May 2007 Exam MFE Solutions 1. Answer = (B)
|
|
- Candace Holt
- 6 years ago
- Views:
Transcription
1 May 007 Exam MFE Soluions. Answer = (B) Le D = he quarerly dividend. Using formula (9.), pu-call pariy adjused for deerminisic dividends, we have = D e D e 50 e = D ( ) Rearranging he equaion yields D = =.477, or D = Answer = (A) Le p be he rue probabiliy of he sock going up. Thus, pus + ( p)ds = e αh S (which is equaion (.3) on p. 347), yielding αh e d p =. u d Because α = 0., h =, u =.433, and d = 0.756, we have p = Answer = (C) Le P denoe he price of he European pu opion. Then, ½ 0.0 ½ P= 98e N( d) 00e N( d ) by formula (.3) wih S = 00, K = 98, δ = %, σ = 50%, r = 5.5%, and T = ½. Here, d is calculaed using formula (.a) and is equal o ; d is from formula (.b) and is equal o From he normal cdf able, N(0.06) = and N( 0.30) = = Thus, P / 0.0/ 98e e 0.38 = Answer = (E) For a special pu opion wih srike price K, he payoff upon immediae exercise is K 50. This value should be compared wih P, he price of he corresponding one-period European pu opion. The value of P can be deermined using pu-call pariy:
2 r δ P = Ke Se + C. Wih S = 50, r = 4%, and δ = 8%, P = K e 50 e + C = K C. K C P K From he able above, we see ha i is no opimal o exercise any of hese special pu opions immediaely. 5. Answer = (D) By (.9), σ opion = σ sock Ω = 0.50 Ω, where Ω is he opion elasiciy. By (.8), Ω = SΔ/C, where Δ is he opion dela, Δ = e N( d) (see page 383). By (.), C = Se N( d) Ke N( d) = SΔ Ke N( d). Thus, Ω = SΔ/C = SΔ/[SΔ Ke N( d) ] = /[ Ke N( d) /(SΔ)] = /{ [ Ke N( d) ]/[S e N( d) ]}. We are given S = 85, K = 80, δ = 0, r = 5.5%, T =. By equaion (.a), d is ; hence, Nd ( ) By equaion (.b), d is ; hence, Nd ( ) Wih hese values, we obain e N( d ) 85 e = 58.74, S Ke N( d) 80 e = Hence, Ω = /{ [ Ke N( d) ]/[S e N( d) ]}.78, and
3 σ opion = σ sock Ω =.39. Remark: To derive he volailiy of an opion by means of Iô s Lemma, see equaions (.8) and (.9). Chaper is no in he syllabus of Exam MFE. 6. Answer = (C) Because of he ideniy Maximum( S (3), S (3)) = Maximum( S (3) S (3), 0) + S (3), he payoff of he claim can be decomposed as he sum of he payoff of he exchange opion in saemen (v) of he problem and he price of sock a ime 3. In a noarbirage model, he price of he claim mus be equal o he sum of he exchange opion price (which is 0) and he prepaid forward price for delivery of sock a ime 3 (which is e δ 3 S (0)). So, he answer is 0 + e = Remark: If one buys e δ 3 share of sock a ime 0 and re-invess all dividends, one will have exacly one share of sock a ime Answer = (E) By formula (4.3), he call opion price is C = P(0, T)[F N(d ) K N(d )], where T =, P(0, T) = P(0, ) = , F = F 0, [P(, )] = P(0, )/P(0, ) = 0.887/ = , K = Wih σ = 0.05, we have ln( F / K) + σ T d = = , σ T d = d σ T = 0.6. Thus, N(d ) N(0.) = 0.583, N(d ) N(0.6) = Hence, C = P(0, T)[F N(d ) K N(d )] = [ ] =
4 Remarks: () The foonoe on page 79 poins ou ha he call opion price formula can also be expressed as C = P(0, )N(d ) KP(0, )N(d ). () The symbol F in he Black formula (4.3) denoes a forward price, bu he same symbol in he Black formula (.7) denoes a fuures price. There is no conradicion because, in he Black model discussed on page 38, he ineres rae is consan. I is saed on page 46 ha if he ineres rae were no random, hen forward and fuures price would be he same. (3) Consider a forward conrac, wih delivery dae T, for an underlying asse whose price a ime T is denoed by S(T). For < T, he ime- prepaid forward price is P F, T [S(T)] = E [e R(, T) S(T)] by risk-neural pricing. Here, we use he noaion in he las paragraph of page 783; E means he condiional expecaion wih respec o he risk-neural probabiliy measure given he informaion up o ime, and R(, T) = T r ( u) du. (4.) Thus, he ime- forward price is P F,T [S(T)] = F, T [S(T)] = E [e R(, T) S(T)]. P(, P(, Noing (4.0), we can rewrie his formula as R(, E [ e S( ] F, T [S(T)] =. R(, E [ e ] If he shor-rae, r(u), is no sochasic, hen he righ-hand side is R(, R(, E [ e S( ] e E [ S( ] = = E R(, R(, [S(T)], E [ e ] e E [] which is he formula for he ime- fuures price of he underlying asse deliverable a ime T. (4) Consider he special case S(T) = P(T, T + s). Then he ime- prepaid forward price of he zero-coupon bond deliverable a ime T is P F, T [P(T, T + s)] = E [ e R(, T) P(T, T + s)] R(, T) = E [ e E T [e R(T, T+s) ]] = E [ e R(, T) e R(T, T+s) ] = E [ e R(, T) R(T, T+s) ] = E [e R(, T+s) ] = P(, T + s), where he hird equaliy is by he law of ieraed expecaions. Thus, he ime- forward price is P F, T [P(T, T + s)] = F, T [P(T, T + s)] = P(, T + s), P(, P(, which is equaion (4.3). 4
5 8. Answer = (C) By formulas (.) and (.a, b), wih δ = 0, he call opion price is rt S(0) N( σ T) S(0) e ( ) e N σ T = S(0) N( σ T) N( σ T) = S(0) N( σ T) where he las equaliy is due o he ideniy N( x) = N(x). By (0.), he random variable ln S( ) Thus, saemen (iii) means ha σ = 0.4, and σ T = = =. Therefore, he opion price is S(0) N() = = [ ] [ ] is normally disribued wih variance σ. 9. Answer = (A) This problem is a modificaion of he example on page 805. Noe ha he example is abou cap paymens on a four-year loan, no a hree-year loan. An ineres rae cap pays he difference beween he realized ineres rae in a period and he cap rae, if he difference is posiive. Observe ha in his problem only r u and r uu are higher han 7.5%. A he u node, i is expeced ha a paymen of 00 (7.704% 7.5%) will be made a he end of he year. Thus, he presen value of he paymen a he node is 00 ( 7.704% 7.5% ) = % A he uu node, i is expeced ha a paymen of 00 (9.89% 7.5%) will be made a he end of he year. Thus, he presen value of he paymen a he node is 00 ( 9.89% 7.5% ) = % 5
6 The ree below corresponds o Figure 4.5 and Figure 4.9 of McDonald (006). Year 0 Year Year 6.000% $ % $ % $0 9.89% $ % $ % $0 By risk-neural pricing, he ime-0 price of he ineres rae cap is / + 6% / / % % = = Remark: The cap paymens are no $0.894 and $ They are $00 (7.704% 7.5%) o be paid one year afer he u node, and $00 (9.89% 7.5%) o be paid one year afer he uu node. One may be emped o pu $00 (7.704% 7.5%) a he uu node and a he ud node, and pu $00 (9.89% 7.5%) a he uuu node and a he uud node. Unforunaely, his can be confusing, because hese cash flows are no pah-independen. For example, if one reaches he ud node via he d node, hen here is no cap paymen because r d is less han 7.5%. 0. Answer = (B) Le y = number of unis of he sock you will buy, z = number of unis of he Call-II opion you will buy. If x or y urns ou o be negaive, his means ha you sell. Dela-neuraliy means = y + z Gamma-neuraliy means = y 0 + z
7 From he second equaion (he gamma-neural equaion), we obain z = 65./ = (This is sufficien o deermine ha (B) is he correc answer.) Subsiuing his in he firs equaion (he dela-neural equaion) yields y = = Answer = (D) Wih u =.8, d = 0.890, h = 0.5, andδ = 0, he risk-neural probabiliy ha he sock price will increase a he end of a period is ( r δ ) h e d e p* = = = (0.5) u d For he wo-period model, he sock prices are S 0 = 70 Su = us0 =.8 70 = 8.67 Sd = ds0 = = 6.30 Suu = usu = = Sud = dsu = = S = ds = = dd d Le P 0, P u, P d, P uu, P ud, P dd denoe he corresponding prices for he American pu opion. The hree prices a he opion expiry dae are P uu = max(k S uu, 0) = max( , 0) = 0, P ud = max(k S ud, 0) = max( , 0) = 6.4, P dd = max(k S dd, 0) = max( , 0) = By he backward inducion formula (0.), he wo prices a ime are P u = max(k S u, e rh [P uu p* + P ud ( p*)]) = max( , e 0.05/ [ ( 0.465)]) = e 0.05/ = 3.35, P d = max(k S d, e rh [P ud p* + P dd ( p*)]) = max( , e 0.05/ [ ( 0.465)]) = max(7.70, 5.7) = Finally, he ime-0 price of he American pu opion is P 0 = max(k S 0, e rh [P u p* + P d ( p*)]) = max(80 70, e 0.05/ [ ( 0.465)]) = max(0, 0.75) =
8 . Answer = (A) Define he funcion f (x, ) = xe (r r*)(t ). Then, G() = f (S(), ). Obviously, f ( x, ) = e (r r*)(t ), f ( x, ) = 0, and f ( x, ) = f(x, )(r r*)( ). x x By Iô s Lemma, we have dg() = e (r r*)(t ) ds() f(s(), )(r* r)d = e (r r*)(t ) S()[0.d + 0.4dZ()] + G()(r* r)d = G()[0.d + 0.4dZ()] + G()( )d = G()[( )d + 0.4dZ()] = G()[0.d + 0.4dZ()]. 3. Answer = (E) In a Vasicek model, zero-coupon bond prices are of he form B(, T) r PT (,, r) = ATe (, ). (4.6) Furhermore, he funcions AT (, ) and B(, are funcions of T. Therefore, we can rewrie formula (4.6) as P(, T, r) = exp α( T ) + β( T ) r. ( [ ]) The firs wo pieces of daa ell us: () () = e α β () () = e α β which, by aking logarihms, are equivalen o = α() + β () = α() + β () 0.05 The soluion of his pair of linear equaions is β () =.3 α () = The las piece of daa says () () r* = β e α Taking logarihms yields = α() + β () r *, or r* = ( )/.3 =
9 Remark: By comparing (4.9) wih (4.6), we see ha he word Vasicek in his problem can be changed o CIR. 4. Answer = (E) This is a one-period binomial model. Le p* be he risk-neural probabiliy of an increase in he sock price. (See page 3.) Then, r e e 45 p* = = = By risk-neural pricing, he price of he sraddle is e r [p* 70 K + ( p*) 45 K ] = e 0.08 [p* ( p*) ] = e 0.08 [p* 0 + ( p*) 5] = e 0.08 [5p* + 5] e 0.08 [ ] = e = Answer = (C) This is a variaion of Example.3 on page 380. Because of he discree dividend, we are o use he version of he Black-Scholes pu opion formula ha is in erms of prepaid forward prices. The prepaid forward price of he sock is P F0,/ ( S ) = 50.50e 0.05/3 = = We apply formula (.a), wih S = , K = 50, r = 0.05, δ = 0, σ = 0.3, and T = ½, o obain d = {ln(48.548/50) + [ (0.3) /] ½}/{0.3 ½} = { }/0.3 = (This is he same as applying he formula for d ha follows (.5) on page 380.) Then, d = = I now follows from he prepaid forward price version of (.3) ha he pu opion price is 50e 0.05/ N(+0.3) N( 0.08) = ( ) ( ) = =
10 6. Answer = (D) r δ Define β =. Then, he formulas on page 403 for h and h are σ and r h = + + σ β β r h = β β + σ. Adding hese wo equaions yields h + h = β. Hence, β = 7/9 or β = 7/8. For r = 5% and σ = 0.3, r h = + + σ β β = Alernaive soluion: The parameers h and h are he posiive and negaive roos, respecively, of he quadraic equaion σ h σ + (r δ )h r = 0; (*) see he sudy noe Some Remarks on Derivaives Markes. Thus, σ h σ σ + (r δ )h r = (h h )(h h ). Consequenly, σ σ r δ = ( h h ) σ 7 = 9. Hence, he quadraic equaion (*) becomes h + ( 7 )h 0.05 = 0, 9 he posiive roo of which is h. Remark: For a posiive δ, he posiive roo h is in fac greaer han. 0
11 7. Answer = (B) In erms of he noaion in Secion 4.5, K = 90 and K = 00. By (.), and (.a, b), saemen (ii) of he problem is δ 4 (0) T = S e Nd ( ) Ke Nd ( ), () where S ( ) 0 = 80, ln( S(0) / K ) + ( r δ + σ ) T d =, σ T and ln( S(0) / K ) + ( r δ σ ) T d = d σ T =. σ T Do noe ha boh d and d depend on K, bu no on K. From he las paragraph on page 383 and from saemen (iii), we have Δ= e N( d) = 0., and hence equaion () becomes 4 = e N( d), or e N( d ) = ( ) /00 = 0.. By (4.5) on page 458, he gap call opion price is S(0) e Nd ( ) Ke Nd ( ) = = 5.. Remark: The payoff of he gap call opion is [S(T) K ] I(S(T) > K ), where I(S(T) > K ) is he indicaor random variable, which akes he value if S(T) > K and he value 0 oherwise. Because he payoff can be expressed as S(T) I(S(T) > K ) K I(S(T) > K ), we can obain he pricing formula (4.5) by showing ha he ime-0 price for he ime-t payoff S(T) I(S(T) > K ) is S(0) e N( d), and he ime-0 price for he ime-t payoff I(S(T) > K ) is e N(d ).
12 Noe ha boh d and d are calculaed using he srike price K. We can use riskneural pricing o verify hese wo resuls: E*[e S(T) I(S(T) > K )] = S(0) e N( d), which is he pricing formula for a European asse-or-nohing (or digial share) call opion, and E*[e I(S(T) > K )] = e N(d ), which is he pricing formula for a European cash-or- nohing (or digial cash) call opion. Here, we follow he noaion on pages 604 and 605 ha he aserisk is used o signify ha he expecaion is aken wih respec o he risk-neural probabiliy measure. Under he risk-neural probabiliy measure, he random variable ln[s(t)/s(0)] is normally disribued wih mean (r δ σ )T and variance σ T. The second expecaion formula, which can be readily simplified as E*[I(S(T) > K )] = N(d ), is paricularly easy o verify: Because an indicaor random variable akes he values and 0 only, we have E*[I(S(T) > K )] = Prob*[S(T) > K ], which is he same as Prob*(ln[S(T)/S(0)] > ln[k /S(0)]). To evaluae his probabiliy, we use a sandard mehod, which is also described on pages 590 and 59. We subrac he mean of ln[s(t)/s(0)] from boh sides of he inequaliy and hen divide by he sandard deviaion of ln[s(t)/s(0)]. The lef-hand side of he inequaliy is now a sandard normal random variable, Z, and he righ-hand side is ln[ K / S(0)] ( r δ σ / ) T ln[ S(0) / K] + ( r δ σ / ) T = σ T σ T = d. Thus, we have E*[I(S(T) > K )] = Prob*[S(T) > K ], = Prob(Z > d ) = N( d ) = N(d ). The firs expecaion formula, E*[e S(T) I(S(T) > K )] = S(0) e N( d), is harder o derive. One mehod is o use formula (8.9), which is in he syllabus of Exam C, bu no in he syllabus of Exam MFE. A more elegan way is he acuarial mehod of Esscher ransforms, which is no par of he syllabus of any acuarial examinaion. I shows ha he expecaion of a produc, E*[e S(T) I(S(T) > K )], can be facorized as a produc of expecaions, E*[e S(T)] E**[I(S(T) > K )], where ** signifies a changed probabiliy measure. I follows from (0.6) and (0.4) ha E*[e S(T)] = e S(0).
13 To evaluae he expecaion E**[I(S(T) > K )], which is Prob**[S(T) > K ], one shows ha, under he probabiliy measure **, he random variable ln[s(t)/s(0)] is normally disribued wih mean (r δ σ )T + σ T = (r δ + σ )T, and variance σ T. Then, wih seps idenical o hose above, we have E**[I(S(T) > K )] = Prob**[S(T) > K ], = Prob(Z > d ) = N( d ) = N(d ). Alernaive soluion: Because he payoff of he gap call opion is [S(T) K ] I(S(T) > K ) = [S(T) K ] I(S(T) > K ) + (K K ) I(S(T) > K ), he price of he gap call opion mus be equal o he sum of he price of a European call opion wih he srike price K and he price of (K K ) unis of he corresponding cash-or-nohing call opion. Thus, wih K = 90, K = 00, and saemen (ii), he price of he gap call opion is 4 + (00 90) e Prob*[S(T) > 00] = 4+ 0 e N( d). On he oher hand, from (ii), (iii), and (.), i follows ha 4 = 80(0.) 00 e N( d). Thus, e N( d) = 0., and he price of he gap call opion is = Answer = (A) In an arbirage-free model, wo asses having he same source of randomness (heir prices driven by he same Brownian moion) mus have he same Sharpe raio; see Secion 0.4. Wih r = 4%, we hus have G 0.04 =, 0. H or G = 0.5H () If f(x) is a wice-differeniable funcion of x, hen Iô s Lemma (page 664) simplifies as df(y()) = f (Y())dY() + f (Y())[dY()], because f (x) = 0. If f(x) = ln x, hen f (x) = /x and f (x) = /x. Hence, 3
14 d(ln[y()]) = dy() + [dy ( )] Y( ). () [ Y ( )] We are given ha dy() = Y()[Gd + HdZ()]. (3) Thus, [dy()] = {Y()[Gd + HdZ()]} = [Y()] H d, (4) by applying he muliplicaion rules (0.7) on pages 658 and 659. Subsiuing (3) and (4) in () and simplifying yields d(ln[y()]) = (G H )d + HdZ(). Comparing his equaion wih he one in (i), we have H = σ, (5) G H = (6) Applying () and (5) o (6) yields a quadraic equaion of σ, σ 0.5σ = 0, whose roos can be found by using he quadraic formula or by facorizing, (σ 0.)(σ 0.4) = 0. By condiion (iii), we canno have σ = 0.4. Thus, σ = 0.. Subsiuing H = 0. in () yields G = = Remark: Exercise 0. on page 675 is o use Iô s Lemma o evaluae d[ln(s)]. 9. Answer = (D) The dela-gamma approximaion is merely he Taylor series approximaion wih up o he quadraic erm. In erms of he Greek symbols, he firs derivaive is Δ, and he second derivaive is Γ. The approximaion formula is P(S + ε) P(S) + ε Δ + ε Γ. (3. & 3.5) Wih P(30) = 4, Δ = 0.8, Γ = 0.0, and ε =.50, we have P(3.5) 4 + (.5)( 0.8) + (.5) (0.) =
The Mathematics Of Stock Option Valuation - Part Four Deriving The Black-Scholes Model Via Partial Differential Equations
The Mahemaics Of Sock Opion Valuaion - Par Four Deriving The Black-Scholes Model Via Parial Differenial Equaions Gary Schurman, MBE, CFA Ocober 1 In Par One we explained why valuing a call opion as a sand-alone
More informationTentamen i 5B1575 Finansiella Derivat. Måndag 27 augusti 2007 kl Answers and suggestions for solutions.
Tenamen i 5B1575 Finansiella Deriva. Måndag 27 augusi 2007 kl. 14.00 19.00. Answers and suggesions for soluions. 1. (a) For he maringale probabiliies we have q 1 + r d u d 0.5 Using hem we obain he following
More informationMAFS Quantitative Modeling of Derivative Securities
MAFS 5030 - Quaniaive Modeling of Derivaive Securiies Soluion o Homework Three 1 a For > s, consider E[W W s F s = E [ W W s + W s W W s Fs We hen have = E [ W W s F s + Ws E [W W s F s = s, E[W F s =
More informationSome Remarks on Derivatives Markets (third edition, 2013)
Some Remarks on Derivaives Markes (hird ediion, 03) Elias S. W. Shiu. The parameer δ in he Black-Scholes formula The Black-Scholes opion-pricing formula is given in Chaper of McDonald wihou proof. A raher
More informationMatematisk statistik Tentamen: kl FMS170/MASM19 Prissättning av Derivattillgångar, 9 hp Lunds tekniska högskola. Solution.
Maemaisk saisik Tenamen: 8 5 8 kl 8 13 Maemaikcenrum FMS17/MASM19 Prissäning av Derivaillgångar, 9 hp Lunds ekniska högskola Soluion. 1. In he firs soluion we look a he dynamics of X using Iôs formula.
More informationINSTITUTE OF ACTUARIES OF INDIA
INSIUE OF ACUARIES OF INDIA EAMINAIONS 23 rd May 2011 Subjec S6 Finance and Invesmen B ime allowed: hree hours (9.45* 13.00 Hrs) oal Marks: 100 INSRUCIONS O HE CANDIDAES 1. Please read he insrucions on
More informationBlack-Scholes Model and Risk Neutral Pricing
Inroducion echniques Exercises in Financial Mahemaics Lis 3 UiO-SK45 Soluions Hins Auumn 5 eacher: S Oriz-Laorre Black-Scholes Model Risk Neural Pricing See Benh s book: Exercise 44, page 37 See Benh s
More informationMarket Models. Practitioner Course: Interest Rate Models. John Dodson. March 29, 2009
s Praciioner Course: Ineres Rae Models March 29, 2009 In order o value European-syle opions, we need o evaluae risk-neural expecaions of he form V (, T ) = E [D(, T ) H(T )] where T is he exercise dae,
More informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 05 h November 007 Subjec CT8 Financial Economics Time allowed: Three Hours (14.30 17.30 Hrs) Toal Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1) Do no wrie your
More informationIntroduction to Black-Scholes Model
4 azuhisa Masuda All righs reserved. Inroducion o Black-choles Model Absrac azuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York, NY 6-439 Email:
More informationUCLA Department of Economics Fall PhD. Qualifying Exam in Macroeconomic Theory
UCLA Deparmen of Economics Fall 2016 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and you are o complee each par. Answer each par in a separae bluebook. All
More informationPricing FX Target Redemption Forward under. Regime Switching Model
In. J. Conemp. Mah. Sciences, Vol. 8, 2013, no. 20, 987-991 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.12988/ijcms.2013.311123 Pricing FX Targe Redempion Forward under Regime Swiching Model Ho-Seok
More informationModels of Default Risk
Models of Defaul Risk Models of Defaul Risk 1/29 Inroducion We consider wo general approaches o modelling defaul risk, a risk characerizing almos all xed-income securiies. The srucural approach was developed
More informationErratic Price, Smooth Dividend. Variance Bounds. Present Value. Ex Post Rational Price. Standard and Poor s Composite Stock-Price Index
Erraic Price, Smooh Dividend Shiller [1] argues ha he sock marke is inefficien: sock prices flucuae oo much. According o economic heory, he sock price should equal he presen value of expeced dividends.
More informationTentamen i 5B1575 Finansiella Derivat. Torsdag 25 augusti 2005 kl
Tenamen i 5B1575 Finansiella Deriva. Torsdag 25 augusi 2005 kl. 14.00 19.00. Examinaor: Camilla Landén, el 790 8466. Tillåna hjälpmedel: Av insiuionen ulånad miniräknare. Allmänna anvisningar: Lösningarna
More informationComputations in the Hull-White Model
Compuaions in he Hull-Whie Model Niels Rom-Poulsen Ocober 8, 5 Danske Bank Quaniaive Research and Copenhagen Business School, E-mail: nrp@danskebank.dk Specificaions In he Hull-Whie model, he Q dynamics
More informationBrownian motion. Since σ is not random, we can conclude from Example sheet 3, Problem 1, that
Advanced Financial Models Example shee 4 - Michaelmas 8 Michael Tehranchi Problem. (Hull Whie exension of Black Scholes) Consider a marke wih consan ineres rae r and wih a sock price modelled as d = (µ
More informationEquivalent Martingale Measure in Asian Geometric Average Option Pricing
Journal of Mahemaical Finance, 4, 4, 34-38 ublished Online Augus 4 in SciRes hp://wwwscirporg/journal/jmf hp://dxdoiorg/436/jmf4447 Equivalen Maringale Measure in Asian Geomeric Average Opion ricing Yonggang
More informationAn Analytical Implementation of the Hull and White Model
Dwigh Gran * and Gauam Vora ** Revised: February 8, & November, Do no quoe. Commens welcome. * Douglas M. Brown Professor of Finance, Anderson School of Managemen, Universiy of New Mexico, Albuquerque,
More informationPricing Vulnerable American Options. April 16, Peter Klein. and. Jun (James) Yang. Simon Fraser University. Burnaby, B.C. V5A 1S6.
Pricing ulnerable American Opions April 16, 2007 Peer Klein and Jun (James) Yang imon Fraser Universiy Burnaby, B.C. 5A 16 pklein@sfu.ca (604) 268-7922 Pricing ulnerable American Opions Absrac We exend
More informationChange of measure and Girsanov theorem
and Girsanov heorem 80-646-08 Sochasic calculus I Geneviève Gauhier HEC Monréal Example 1 An example I Le (Ω, F, ff : 0 T g, P) be a lered probabiliy space on which a sandard Brownian moion W P = W P :
More informationMA Advanced Macro, 2016 (Karl Whelan) 1
MA Advanced Macro, 2016 (Karl Whelan) 1 The Calvo Model of Price Rigidiy The form of price rigidiy faced by he Calvo firm is as follows. Each period, only a random fracion (1 ) of firms are able o rese
More informationOption pricing and hedging in jump diffusion models
U.U.D.M. Projec Repor 21:7 Opion pricing and hedging in jump diffusion models Yu Zhou Examensarbee i maemaik, 3 hp Handledare och examinaor: Johan ysk Maj 21 Deparmen of Mahemaics Uppsala Universiy Maser
More informationCh 6. Option Pricing When Volatility is Non-Constant
Ch 6. Opion Pricing When Volailiy is Non-Consan I. Volailiy Smile II. Opion Pricing When Volailiy is a Funcion of S and III. Opion Pricing Under Sochasic Volailiy Process I is convincingly believed ha
More informationYou should turn in (at least) FOUR bluebooks, one (or more, if needed) bluebook(s) for each question.
UCLA Deparmen of Economics Spring 05 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and each par is worh 0 poins. Pars and have one quesion each, and Par 3 has
More informationAlexander L. Baranovski, Carsten von Lieres and André Wilch 18. May 2009/Eurobanking 2009
lexander L. Baranovski, Carsen von Lieres and ndré Wilch 8. May 2009/ Defaul inensiy model Pricing equaion for CDS conracs Defaul inensiy as soluion of a Volerra equaion of 2nd kind Comparison o common
More informationAvailable online at ScienceDirect
Available online a www.sciencedirec.com ScienceDirec Procedia Economics and Finance 8 ( 04 658 663 s Inernaional Conference 'Economic Scienific Research - Theoreical, Empirical and Pracical Approaches',
More informationFinal Exam Answers Exchange Rate Economics
Kiel Insiu für Welwirhschaf Advanced Sudies in Inernaional Economic Policy Research Spring 2005 Menzie D. Chinn Final Exam Answers Exchange Rae Economics This exam is 1 ½ hours long. Answer all quesions.
More informationEconomic Growth Continued: From Solow to Ramsey
Economic Growh Coninued: From Solow o Ramsey J. Bradford DeLong May 2008 Choosing a Naional Savings Rae Wha can we say abou economic policy and long-run growh? To keep maers simple, le us assume ha he
More informationInterest Rate Products
Chaper 9 Ineres Rae Producs Copyrigh c 2008 20 Hyeong In Choi, All righs reserved. 9. Change of Numeraire and he Invariance of Risk Neural Valuaion The financial heory we have developed so far depends
More informationValuation and Hedging of Correlation Swaps. Mats Draijer
Valuaion and Hedging of Correlaion Swaps Mas Draijer 4298829 Sepember 27, 2017 Absrac The aim of his hesis is o provide a formula for he value of a correlaion swap. To ge o his formula, a model from an
More informationIJRSS Volume 2, Issue 2 ISSN:
A LOGITIC BROWNIAN MOTION WITH A PRICE OF DIVIDEND YIELDING AET D. B. ODUOR ilas N. Onyango _ Absrac: In his paper, we have used he idea of Onyango (2003) he used o develop a logisic equaion used in naural
More informationOptimal Early Exercise of Vulnerable American Options
Opimal Early Exercise of Vulnerable American Opions March 15, 2008 This paper is preliminary and incomplee. Opimal Early Exercise of Vulnerable American Opions Absrac We analyze he effec of credi risk
More informationLIDSTONE IN THE CONTINUOUS CASE by. Ragnar Norberg
LIDSTONE IN THE CONTINUOUS CASE by Ragnar Norberg Absrac A generalized version of he classical Lidsone heorem, which deals wih he dependency of reserves on echnical basis and conrac erms, is proved in
More informationt=1 C t e δt, and the tc t v t i t=1 C t (1 + i) t = n tc t (1 + i) t C t (1 + i) t = C t vi
Exam 4 is Th. April 24. You are allowed 13 shees of noes and a calculaor. ch. 7: 137) Unless old oherwise, duraion refers o Macaulay duraion. The duraion of a single cashflow is he ime remaining unil mauriy,
More informationCENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T. J. KEHOE MACROECONOMICS I WINTER 2011 PROBLEM SET #6
CENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T J KEHOE MACROECONOMICS I WINTER PROBLEM SET #6 This quesion requires you o apply he Hodrick-Presco filer o he ime series for macroeconomic variables for he
More informationBond Prices and Interest Rates
Winer erm 1999 Bond rice Handou age 1 of 4 Bond rices and Ineres Raes A bond is an IOU. ha is, a bond is a promise o pay, in he fuure, fixed amouns ha are saed on he bond. he ineres rae ha a bond acually
More informationForeign Exchange, ADR s and Quanto-Securities
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Foreign Exchange, ADR s and Quano-Securiies These noes consider foreign exchange markes and he pricing of derivaive
More informationOn the multiplicity of option prices under CEV with positive elasticity of variance
Rev Deriv Res (207) 20: 3 DOI 0.007/s47-06-922-2 On he mulipliciy of opion prices under CEV wih posiive elasiciy of variance Dirk Veesraeen Published online: 4 April 206 The Auhor(s) 206. This aricle is
More informationExtended MAD for Real Option Valuation
Exended MAD for Real Opion Valuaion A Case Sudy of Abandonmen Opion Carol Alexander Xi Chen Charles Ward Absrac This paper exends he markeed asse disclaimer approach for real opion valuaion. In sharp conras
More informationQuanto Options. Uwe Wystup. MathFinance AG Waldems, Germany 19 September 2008
Quano Opions Uwe Wysup MahFinance AG Waldems, Germany www.mahfinance.com 19 Sepember 2008 Conens 1 Quano Opions 2 1.1 FX Quano Drif Adjusmen.......................... 2 1.1.1 Exensions o oher Models.......................
More informationOption Valuation of Oil & Gas E&P Projects by Futures Term Structure Approach. Hidetaka (Hugh) Nakaoka
Opion Valuaion of Oil & Gas E&P Projecs by Fuures Term Srucure Approach March 9, 2007 Hideaka (Hugh) Nakaoka Former CIO & CCO of Iochu Oil Exploraion Co., Ld. Universiy of Tsukuba 1 Overview 1. Inroducion
More informationVolatility and Hedging Errors
Volailiy and Hedging Errors Jim Gaheral Sepember, 5 1999 Background Derivaive porfolio bookrunners ofen complain ha hedging a marke-implied volailiies is sub-opimal relaive o hedging a heir bes guess of
More informationBlack-Scholes and the Volatility Surface
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Black-Scholes and he Volailiy Surface When we sudied discree-ime models we used maringale pricing o derive he Black-Scholes
More informationSingle Premium of Equity-Linked with CRR and CIR Binomial Tree
The 7h SEAMS-UGM Conference 2015 Single Premium of Equiy-Linked wih CRR and CIR Binomial Tree Yunia Wulan Sari 1,a) and Gunardi 2,b) 1,2 Deparmen of Mahemaics, Faculy of Mahemaics and Naural Sciences,
More informationAn Incentive-Based, Multi-Period Decision Model for Hierarchical Systems
Wernz C. and Deshmukh A. An Incenive-Based Muli-Period Decision Model for Hierarchical Sysems Proceedings of he 3 rd Inernaional Conference on Global Inerdependence and Decision Sciences (ICGIDS) pp. 84-88
More informationForwards and Futures
Handou #6 for 90.2308 - Spring 2002 (lecure dae: 4/7/2002) orward Conrac orward and uure A ime (where 0 < ): ener a forward conrac, in which you agree o pay O (called "forward price") for one hare of he
More informationDynamic Programming Applications. Capacity Expansion
Dynamic Programming Applicaions Capaciy Expansion Objecives To discuss he Capaciy Expansion Problem To explain and develop recursive equaions for boh backward approach and forward approach To demonsrae
More informationPricing formula for power quanto options with each type of payoffs at maturity
Global Journal of Pure and Applied Mahemaics. ISSN 0973-1768 Volume 13, Number 9 (017, pp. 6695 670 Research India Publicaions hp://www.ripublicaion.com/gjpam.hm Pricing formula for power uano opions wih
More information(c) Suppose X UF (2, 2), with density f(x) = 1/(1 + x) 2 for x 0 and 0 otherwise. Then. 0 (1 + x) 2 dx (5) { 1, if t = 0,
:46 /6/ TOPIC Momen generaing funcions The n h momen of a random variable X is EX n if his quaniy exiss; he momen generaing funcion MGF of X is he funcion defined by M := Ee X for R; he expecaion in exiss
More informationA pricing model for the Guaranteed Lifelong Withdrawal Benefit Option
A pricing model for he Guaraneed Lifelong Wihdrawal Benefi Opion Gabriella Piscopo Universià degli sudi di Napoli Federico II Diparimeno di Maemaica e Saisica Index Main References Survey of he Variable
More informationHull-White one factor model Version
Hull-Whie one facor model Version 1.0.17 1 Inroducion This plug-in implemens Hull and Whie one facor models. reference on his model see [?]. For a general 2 How o use he plug-in In he Fairma user inerface
More information1 Purpose of the paper
Moneary Economics 2 F.C. Bagliano - Sepember 2017 Noes on: F.X. Diebold and C. Li, Forecasing he erm srucure of governmen bond yields, Journal of Economerics, 2006 1 Purpose of he paper The paper presens
More informationQuestion 1 / 15 Question 2 / 15 Question 3 / 28 Question 4 / 42
Deparmen of Applied Economics Johns Hopkins Universiy Economics 602 Macroeconomic Theory and olicy Final Exam rofessor Sanjay Chugh Fall 2008 December 8, 2008 NAME: The Exam has a oal of four (4) quesions
More informationJarrow-Lando-Turnbull model
Jarrow-Lando-urnbull model Characerisics Credi raing dynamics is represened by a Markov chain. Defaul is modelled as he firs ime a coninuous ime Markov chain wih K saes hiing he absorbing sae K defaul
More informationLecture Notes to Finansiella Derivat (5B1575) VT Note 1: No Arbitrage Pricing
Lecure Noes o Finansiella Deriva (5B1575) VT 22 Harald Lang, KTH Maemaik Noe 1: No Arbirage Pricing Le us consider a wo period marke model. A conrac is defined by a sochasic payoff X a bounded sochasic
More informationMacroeconomics II A dynamic approach to short run economic fluctuations. The DAD/DAS model.
Macroeconomics II A dynamic approach o shor run economic flucuaions. The DAD/DAS model. Par 2. The demand side of he model he dynamic aggregae demand (DAD) Inflaion and dynamics in he shor run So far,
More information7 pages 1. Hull and White Generalized model. Ismail Laachir. March 1, Model Presentation 1
7 pages 1 Hull and Whie Generalized model Ismail Laachir March 1, 212 Conens 1 Model Presenaion 1 2 Calibraion of he model 3 2.1 Fiing he iniial yield curve................... 3 2.2 Fiing he caple implied
More informationMORNING SESSION. Date: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Quaniaive Finance and Invesmen Core Exam QFICORE MORNING SESSION Dae: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucions 1. This examinaion
More informationOnline Appendix. Using the reduced-form model notation proposed by Doshi, el al. (2013), 1. and Et
Online Appendix Appendix A: The concep in a muliperiod framework Using he reduced-form model noaion proposed by Doshi, el al. (2013), 1 he yearly CDS spread S c,h for a h-year sovereign c CDS conrac can
More informationR e. Y R, X R, u e, and. Use the attached excel spreadsheets to
HW # Saisical Financial Modeling ( P Theodossiou) 1 The following are annual reurns for US finance socks (F) and he S&P500 socks index (M) Year Reurn Finance Socks Reurn S&P500 Year Reurn Finance Socks
More informationa. If Y is 1,000, M is 100, and the growth rate of nominal money is 1 percent, what must i and P be?
Problem Se 4 ECN 101 Inermediae Macroeconomics SOLUTIONS Numerical Quesions 1. Assume ha he demand for real money balance (M/P) is M/P = 0.6-100i, where is naional income and i is he nominal ineres rae.
More informationEcon 546 Lecture 4. The Basic New Keynesian Model Michael Devereux January 2011
Econ 546 Lecure 4 The Basic New Keynesian Model Michael Devereux January 20 Road map for his lecure We are evenually going o ge 3 equaions, fully describing he NK model The firs wo are jus he same as before:
More informationCoupling Smiles. November 18, 2006
Coupling Smiles Valdo Durrleman Deparmen of Mahemaics Sanford Universiy Sanford, CA 94305, USA Nicole El Karoui Cenre de Mahémaiques Appliquées Ecole Polyechnique 91128 Palaiseau, France November 18, 2006
More informationOn Monte Carlo Simulation for the HJM Model Based on Jump
On Mone Carlo Simulaion for he HJM Model Based on Jump Kisoeb Park 1, Moonseong Kim 2, and Seki Kim 1, 1 Deparmen of Mahemaics, Sungkyunkwan Universiy 44-746, Suwon, Korea Tel.: +82-31-29-73, 734 {kisoeb,
More informationOPTIMUM FISCAL AND MONETARY POLICY USING THE MONETARY OVERLAPPING GENERATION MODELS
Kuwai Chaper of Arabian Journal of Business and Managemen Review Vol. 3, No.6; Feb. 2014 OPTIMUM FISCAL AND MONETARY POLICY USING THE MONETARY OVERLAPPING GENERATION MODELS Ayoub Faramarzi 1, Dr.Rahim
More informationCHAPTER 3 How to Calculate Present Values. Answers to Practice Questions
CHAPTER 3 How o Calculae Presen Values Answers o Pracice Quesions. a. PV $00/.0 0 $90.53 b. PV $00/.3 0 $9.46 c. PV $00/.5 5 $ 3.5 d. PV $00/. + $00/. + $00/. 3 $40.8. a. DF + r 0.905 r 0.050 0.50% b.
More informationChanges of Numeraire for Pricing Futures, Forwards, and Options
Changes of Numeraire for Pricing Fuures, Forwards, and Opions Mark Schroder Michigan Sae Universiy A change of numeraire argumen is used o derive a general opion pariy, or equivalence, resul relaing American
More informationFINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004
FINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004 This exam has 50 quesions on 14 pages. Before you begin, please check o make sure ha your copy has all 50 quesions and all 14 pages.
More informationSynthetic CDO s and Basket Default Swaps in a Fixed Income Credit Portfolio
Synheic CDO s and Baske Defaul Swaps in a Fixed Income Credi Porfolio Louis Sco June 2005 Credi Derivaive Producs CDO Noes Cash & Synheic CDO s, various ranches Invesmen Grade Corporae names, High Yield
More informationSTATIONERY REQUIREMENTS SPECIAL REQUIREMENTS 20 Page booklet List of statistical formulae New Cambridge Elementary Statistical Tables
ECONOMICS RIPOS Par I Friday 7 June 005 9 Paper Quaniaive Mehods in Economics his exam comprises four secions. Secions A and B are on Mahemaics; Secions C and D are on Saisics. You should do he appropriae
More informationMoney in a Real Business Cycle Model
Money in a Real Business Cycle Model Graduae Macro II, Spring 200 The Universiy of Nore Dame Professor Sims This documen describes how o include money ino an oherwise sandard real business cycle model.
More informationFinancial Econometrics (FinMetrics02) Returns, Yields, Compounding, and Horizon
Financial Economerics FinMerics02) Reurns, Yields, Compounding, and Horizon Nelson Mark Universiy of Nore Dame Fall 2017 Augus 30, 2017 1 Conceps o cover Yields o mauriy) Holding period) reurns Compounding
More informationPrinciples of Finance CONTENTS
Principles of Finance CONENS Value of Bonds and Equiy... 3 Feaures of bonds... 3 Characerisics... 3 Socks and he sock marke... 4 Definiions:... 4 Valuing equiies... 4 Ne reurn... 4 idend discoun model...
More informationOn the Impact of Inflation and Exchange Rate on Conditional Stock Market Volatility: A Re-Assessment
MPRA Munich Personal RePEc Archive On he Impac of Inflaion and Exchange Rae on Condiional Sock Marke Volailiy: A Re-Assessmen OlaOluwa S Yaya and Olanrewaju I Shiu Deparmen of Saisics, Universiy of Ibadan,
More informationFinancial Econometrics Jeffrey R. Russell Midterm Winter 2011
Name Financial Economerics Jeffrey R. Russell Miderm Winer 2011 You have 2 hours o complee he exam. Use can use a calculaor. Try o fi all your work in he space provided. If you find you need more space
More informationCredit Spread Option Valuation under GARCH. Working Paper July 2000 ISSN :
Credi Spread Opion Valuaion under GARCH by Nabil ahani Working Paper -7 July ISSN : 6-334 Financial suppor by he Risk Managemen Chair is acknowledged. he auhor would like o hank his professors Peer Chrisoffersen
More informationANSWER ALL QUESTIONS. CHAPTERS 6-9; (Blanchard)
ANSWER ALL QUESTIONS CHAPTERS 6-9; 18-20 (Blanchard) Quesion 1 Discuss in deail he following: a) The sacrifice raio b) Okun s law c) The neuraliy of money d) Bargaining power e) NAIRU f) Wage indexaion
More informationFinancial Markets And Empirical Regularities An Introduction to Financial Econometrics
Financial Markes And Empirical Regulariies An Inroducion o Financial Economerics SAMSI Workshop 11/18/05 Mike Aguilar UNC a Chapel Hill www.unc.edu/~maguilar 1 Ouline I. Hisorical Perspecive on Asse Prices
More informationUnemployment and Phillips curve
Unemploymen and Phillips curve 2 of The Naural Rae of Unemploymen and he Phillips Curve Figure 1 Inflaion versus Unemploymen in he Unied Saes, 1900 o 1960 During he period 1900 o 1960 in he Unied Saes,
More informationA Theory of Tax Effects on Economic Damages. Scott Gilbert Southern Illinois University Carbondale. Comments? Please send to
A Theory of Tax Effecs on Economic Damages Sco Gilber Souhern Illinois Universiy Carbondale Commens? Please send o gilbers@siu.edu ovember 29, 2012 Absrac This noe provides a heoreical saemen abou he effec
More informationHEDGING VOLATILITY RISK
HEDGING VOLAILIY RISK Menachem Brenner Sern School of Business New York Universiy New York, NY 00, U.S.A. Email: mbrenner@sern.nyu.edu Ernes Y. Ou ABN AMRO, Inc. Chicago, IL 60604, U.S.A. Email: Yi.Ou@abnamro.com
More information(1 + Nominal Yield) = (1 + Real Yield) (1 + Expected Inflation Rate) (1 + Inflation Risk Premium)
5. Inflaion-linked bonds Inflaion is an economic erm ha describes he general rise in prices of goods and services. As prices rise, a uni of money can buy less goods and services. Hence, inflaion is an
More informationFundamental Basic. Fundamentals. Fundamental PV Principle. Time Value of Money. Fundamental. Chapter 2. How to Calculate Present Values
McGraw-Hill/Irwin Chaper 2 How o Calculae Presen Values Principles of Corporae Finance Tenh Ediion Slides by Mahew Will And Bo Sjö 22 Copyrigh 2 by he McGraw-Hill Companies, Inc. All righs reserved. Fundamenal
More informationAspects of Some Exotic Options
Aspecs of Some Exoic Opions Nadia Theron Assignmen presened in parial fulfilmen of he requiremens for he degree of MASTER OF COMMERCE in he Deparmen of Saisics and Acuarial Science, Faculy of Economic
More informationSpring 2011 Social Sciences 7418 University of Wisconsin-Madison
Economics 32, Sec. 1 Menzie D. Chinn Spring 211 Social Sciences 7418 Universiy of Wisconsin-Madison Noes for Econ 32-1 FALL 21 Miderm 1 Exam The Fall 21 Econ 32-1 course used Hall and Papell, Macroeconomics
More informationCHAPTER CHAPTER18. Openness in Goods. and Financial Markets. Openness in Goods, and Financial Markets. Openness in Goods,
Openness in Goods and Financial Markes CHAPTER CHAPTER18 Openness in Goods, and Openness has hree disinc dimensions: 1. Openness in goods markes. Free rade resricions include ariffs and quoas. 2. Openness
More informationLIBOR MARKET MODEL AND GAUSSIAN HJM EXPLICIT APPROACHES TO OPTION ON COMPOSITION
LIBOR MARKET MODEL AND GAUSSIAN HJM EXPLICIT APPROACHES TO OPTION ON COMPOSITION MARC HENRARD Absrac. The win brohers Libor Marke and Gaussian HJM models are invesigaed. A simple exoic opion, floor on
More informationCHAPTER CHAPTER26. Fiscal Policy: A Summing Up. Prepared by: Fernando Quijano and Yvonn Quijano
Fiscal Policy: A Summing Up Prepared by: Fernando Quijano and vonn Quijano CHAPTER CHAPTER26 2006 Prenice Hall usiness Publishing Macroeconomics, 4/e Olivier lanchard Chaper 26: Fiscal Policy: A Summing
More informationAsymmetry and Leverage in Stochastic Volatility Models: An Exposition
Asymmery and Leverage in Sochasic Volailiy Models: An xposiion Asai, M. a and M. McAleer b a Faculy of conomics, Soka Universiy, Japan b School of conomics and Commerce, Universiy of Wesern Ausralia Keywords:
More informationFAIR VALUATION OF INSURANCE LIABILITIES. Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004
FAIR VALUATION OF INSURANCE LIABILITIES Pierre DEVOLDER Universié Caholique de Louvain 03/ 09/004 Fair value of insurance liabiliies. INTRODUCTION TO FAIR VALUE. RISK NEUTRAL PRICING AND DEFLATORS 3. EXAMPLES
More informationResearch Article A General Gaussian Interest Rate Model Consistent with the Current Term Structure
Inernaional Scholarly Research Nework ISRN Probabiliy and Saisics Volume 212, Aricle ID 67367, 16 pages doi:1.542/212/67367 Research Aricle A General Gaussian Ineres Rae Model Consisen wih he Curren Term
More informationECON Lecture 5 (OB), Sept. 21, 2010
1 ECON4925 2010 Lecure 5 (OB), Sep. 21, 2010 axaion of exhausible resources Perman e al. (2003), Ch. 15.7. INODUCION he axaion of nonrenewable resources in general and of oil in paricular has generaed
More informationSupplement to Models for Quantifying Risk, 5 th Edition Cunningham, Herzog, and London
Supplemen o Models for Quanifying Risk, 5 h Ediion Cunningham, Herzog, and London We have received inpu ha our ex is no always clear abou he disincion beween a full gross premium and an expense augmened
More informationPricing options on defaultable stocks
U.U.D.M. Projec Repor 2012:9 Pricing opions on defaulable socks Khayyam Tayibov Examensarbee i maemaik, 30 hp Handledare och examinaor: Johan Tysk Juni 2012 Deparmen of Mahemaics Uppsala Universiy Pricing
More informationFair Valuation of Participating Policies in Stochastic Interest Rate Models: Two-dimensional Cox-Ross-Rubinstein Approaches
Fair Valuaion of aricipaing olicies in Sochasic Ineres Rae Models: Two-dimensional Cox-Ross-Rubinsein Approaches Liao, Szu-Lang Deparmen of Money and anking, Naional Chengchi Universiy, Taipei, Taiwan,
More informationIntroduction. Enterprises and background. chapter
NACE: High-Growh Inroducion Enerprises and background 18 chaper High-Growh Enerprises 8 8.1 Definiion A variey of approaches can be considered as providing he basis for defining high-growh enerprises.
More informationAppendix B: DETAILS ABOUT THE SIMULATION MODEL. contained in lookup tables that are all calculated on an auxiliary spreadsheet.
Appendix B: DETAILS ABOUT THE SIMULATION MODEL The simulaion model is carried ou on one spreadshee and has five modules, four of which are conained in lookup ables ha are all calculaed on an auxiliary
More information4452 Mathematical Modeling Lecture 17: Modeling of Data: Linear Regression
Mah Modeling Lecure 17: Modeling of Daa: Linear Regression Page 1 5 Mahemaical Modeling Lecure 17: Modeling of Daa: Linear Regression Inroducion In modeling of daa, we are given a se of daa poins, and
More informationVaR and Low Interest Rates
VaR and Low Ineres Raes Presened a he Sevenh Monreal Indusrial Problem Solving Workshop By Louis Doray (U de M) Frédéric Edoukou (U de M) Rim Labdi (HEC Monréal) Zichun Ye (UBC) 20 May 2016 P r e s e n
More information