } (5.88) ds = μ d + σ dw (5.9) d[e r S ]=e r [μ rs ]d + e r σ dw (5.9) dw = dx + dw μ rs (5.97) dx = d σ (5.98) d[e r S ]=e r σ dw (5.) d e r F (S,) = e r σ F s dw Chaper 7 (page 8) R =(+r ) (page 8) R =(+r ) (page 8) B s = B(, 3 ) (page 8) B = B(,T) (page 8) B s 3 = B( 3,T) R R u R R u R R d R R d (7.6) B s = (F L u ) (F L u ) (F L d ) (F L d ) B B uu 3 B ud 3 B du 3 B dd 3 C C uu 3 C ud 3 C du 3 C dd 3 ψ uu ψ ud ψ du ψ dd

12 (7.3) = R R u ψ uu + R R u ψ ud + R R d ψ du + R R d ψ dd (7.4) Q ij =(+r )( + r i )ψ ij (7.5) = Q uu + Q ud + Q du + Q dd (7.6) Q ij > (7.8) B s = E Q ( + r )( + r ) (7.) B = E Q B 3 ( + r )( + r ) (7.) = E Q ( + r )( + r ) [F L ] (7.3) C = E Q ( + r )( + r ) C 3 (7.3) F = h ie Q L E Q ( + r )( + r ) (+r )(+r ) (7.36) B s = ψ uu + ψ ud + ψ du + ψ dd (7.38) π ij = B s ψ ij (7.39) = π uu + π ud + π du + π dd (7.46) F = E π [L ] (7.5) C 3 = N max[l K, ] (7.53) C = E Q ( + r )( + r ) max[l K, ] (7.55) C = B s E π [max[l K, ]] C (7.56) = E S CT S S T C(K) (7.57) = E S (ST K) + S S T ST (page 9) y =log K (page 9) C(K) S = Z (7.63) F (; T,S)= S T ( F (y))e y dy P (, T ) P (, S) (7.64) = T <T <T < <T M (7.65) i = T i+ T i,i=,,,...,m (7.66) L n () = P (, T n) P (, T n+ ) n P (, T n+ ) ny (7.7) P (T i,t n+ )= + j L j (T i ) j=

13 n Y (7.7) P (, T n )=P(, T l ) + j L j (T i ), T l < T l j=l l Y (7.75) B = P (, T l ) ( + j L j (T j )) (7.77) D n () = j= Q n j=l +δ jl j() Q l j= ( + jl j (T J )) (7.78) dl n () =μ n ()L n ()d + L n ()σn()dw τ, T n,n=,,...,m nx j L j ()σ (7.3) μ n () = n()σ τ j () + j L j () j=l (7.5) V = E Q he i T +δ r udu (F L T )Nδ (7.) V = E π [B(, T + δ)(f L T )Nδ] (7.) V = B(, T + δ)e π [(F L T )Nδ] (7.) F = E π [L T ] (page 96) C T =max[l T δ K, ] (7.3) C = E Q he i T +δ r u du [L T δ K, ] Chaper 8 (8.3) B(, T )=e R(T,)(T ), < T (8.) B(, T )=E Q he T rsdsi h log E Q e i T r sds (8.) R(, T )= T (8.33) B(, T )=e T F (,s)ds (8.4) log B(, T ) log B(, T + ) F (, T ) = lim (page 3) log B(, T ) log B(, U) F (, T, U) = U T Chaper 9 (9.4) db = μ(, T, B )B d + σ(, T, B )B dv T (9.5) db = r B d + σ(, T, B )B dw T σ(, T, B(, T )) (9.) df (, T )=σ(, T, B(, T )) d + T (9.) df (, T )=a(f(, T ),)d + b(f (, T ),)dw (9.5) r = F (, ) σ(, T, B(, T )) T dw 3

14 Z " Z # T Z (page 35) F (, T )=F(,T)+ b(s, T ) b(s, u)du ds + b(s, T )dw s s Z Z Z (9.6) r = F (,)+ b(s, ) b(s, u)du ds + b(s, )dw s s (9.33) df (, T )=b (T )d + bdw (9.34) db(, T )=r B(, T )d + b(t )B(, T )dw (9.35) r = F (,)+ b + bw (9.36) dr =(F (,)+b )d + bdw (9.37) F (,)= F(,) Paul Wilmo Inroduces Quaniaive Finance, nd Ediion, P. Wilmo Chaper 6 (6.6) (6.7) (6.8) (6.9) (6.) Chaper 8 V + σ S V V + rs rv = S S V + σ S V V +(r D)S rv = S S V + σ S V S +(r r f)s V rv = S V + σ S V V +(r + q)s rv = S S V + σ F V + rv = F e r(t ) (8.7) V (S, ) = σ p π(t ) Z Call opion value Se D(T ) N(d ) Ee r(t ) N(d ) d = log(s/e)+(r D+ σ )(T ) σ T d = log(s/e)+(r D σ )(T ) σ T = d σ T e (log(s/s )+(r σ )(T )) /σ (T ) Payoff(S ) ds S Pu opion value Se D(T ) N( d )+Ee r(t ) N( d ) Binary call opion value e r(t ) N(d ) Binary pu opion value e r(t ) ( N(d )) 4

15 Dela of common conracs Call e D(T ) N(d ) Pu e D(T ) (N(d ) ) Binary call e r(t ) N (d ) σs T Binary pu e r(t ) N (d ) σs T N (x) = π e x Gamma of common conracs Call Pu e D(T ) N (d ) σs T e D(T ) N (d ) σs T Binary call e r(t ) d N (d ) Binary Pu σ S (T ) e r(t ) d N (d ) σ S (T ) Theas of common conracs Call σse D(T ) N (d ) + DSN(d T )e D(T ) ree r(t ) N(d ) Pu σse D(T ) N ( d ) DSN( d T )e D(T ) + ree r(t ) N( d ) ³ Binary call re r(t ) N(d )+e r(t ) N (d ) d (T ) r D ³ σ T Binary pu re r(t ) ( N(d )) e r(t ) N (d ) d (T ) r D σ T Speedofcommonconracs Call e D(T ) N (d ) σ S (T ) d + σ T Pu e D(T ) N (d ) σ S (T ) d + σ T ³ Binary call e r(t ) N (d ) σ S 3 (T ) d + d d ³ σ T e Binary pu r(t ) N (d ) σ S 3 (T ) d + dd σ T Vegas of common conracs Call S T e D(T ) N (d ) Pu S T e D(T ) N (d ) Binary call e r(t ) N (d ) d σ Binary pu e r(t ) N (d ) d σ Rhos of common conracs Call E(T )e r(t ) N(d ) Pu E(T )e r(t ) N( d ) Binary call (T )e r(t ) N(d )+ T σ e r(t ) N (d ) Binary pu (T )e r(t ) ( N(d )) T σ e r(t ) N (d ) Sensiiviy o dividend for common conracs Call (T )Se D(T ) N(d ) Pu (T )Se D(T ) N( d ) Binary call T σ e r(t ) N (d ) Binary pu T σ e r(t ) N (d ) 5

16 Chaper 9 This maerial was removed from he syllabus for 6. Chaper Secion.4, one ime sep mark-o-marke profi (using Black-Scholes wih σ = σ) = (σ σ )S Γ i d +( i a )((μ r + D)Sd + σsdx) Secion.5, mark-o-marke profi from oday o omorrow = σ σ S Γ i d Chaper 4 (4.) V = Pe y(t ) + Chaper 6 (6.4) V + w V r NX C i e y( i ) +(u λw) V r (6.5) dv rv d = w V (dx + λd) r 6.6 NAMED MODELS 6.6. Vasicek dr =(η γr)d + β / dx rv = A(;T ) rb(;t ) The value of a zero coupon bond is given by e B = γ ( e γ(t ) ) A = γ (B(; T ) T + )(ηγ βb(; T ) β) 4γ 6.6. Cox, Ingersoll & Ross dr =(η γr)d + αrdx A(;T ) rb(;t ) The value of a zero coupon bond is given by e α A = aψ log(a B)+ψ b log((b + b)/b) aψ log a (e ψ (T ) ) B(; T )= (γ + ψ )(e ψ(t ) ) + ψ ψ = p γ +α and ψ = η a + b b, a = ±γ + p γ +α α 6

17 6.6.3 Ho & Lee dr = η()d + β / dx A(;T ) rb(;t ) The value of a zero coupon bond is given by e B = T A = η(s)(t s)ds + β(t )3 6 η() = log Z M( ; )+β( ) Secion 6.7 Forward price = S Z Z + w Z +(u λw) Z r r wih Z(r, T )= rz = Secion 6.8 Fuures price F (S, r, ) = S p(r, ) ³ p p + w p r +(μ λw) p rp w + pσβ p r r q r = wih p(r, T )= Chaper 7 Secion7.Ho&Lee dr = η()d + cdx A(;T ) r(t ) Z(r, : T )=e A(; T )=log ZM ( ; T ) Z M ( ; ) (T ) log(z M( ; )) c ( )(T ) Secion 7.3 The Exended Vasicek Model of Hull & Whie dr =(η γr)d + cdx dr =(η() γr)d + cdx A(;T ) rb(;t ) Z(r, : T )=e B(; T )= γ(t ³ e ) γ ZM ( ; T ) A(; T )=log Z M ( B(; T ) ³ ; ) log(z M( ; )) c 4γ 3 e γ(t ) ) e ³e γ( γ( ) 7

18 Chaper 8 Secion 8.3. Bond Opions - Marke Pracice Call opion price: e r(t ) (FN(d ) EN(d )) Pu opion price: e r(t ) (EN( d ) FN( d )) d = log(f/e)+ σ (T ) σ T d = d σ T Secion Caps and Floors - Marke Pracice Caple price: e r ( i ) (F (, i, i )N(d ) r cn(d )) Floorle price: e r ( i ) ( F (, i, i )N( d )+r c N( d )) d = log(f/r c)+ σ i σ i d = d σ p i Secion 8.6. Swapions, Capions and Floorions - Marke Pracice Ã ) Price of a payer swapion: e r(t +! (Ts T ) F F (FN(d ) EN(d )) Ã ) Price of a receiver swapion: e r(t +! (Ts T ) F F (EN( d ) FN( d )) d = log(f/e)+ σ (T ) σ T d = d σ T Chaper 9 (9.) Z(; T )=e T F (;s)ds (9.) dz(; T )=μ(, T )Z(; T )d + σ(, T )Z(; T )dx (nn) F (; T )= log Z(; T ) T 8

19 (9.3) df (; T )= T σ (, T ) μ(, T ) Ã Z! T (9.5) df (; T )=ν(, T ) ν(, s)ds d σ(, T )dx T d + ν(, T )dx Ã X N Z! T NX (9.6) df (; T )= ν i (, T ) ν i (, s)ds d + ν i (, T )dx i Xi (9.7) dz i = rz i d + Z i a ij dx j j= ix (9.8) dz i =(+τf i )dz i+ + τz i+ df i + τσ i F i Z i+ a i+,j ρ ij d ix (9.9) df i = j= σ j F j τρ ij +τf j σ i F i d + σ i F i dx i FAQ s in Opion Pricing Theory, P. Carr (38) β T =[V (S, ; σ h ) V (S, ; σ i )]e rt + S T f (S T ) f(s T )+ e r(t ) (σ σh) S (39) N T = f (S T ) (4) P &L T N T S T β T f(s T )=[V (S, ; σ i ) V (S, ; σ h )]e rt + The Handbook of Fixed Income Securiies, 8h ed., F. Fabozzi j= e r(t ) (σ h σ )S S V (S,; σ h )d S V (S,; σ h )d Page Y d = (F P ) F 36 Managing Invesmen Porfolios, a dynamic process, Maginn, e al Chaper 5 (5-) U m = E(R m ).5R A σ m (5-) SFRaio = E(R P ) R L (5-3) σ P E(R new ) R F σ new > E(Rp ) R F (5-4) U ALM m = E(SR m ).5R A σ (SR m ) σ p Corr(R new,r p ) 9

20 Modern Invesmen Managemen: (QFIC 6-3) Chaper An Equilibrium Approach, B. Lierman (.) R L, R f, = β(r B, R f, )+ε (.) SR i = μ i R f σ i (.3) S A L (.4) F A /L (.6) RACS = E [A ( + R A,+ ) L ( + R L,+ ) (A L )(+R f )] σ [A ( + R A,+ ) L ( + R L,+ )] (.8) (.9) σ [S + ] A β L A σ B ρσ Bσ E σ E + σ B ρσ Bσ E (.) μ B β L + L R f ( β) A A μ E μ B (.) A + = A ( + R A,+ ) pl ( + R L,+ ) and L + = L ( + R L,+ )( p) (.3) E [R x,+ ]= E [F +]( p)+p F +E [Rx ] +E [Rx ] p (.4) E [F ]= F + p p E [R x ]+p (.A.) R L, R f, = β (R B, R f, )+ε (.A.) (.A.4) r(t ) V = Ce dv V (.A.4) α = = dc C (T )dr + rd β L A σ B ρσ Eσ B σ E + σ B ρσ Eσ B (.A.6) α = (.A.) E [F ]= μ B β L + L [R f ( β)+η] A A μ E μ B +μx F + p p +μx p μ x + p

21 Analysis of Financial Time Series, hird ediion, R. Tsay (QFIC -3) Chaper 3 (3.) μ = E(r F ), σ = Var(r F )=E[(r μ ) F ] px qx (3.3) r = μ + a, μ = φ i y i θ i a i, y = r φ (3.4) σ = Var(r F )=Var(a F ) (p4) a = α + α a + + α m a m + e. = m +,...,T (p4) F = (SSR SSR )/m SSR /(T m ) (3.5) a = σ, σ = α + α a + + α m a m (p7) a = σ, σ = α + α a (p8) E(a 4 )= (p) 3α ( + α ) ( α )( 3α ) kx β i x i f(a,...,a T α) =f(a T F T )f(a T F T ) f(a m+ F m )f(a,...,a m α) TY = p exp a πσ σ f(a,...,a m α) =m+ (p) (a m+,...,a T α, α,...,α m )= Γ[(v +)/] (3.7) f( v) = Γ(v/) p (v )π (3.8) (a m+,...,a T α, A m )= + TX =m+ TX =m+ v v + (p) (a m+,...,a T α, υ, A m )=(T m) ln(σ )+ (v+)/,v> ln + ½ ln Γ υ + o.5ln[(υ )π] + (a m+,...,a T α, A m ) ξ + f[ξ( + ω) v] if < ω/ ξ (3.9) g( ξ,v) = ξ + f[( + ω)/ξ v] if ω/ ξ a a σ (v )σ + ln(σ ) h ³ υ i ln Γ

22 (p) = Γ[(υ )/] υ ξ, = ξ + ξ πγ(υ/) ξ (3.) f(x) = v exp x/λ v, <x<, <v λ (+/v) Γ(/v) mx (3.) σh( ) =α + α i σh( i). where σh( i) =a h+ i if i (3.4) GARCH(m, s) :a = σ,σ = α + mx α i a i + max(m,s) X (3.5) GARCH(m, s) :a = α + (α i + β i )a i + η sx β j σ j j= sx β j η j (p3) E(a 4 ) [E(a )] = 3[ (α + β ) ] (α + β ) α > 3 (3.7) σh( ) =α +(α + β )σh( ), > (p33) σh ( ) =α [ (α + β ) ] +(α + β ) σh α β () (3.) σh( ) =σh() + ( )α, (3.3) GARCH(, ) M : r = μ + cσ + a,a = σ,σ = α + α a + β σ (3.4) g( )=θ + γ[ E( )] ½ (θ + γ) γe( (p43) g( )= ) if (θ γ) γe( ) if < (p43) E( )= υ Γ[(υ +)/] (υ )Γ(υ/) π (3.5) EGARCH(m, s) :a = σ, ln(σ )=α + +β B + + β s B s α B α m B m g( ) (3.6) a = σ, ( αb)ln(σ )=( α)α + g( ) j= ½ (3.7) ( αb)ln(σ α +(γ + θ) )= if α +(γ θ)( = ) if < exp (γ + θ) a if a (p44) σ = σ α σ exp(α ) exp (γ θ) a if a < σ (p48) σh+ = σ α h exp[( α )α ]exp[g( h+ )] ³ (p48) E{exp[g( )]} =exp γ p h i /π e (θ+γ)/ Φ(θ + γ)+e (θ γ)/ Φ(γ θ) (p48) ˆσ h(j) =ˆσ α h (j ) exp(ω) exp[(θ + γ) /Φ(θ + γ)+exp[(θ γ) /Φ(γ θ) ª

23 Frequenly Asked Quesions in Quanaive Finance, P. Wilmo Q3 - Jensen s Inequaliy (3-5) If he payoff is a convex funcion, E[P (S T )] P (E[S T ]) E[f(S)] = E f( S + ) = E f( S)+ f ( S)+ f ( S)+ f( S)+ f ( S)E[ ] = f(e[s]) + f (E[S])E[ ] The LHS is greaer han he RHS by approximaely f (E[S])E[ ] Q6 - Girsanov s Theorem (3-5) The Theorem is: Le W be a Brownian moion wih measure P and sample space Ω. If γ is a previsible process saisfying R i T he consrain E P hexp γ < hen here exiss an equivalen measure Q on Ω such ha W = ³ W + R γ sds is a Brownian moion. Chaper 6 - The Mos Popular Probabiliy Disribuions Normal or Guassian: f(x) = (x a) exp πb b, <x<,μ= a, σ = b Lognormal: f(x) = Poisson: πbx exp (ln(x) a),x>,μ=exp a + b b,σ =exp a + b exp(b ) p(x) = e a a x,x=,,...,μ= a, σ = a x! Chi square (noe ha his is a generalizaion of he usual chi-square disribuion): f(x) = e (x+a)/ v/ X i= x i +v/ a i i j!γ(i + v/),x,μ= v + a, σ =(v +a) Gumbel: f(x) = b exp a x b exp ³ e a x b, <x<,μ= a + γb,σ = 6 π b whereγ = Weibull: f(x) = c c c! x a x a c + c + c + exp,x>a,μ= a+bγ,σ = b ÃΓ Γ b b b c c c Suden s : f(x) = Γ c+ b πcγ c Ã + x a b c! c+, <x<, μ= a for c>, σ = 3 c b for c>. c

24 Pareo: f(x) = bab,x a, μ = ab xb+ a for b>, σ = Uniform: f(x) = b a, a<x<b, μ = a + b, (b σ a) = a b for b> (b )(b ) Inverse normal: r Ã b f(x) = πx 3 exp b! x a, x>, μ = a, σ = a3 x a b Gamma: f(x) = bγ(c) c x a a x exp b b, x a, μ = a + bc, σ = b c Logisic f(x) = exp x a b b +exp x a, <x<, μ= a, σ = 3 π b b Laplace: f(x) = b exp x a, <x<, μ = a, σ =b b Cauchy: f(x) = ³ πb + x a, <x<, momens do no exis b Bea: f(x) = Exponenial: Γ(c + d) Γ(c)Γ(d)(b a) c+d (x a)c (b x) d ad + bc,a x b, μ = c + d, cd(b a) σ = (c + d +)(c + d) f(x) = b exp a x b Lévy:,x a, μ = a + b, σ = b no closed form for densiy funcion, μ = μ, varianceisinfinie unless α =,wheniisσ =v 4

25 Risk Managemen and Financial Insiuions, second ediion, J. Hull Chaper 9-Volailiy (9.) Prob(υ >x)=kx α (9.5) σ n = (page 86) mx α i u n i mx α i = (9.6) σ n = γv L + (9.7) σ n = ω + mx α i u n i mx α i u n i (9.8) σ n = λσ n +( λ)u n (9.9) σ n = γv L + αu n + βσ n (9.) σ n = ω + αu n + βσ n (9.4) E[σn+] =V L +(α + β) (σn V L ) (9.5) σ(t ) = 5 ½V L + ¾ e at [V () V L ) at (9.6) σ(t ) σ() e at σ() at σ(t ) 5