Quantitative Finance and Investment Core Formula Sheet. Spring 2017

Transcript

1 Quaniaive Finance and Invesmen Core Formula Shee Spring 7 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. A new ex Problems and Soluions in Mahemaical Finance: Sochasic Calculus The Wiley Finance Series by Eric Chin, Dian Nel and Sverrir Olafsson has been added o he syllabus among oher changes, bu formulas from his are no on he formula shee.

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 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 E Q C+ C n i + i f Chaper 4 Qup C up + + Q downc+ down C u Q up + C d Q down gsdfs + r i i fsds n i + i g 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. ES I u = S fs I u ds, 5.8 P F = +a = p 5.9 P F = a = p u < 5.44 P X +s x +s x,..., x = P X +s x +s x 5.45 r + r = Er + r I + σi, W 5.48 dr = µr, d + σr, dw r+ α r 5.49 = + β R σ W + + R + α r + β R σ W+ Chaper E S T = ES 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 Nµ, σ 6.46 X +T = X Z = X µ +T 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 = fs T 6.7 B T = e rsds 6.7 M = E P GT B T I 6.6 M k = M E M k = M 6. C T = C + Chaper 7 k H i Z i Z i D s ds + gc s dm s 7.3 W k = S k S k E k S k S k 7.6 W k = k W i 7.8 E k W k = W k 7.9 V k = E W k n 7.3 V = E W k = k= n 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 V max > A 3, < A 3 < 7.36 E W k = σ kh 7.56 S k S k = E k S k S k + σ k W k Chaper dn = 8.8 M = N λ 8.9 EM = { wih probabiliy λd wih probabiliy λd 8.5 Ee iux = expφu 8.6 φu = iγu σ u X = r q + ω + Z page 9 dνy = kydy ky = e λpy νy I y> + e λn y ν y θ nn λ p = σ 4 + σ θ ν σ nn λ n = θ σ 4 + σ ν 8. σ k W k = + θ σ e iuy iuyi { y } dνy I y< ω wih probabiliy p ω wih probabiliy p. ω m 8. Eσ k W k = σkh m 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 8.58 J = N λ. wih probabiliy p m 8.6 ds = as, 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 { ui S 8.78 S i+ = i wih probabiliy p i d i S 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 = + µ, 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 log d = Z log u d + n log d Chaper lim n E n k= T σs k, k W k W k σs u, udw u = 9.38 S k S k = as k, kh + σs k, kw k W k, k =,,..., n T 9.39 E σs, d < n σs k, kw k W k k= 9.74 lim n E 9.76 If x dx = n x T T x i+ T = i= dx exiss, hen d = T dx = 9.79 dw = d 9.85 E s σ u dw u = d s lim E n σ u dw u, σs, dw n x i+ i= < s < dx = T T 9.3 E fw, dw gw, dw = E fw, gw, d T T 9.33 E fw, dw = E fw, d 5

6 Chaper page 7 ds = a d + σ dw,.36 df = F S ds + F.37 df =.64 d + F σ d S F a + F S + F σ S F s ds u = F S, F S, d + F σ dw S F u + F ssσu du.69 df = F d + F s ds + F s ds + Fss ds + F ss ds + F ss ds ds n.79 Y = N i P i.8 dy = n N i dp i + n n dn i P i + dn i dp i.8 ds = a d + σ dw + dj,.8 E J = k.83 J = N λ h a i p i k.84 a = α + λ a i p i k.85 df S, = F + λ F S + a i, F S, p i + F ssσ d + F s ds + dj F k.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 = S e r σ T e σw T.4 Z = e σw.5 x = EZ = e σ.55 S = e rt E S T 6

7 .7 ds = µs d + σ S dw,,.74 ds = λµ S d + σs dw.78 ds = µs d + σdw.79 ds = µd + σ dw page 9 dσ = λσ σ d + ασ dw { wih probabiliy λd page 93 dn = wih probabiliy λd.83 ds S = µ λκd + σdw + e J dn.84 page 94 ds S = µ λ κ d + σd W + e J dn S = S e r q+ω+x;σ,ν,θ page 94 fx; σ, ν, θ = 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 = as, d + σs, 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 rf 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 rs d F ss σs dw 7

8 page dw = σdw + µ rd.9 a F + a F s S + a F + a 3 F ss =, S, T.3 F S T, T = GS 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, = αs S +.3 α + β.53 F S, = S 4 3,, S Chaper 3 3. as, = µs 3. σs, = σs,, 3.3 rf + rf s S + F + σ F ss S =, S, T 3.4 F T = maxs T K, 3.6 F S, = S Nd Ke rt Nd 3.7 d = ln S K + r + σ T σ T 3.8 d = d σ T 3.9 Nd = d 3. as, = µs π e x dx 3.3 σs, = σs, S,, 3.4 rf + rf s S + F + σs, F ss S =, S, T 3.5 F T = maxs T K, 3.34 rf rf s S δ F F ssσ = 3.35 F S T, S T, T = max, maxs 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 Ez = dpz = 4.9 E z Ez = z dpz 4.9 E S + = S + R 4.3 E Q S + = S + r 4.4 z N, 4.4 dpz = π e z dz 4.43 ξz = e zµ µ z Ez dpz 4.44 dpz ξz = π e z +µz µ dz 4.45 dqz = π e z µ dz 4.47 dqz = ξz dpz 4.48 ξz dqz = dpz 9

10 4.53 fz, z = π Ω e σ 4.54 Ω = σ σ σ z µ, z µ Ω z µ z µ 4.55 Ω = σ σ σ 4.56 dpz, z = fz, z dz dz { 4.57 ξz, z = exp z, z Ω 4.58 dqz, z = ξz, z dpz, z 4.59 dqz, z = 4.6 ξz = e z Ω µ+ µ Ω µ 4.69 dqz dpz = ξz 4.74 dqz = ξz dpz π Ω e z, z 4.75 dpz = ξz dqz 4.76 ξ = e XudWu du X u,, T 4.77 E e Xdu u <,, T u 4.83 E ξ s X s dw s I u = ξ s X s dw s 4.84 W = W 4.85 QA = E P A ξ T 4.86 dw = dw X d 4.93 dq = ξ T dp X u du,, T 4. A A A n = Ω 4.3 A + A +... An = Ω 4.7 E P Z Ai = QA i 4.38 E P gx = gxfxdx µ page 48 gx = Z hx 4.4 E P gx = hx fxdx = E Q hx Ω Ω µ + µ, µ Ω µ µ Ω z z dz dz }

11 Chaper 5 5. Y Nµ, σ 5.4 Mλ = Ee Yλ 5. Mλ = e λµ+ σ λ 5.5 S = S e Y,, 5.5 ES 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 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 Nρ, 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 5.58 C = E Q e rt max{s T K, } 5.88 ds = µ d + σ dw 5.9 de 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 de 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 7.6 B s B = B uu 3 C R R u R R u R R d R R d F L u F L u F L d F L d C uu 3 B ud 3 C ud 3 B du 3 C du 3 B dd 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 F = E 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 maxl K, 7.53 C = E Q + r + r maxl K, 7.55 C = B s E π maxl K, C 7.56 = E S CT S S T CK 7.57 = E S ST K + S S T ST page 9 y = log K page 9 CK S = 7.63 F ; T, S = S T F ye 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+ n 7.7 P T i, T n+ = + j L j T i j=

13 n 7.7 P, T n = P, T l + j L j T i, T l < T l j=l l 7.75 B = P, T l + j L j T j 7.77 D n = j= n j=l +δ jl j l j= + jl j T J 7.78 dl n = µ n L n d + L n σndw τ, T n, n =,,..., M n j L j σ 7.3 µ n = nσ τ j + j L j j=l 7.5 V = E Q e T +δ r udu F L T Nδ 7. V = E π B, T + δf L T Nδ 7. V = B, T + δe π F L T Nδ 7. F = E π L T page 96 C T = maxl T δ K, 7.3 C = E Q e T +δ r udu L T δ K, Chaper B, T = e RT,T, < T 8. B, T = E Q e rsds log E Q e T rsds 8. R, T = T 8.33 B, T = e F,sds 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 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 = af, T, d + bf, T, dw 9.5 r = F, σ, T, B, T T dw 3

14 page 35 F, T = F, T r = F, df, T = b T d + bdw T bs, T bs, udu ds + bs, bs, udu ds + s 9.34 db, T = r B, T d bt B, T dw 9.35 r = F, + b + bw 9.36 dr = F, + b d + bdw 9.37 F, = F, s bs, dw s bs, T dw s Paul Wilmo Inroduces Quaniaive Finance, nd Ediion, P. Wilmo Chaper Chaper 8 V + σ S V V + rs S S rv = V + σ S V V + r DS S S rv = V + σ S V S + r r f S V S rv = V + σ S V V + r + qs S S rv = V + σ F V F + rv = e rt 8.7 V S, = σ πt Call opion value Se DT Nd Ee rt Nd d = logs/e+r D+ σ T σ T d = logs/e+r D σ T σ T = d σ T e logs/s +r σ T /σ T PayoffS ds S Pu opion value Se DT N d + Ee rt N d Binary call opion value e rt Nd Binary pu opion value e rt Nd 4

15 Dela of common conracs Call e DT Nd Pu e DT Nd Binary call e rt N d σs T Binary pu e rt N d σs T N x = π e x Gamma of common conracs Call Pu e DT N d σs T e DT N d σs T Binary call e rt d N d Binary Pu σ S T e rt d N d σ S T Theas of common conracs Call σse DT N d + DSNd T e DT ree rt Nd Pu σse DT N d DSN d T e DT + ree rt N d Binary call re rt Nd + e rt N d d Binary pu re rt Nd e rt N d T r D σ T d T r D σ T Speed of common conracs Call e DT N d σ S T d + σ T Pu e DT N d σ S T d + σ T Binary call e rt N d σ S 3 T d + dd σ T Binary pu e rt N d σ S 3 T d + dd σ T Vegas of common conracs Call S T e DT N d Pu S T e DT N d Binary call e rt N d d σ Binary pu e rt N d d σ Rhos of common conracs Call ET e rt Nd Pu ET e rt N d Binary call T e rt Nd + T σ e rt N d Binary pu T e rt Nd T σ e rt N d Sensiiviy o dividend for common conracs Call T Se DT Nd Pu T Se DT N d Binary call Binary pu T σ e rt N d T σ e rt 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 + DSd + σsdx Secion.5, mark-o-marke profi from oday o omorrow = σ σ S Γ i d Chaper 4 4. V = P e yt + Chaper V + w V r N C i e yi + u λw V r rv = 6.5 dv rv d = w V dx + λd r 6.6 NAMED MODELS 6.6. Vasicek dr = η γrd + β / dx A;T rb;t The value of a zero coupon bond is given by e B = γ e γt A = γ B; T T + ηγ βb; T β 4γ 6.6. Cox, Ingersoll & Ross dr = η γrd + αrdx A;T rb;t The value of a zero coupon bond is given by e α A = aψ loga B + ψ b logb + b/b aψ log a e ψt B; T = γ + ψ e ψt + ψ ψ = γ + α and ψ = η a + b b, a = ±γ + γ + α α 6

17 6.6.3 Ho & Lee dr = ηd + β / dx A;T rb;t The value of a zero coupon bond is given by e B = T A = ηst sds + βt 3 6 η = log Z M ; + β Secion 6.7 Forward price = S Z Z + w Z r wih Zr, T = Secion u λw Z r rz = Fuures price F S, r, = S pr, p p + w p r + µ λw p rp w + pσβ p r r q r = wih pr, T = Chaper 7 Secion 7. Ho & Lee dr = ηd + cdx A;T rt Zr, : T = e A; T = log ZM ; T Z M ; T logz M ; c T Secion 7.3 The Exended Vasicek Model of Hull & Whie dr = η γrd + cdx dr = η γrd + cdx A;T rb;t Zr, : T = e B; T = γt e γ ZM ; T A; T = log Z M B; T ; logz M ; c 4γ 3 e γt e γ e γ 7

18 Chaper 8 Secion 8.3. Bond Opions - Marke Pracice Call opion price: e rt F Nd ENd Pu opion price: e rt EN d F N d d = logf/e + σ T σ T d = d σ T Secion Caps and Floors - Marke Pracice Caple price: e r i F, i, i Nd r c Nd Floorle price: e r i F, i, i N d + r c N d d = logf/r c + σ i σ i d = d σ i Secion 8.6. Swapions, Capions and Floorions - Marke Pracice Price of a payer swapion: e rt + Ts T F F F Nd ENd Price of a receiver swapion: e rt + Ts T F F EN d F N d d = logf/e + σ T σ T d = d σ T Chaper 9 9. Z; T = e F ;sds 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 T 9.5 df ; T = ν, T ν, sds d σ, T dx T d + ν, T dx N T N 9.6 df ; T = ν i, T ν i, sds d + ν i, T dx i i 9.7 dz i = rz i d + Z i a ij dx j j= i 9.8 dz i = + τf i dz i+ + τz i+ df i + τσ i F i Z i+ a i+,j ρ ij d i 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 fs T + e rt σ σh S 39 N T = f S T 4 P &L T N T S T β T fs T = V S, ; σ i V S, ; σ h e rt + The Handbook of Fixed Income Securiies, 8h ed., F. Fabozzi j= e rt 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 = ER m.5r A σ m 5- SFRaio = ER P R L 5-3 σ P ER new R F σ new > ERp R F 5-4 U ALM m = ESR m.5r A σ SR m σ p CorrR new, R p 9

20 Modern Invesmen Managemen: QFIC 6-3 Chaper An Equilibrium Approach, B. Lierman. R L, R f, = βr 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..a..a.4 R L, R f, = β R B, R f, + ε rt 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 = Er F, σ = V arr F = Er µ F p q 3.3 r = µ + a, µ = φ i y i θ i a i, y = r φ 3.4 σ = V arr F = V ara F p4 p4 a = α + α a + + α m a m + e. = m +,..., T F = SSR SSR /m SSR /T m 3.5 a = σ ɛ, σ = α + α a + + α m a m p7 a = σ ɛ, σ = α + α a p8 Ea 4 = p 3α + α α 3α =m+ k β i x i fa,..., a T α = fa T F T fa T F T fa m+ F m fa,..., a m α T = exp a πσ σ fa,..., a m α p la m+,..., a T α, α,..., α m = Γv + / 3.7 fɛ v = Γv/ v π 3.8 la m+,..., a T α, A m = + ɛ T =m+ T =m+ v v + p la m+,..., a T α, υ, A m = T m lnσ + v+/, v > ln + { ln Γ υ + }.5 lnυ π + la m+,..., a T α, A m ξ + ϱfξϱɛ + ω v if ɛ < ω/ϱ ξ 3.9 gɛ ξ, v = ξ + ϱfϱɛ + ω/ξ v if ɛ ω/ϱ ξ a a σ v σ + lnσ υ ln Γ

22 p ϖ = Γυ / υ ξ, ϱ = ξ + ξ πγυ/ ξ ϖ 3. fx = v exp x/λ v, < x <, < v λ +/v Γ/v m 3. σhl = α + α i σhl i. where σhl i = a h+l i if l i 3.4 GARCHm, s : a = σ ɛ, σ = α + m α i a i + maxm,s 3.5 GARCHm, s : a = α + α i + β i a i + η p3 Ea 4 Ea = 3 α + β α + β α > σ hl = α + α + β σ hl, l > p33 σ hl = α α + β l α β + α + β l σ h 3. σ hl = σ h + l α, l s β j σ j j= s β j η j 3.3 GARCH, M : r = µ + cσ + a, a = σ ɛ, σ = α + α a + β σ 3.4 gɛ = θɛ + γ ɛ E ɛ { θ + γɛ γe ɛ p43 gɛ = if ɛ θ γɛ γe ɛ if ɛ < p43 E ɛ = υ Γυ + / υ Γυ/ π 3.5 EGARCHm, s : a = σ ɛ, lnσ = α + + β B + + β s B s α B α m B m gɛ 3.6 a = σ ɛ, αb lnσ = αα + gɛ j= { 3.7 αb lnσ α + γ + θɛ = if ɛ α + γ θ ɛ = if ɛ < exp γ + θ a if a p44 σ = σ α σ expα exp γ θ a if a < σ p48 p48 p48 σh+ = σ α h exp α α expgɛ h+ E{expgɛ} = exp γ /π e θ+γ/ Φθ + γ + e θ γ/ Φγ θ ˆσ hj = ˆσ α h j expω { expθ + γ /Φθ + γ + expθ γ /Φγ θ }

23 Frequenly Asked Quesions in Quanaive Finance, P. Wilmo Q3 - Jensen s Inequaliy 3-5 If he payoff is a convex funcion, EP S T P ES T EfS = E f S + ɛ = E f S + ɛf S + ɛ f S + f S + f SEɛ = fes + f ESEɛ The LHS is greaer han he RHS by approximaely f ESEɛ 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 T he consrain E P exp γ < hen here exiss an equivalen measure Q on Ω such ha W = W + γ sds is a Brownian moion. Chaper 6 - The Mos Popular Probabiliy Disribuions Normal or Guassian: fx = x a exp πb b, < x <, µ = a, σ = b Lognormal: fx = Poisson: πbx exp lnx a b px = e a a x, x =,,..., µ = a, σ = a x!, x >, µ = exp a + b, σ = exp a + b expb Chi square noe ha his is a generalizaion of he usual chi-square disribuion: fx = e x+a/ v/ i= x i +v/ a i i j!γi + v/, x, µ = v + a, σ = v + a Gumbel: fx = b exp a x b exp e a x b, < x <, µ = a + γb, σ = 6 π b whereγ = Weibull: fx = c c c x a x a c + c + c + exp, x > a, µ = a+bγ, σ = b Γ Γ b b b c c c Suden s : fx = Γ c+ b πcγ c + x a b c c+, < x <, µ = a for c >, σ = 3 c b for c >. c

24 Pareo: fx = bab, x a, µ = ab xb+ Uniform: a for b >, σ = fx = b a, a < x < b, µ = a + b, b σ a = a b b b for b > Inverse normal: b fx = πx 3 exp b x a, x >, µ = a, σ = a3 x a b Gamma: fx = bγc c x a a x exp b b, x a, µ = a + bc, σ = b c Logisic fx = exp x a b b + exp x a, < x <, µ = a, σ = 3 π b b Laplace: fx = b exp x a, < x <, µ = a, σ = b b Cauchy: fx = πb + x a, < x <, momens do no exis b Bea: fx = Exponenial: Γc + d ΓcΓdb a c+d x ac b x d ad + bc, a x b, µ = c + d, cdb a σ = c + d + c + d fx = b exp a x b Lévy:, x a, µ = a + b, σ = b no closed form for densiy funcion, µ = µ, variance is infinie unless α =, when i is σ = v 4

25 Risk Managemen and Financial Insiuions, second ediion, J. Hull Chaper 9-Volailiy 9. Probυ > x = Kx α 9.5 σ n = page 86 m α i u n i m α i = 9.6 σ n = γv L σ n = ω + m α i u n i m α i u n i 9.8 σ n = λσ n + λu n 9.9 σ n = γv L + αu n + βσ n 9. σ n = ω + αu n + βσ n 9.4 Eσn+ = V L + α + β σn V L 9.5 σt = 5 {V L + } e at V V L at 9.6 σt σ e at σ at σt 5