May 007 Exam MFE Soluions. Answer = (B) Le D = he quarerly dividend. Using formula (9.), pu-call pariy adjused for deerminisic dividends, we have 0.0 0.05 0.03 4.50 =.45 + 5.00 D e D e 50 e = 54.45 D ( 0.99005 + 0.9753) 50 0.970446. Rearranging he equaion yields D.96356 = 54.45 4.50 48.53 =.477, or D = 0.73.. 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 = 0.5. 3. Answer = (C) Le P denoe he price of he European pu opion. Then, 0.055 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 0.975589 0.30; d is from formula (.b) and is equal o 0.055995 0.06. From he normal cdf able, N(0.06) = 0.539 and N( 0.30) = 0.679 = 0.38. Thus, P 0.055 / 0.0/ 98e 0.539 00e 0.38 =.93.90. 4. 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:
r δ P = Ke Se + C. Wih S = 50, r = 4%, and δ = 8%, 0.04 0.08 P = K e 50 e + C = 0.9608K 46.558 + C. K C P K 50 40 9..396 0 50 4.9 6.794 0 60 0.7.0 0 70 0.00.00 0 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 0.48 0.48; hence, Nd ( ) 0.6844. By equaion (.b), d is 0.088 0.0; hence, Nd ( ) 0.490. Wih hese values, we obain e N( d ) 85 e 0 0.6844 = 58.74, S Ke N( d) 80 e 0.055 0.490 = 37.537. Hence, Ω = /{ [ Ke N( d) ]/[S e N( d) ]}.78, and
σ opion = σ sock Ω 0.50.78 =.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 0. 3 00 = 58.6. 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 3. 7. 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, ) = 0.9434, F = F 0, [P(, )] = P(0, )/P(0, ) = 0.887/0.9434 = 0.934598, K = 0.959. Wih σ = 0.05, we have ln( F / K) + σ T d = = 0.0 0., σ T d = d σ T 0. 0.05 = 0.6. Thus, N(d ) N(0.) = 0.583, N(d ) N(0.6) = 0.5636. Hence, C = P(0, T)[F N(d ) K N(d )] = 0.9434[0.9346 0.583 0.959 0.5636] = 0.9434 0.03 0.0 3
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
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 = 0.4 0 = =. Therefore, he opion price is S(0) N() = 00 0.843 = 68.6. [ ] [ ] 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% ) = 0.894. + 7.704% 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% ) =.7668. + 9.89% 5
The ree below corresponds o Figure 4.5 and Figure 4.9 of McDonald (006). Year 0 Year Year 6.000% $0 7.704% $0.894 4.673% $0 9.89% $.7668 6.000% $0 3.639% $0 By risk-neural pricing, he ime-0 price of he ineres rae cap is / + 6% 0.894 / / +.7668 + 6% + 7.704% = 0.08934 + 0.47649 = 0.56587 0.57 Remark: The cap paymens are no $0.894 and $.7668. 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 000 0.585 = y + z 0.7773. Gamma-neuraliy means 000 0.065 = y 0 + z 0.0746. 6
From he second equaion (he gamma-neural equaion), we obain z = 65./0.0746 = 87.654 87.7. (This is sufficien o deermine ha (B) is he correc answer.) Subsiuing his in he firs equaion (he dela-neural equaion) yields y = 58.5 87.7 0.7773 = 95.8.. 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 0.05 0.5 e d e 0.890 p* = = = 0.465. (0.5) u d.8 0.890 For he wo-period model, he sock prices are S 0 = 70 Su = us0 =.8 70 = 8.67 Sd = ds0 = 0.890 70 = 6.30 Suu = usu =.8 8.67 = 97.63 Sud = dsu = 0.890 8.67 = 73.58 S = ds = 0.890 6.30 = 55.45 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(80 97.63, 0) = 0, P ud = max(k S ud, 0) = max(80 73.58, 0) = 6.4, P dd = max(k S dd, 0) = max(80 55.45, 0) = 4.55. 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(80 8.67, e 0.05/ [0 0.465 + 6.4 ( 0.465)]) = e 0.05/ 6.4 0.535 = 3.35, P d = max(k S d, e rh [P ud p* + P dd ( p*)]) = max(80 6.30, e 0.05/ [6.4 0.465 + 4.55 ( 0.465)]) = max(7.70, 5.7) = 7.70. 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/ [3.35 0.465 + 7.70 ( 0.465)]) = max(0, 0.75) = 0.75. 7
. 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() + 0 + f(s(), )(r* r)d = e (r r*)(t ) S()[0.d + 0.4dZ()] + G()(r* r)d = G()[0.d + 0.4dZ()] + G()(0.0 0.08)d = G()[(0. + 0.0)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: () () 0.04 0.9445 = e α β () () 0.05 0.93 = e α β which, by aking logarihms, are equivalen o 0.057 = α() + β () 0.04 0.0703 = α() + β () 0.05 The soluion of his pair of linear equaions is β () =.3 α () = 0.0043 The las piece of daa says () () r* 0.8960 = β e α Taking logarihms yields 0.098 = α() + β () r *, or r* = (0.098 0.0043)/.3 = 0.08. 8
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 0.08 60 e 45 60 e 45 p* = = = 0.79988896 0.8. 70 45 70 45 By risk-neural pricing, he price of he sraddle is e r [p* 70 K + ( p*) 45 K ] = e 0.08 [p* 70 50 + ( p*) 45 50 ] = e 0.08 [p* 0 + ( p*) 5] = e 0.08 [5p* + 5] e 0.08 [5 0.8 + 5] = e 0.08 7 = 5.693 5.70. 5. 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 = 50.5 0.98347 = 48.548. We apply formula (.a), wih S = 48.548, K = 50, r = 0.05, δ = 0, σ = 0.3, and T = ½, o obain d = {ln(48.548/50) + [0.05 0 + (0.3) /] ½}/{0.3 ½} = { 0.0995 + 0.0475}/0.3 = 0.08740 0.08. (This is he same as applying he formula for d ha follows (.5) on page 380.) Then, d = 0.08740 0.3 = 0.939 0.3. I now follows from he prepaid forward price version of (.3) ha he pu opion price is 50e 0.05/ N(+0.3) 48.548N( 0.08) = (50 0.97535 0.557) (48.548 0.468) = 6.9039.745 = 4.894 4.9. 9
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 7 7 0.05 = + +.5. 8 8 0.3 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 0.3 0.3 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
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 = 80 0. 00 e N( d), or e N( d ) = (80 0. 4) /00 = 0.. By (4.5) on page 458, he gap call opion price is S(0) e Nd ( ) Ke Nd ( ) = 80 0. 90 0. = 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 ).
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).
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 4 + 0 0. = 5.. 8. 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 0.07 0.04 G 0.04 =, 0. H or G = 0.5H + 0.04. () 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
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 = 0.06. (6) Applying () and (5) o (6) yields a quadraic equaion of σ, σ 0.5σ + 0.0 = 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 = 0.5 0. + 0.04 = 0.065. 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.) = 3.695 3.70. 4