MAFS Quantitative Modeling of Derivative Securities

Transcription

1 MAFS 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 = E [ W W s + W s s s Fs = s + W s s s = W s s Also, E [ W < is observed Therefore, W is a P, {F } 0 - maringale b From he lecure noe p8, we obain E [expσw W s F s = σ s For > s, consider [ E exp σw σ = E [expσw W s σ s + expσw s σ s F s = E [expσw W s F s σ s + expσw s σ s = exp σw s σ s [ Also, E expσw σ < is observed Therefore, expσw σ is a P, {F } 0 -maringale We have E [Zu Z du = E[Zu Z du = 0 1

2 and var σ = σ E = σ E = σ = σ [Zu Z du [Zu Z du = σ u du = σ T 3 /3 [Zu Z[Zv Z dudv E[{Zu Z}{Zv Z} dudv [minu, v dudv u dv + v dv v du 3 By virue of he properies of normal disribuion and he definiion of a Brownian moion, we observe ha σ 1 dz 1 +σ dz is a Brownian moion wih mean 0 and variance rae σ, where σ = σ1 + σ + ρ 1 σ 1 σ Define Z = σ 1Z 1 + σ Z, σ which is seen o be a Brownian moion wih zero mean and uni variance rae Noe ha dz 1 dz = ρ 1 d in he mean square sense For f = S 1 S, i follows ha df = S 1 ds + S ds 1 + ds 1 ds = S 1 S µ d + σ dz + S S 1 µ 1 d + σ 1 dz 1 + S 1 S σ 1 σ dz 1 dz = fµ 1 + µ + ρ 1 σ 1 σ d + fσ 1 dz 1 + σ dz = fµ d + fσ dz Alernaively, we may consider S 1 = S 1 0 exp µ 1 σ 1 S = S 0 exp µ σ so ha f = S 1 S = S 1 0S 0 exp = S 1 0S 0 exp µ 1 + µ σ 1 σ + σ 1 Z 1 + σ Z + σ 1 Z 1 + σ Z µ 1 + µ + ρ 1 σ 1 σ σ 1 + σ + ρ 1 σ 1 σ From S = S 0 exp µ σ + σ Z, we deduce [ S 1 = 1 S 0 exp µ + σ σ Z, + σz

3 The corresponding dynamic equaion is seen o be ds 1 S 1 = µ + σ d σ dz From he firs resul, for g = S 1 /S, i follows ha dg = g µ d + g σ dẑ, where µ = µ 1 µ ρ 1 σ 1 σ + σ, σ = σ 1 + σ ρ 1 σ 1 σ and Ẑ = σ 1Z 1 σ Z σ 4 Firs, we have var P = /3 By solving he following equaions: E P [ = P [ω P [ω + 4 P [ω 3 = 35, var P = 35 P [ω P [ω P [ω3 = /3, and P [ω 1 + P [ω + P [ω 3 = 1, We obain he unique soluion: P [ω1 = 5/4, P [ω = 1/1, P [ω 3 = 17/4 Since P [ω i, i = 1,, 3 are all posiive, we do obain he required probabiliy measure P and i is unique 5 Le γ = µ µ σ and consider he Radon-Nikodym derivaive: where d P dp = ρ ρ = exp γs dzs 1 0 Under he measure P, he sochasic process Z = Z + 0 γs ds 0 γs ds is P -Brownian by he Girsanov Theorem I is seen ha when we se γ = µ µ, σ hen µ d + σ d Z = µ d + σdz + γ d = µ d + σ dz Therefore, S is governed by ds S = µ d + σ d Z under he measure P 3

4 6 For s <, we consider E P [exp µz P µ F s = E P [exp µz P s µ s exp µ Z P Z P s µ s s F }{{} s normal wih variance s Recall ha he expecaion of a normal random variable wih variance µ s is µ exp s We hen have E P [exp µz P µ F s = exp µz P s µ s µ exp s exp µ s = exp µz P s µ s 7 Le K be he delivery price of he commodiy forward, hen he value of he forward conrac is given by f = S Ke rτ = S Bτ; K; where Bτ; K denoes he price of a T -mauriy bond wih par value K a ime o expiry τ, hence he hedge raio is always one The forward conrac is an agreemen where he holder agrees o buy he commodiy a he delivery ime T for he delivery price K I can be replicaed by holding one uni of he commodiy and shoring one uni of a bond wih par value K, implying he hedge raio is one Seing f = 0, we ge K = Se rτ I follows ha he forward price F S, τ is given by F S, τ = Se rτ = S/B, T, τ = T 8 When he self-financing rading sraegy is adoped, he purchase of addiional unis of asse is financed by he sale of he riskless asse, hence Sd + dm = 0 and i follows ha dπ = ds + Sd + rmd + dm = ds + rmd By subsiuing ds = ρs d + σs dz ino he above equaion, we obain On he oher hand, Io s Lemma gives dv = d + S = dπ = ρs + rm d + σs dz σ ds + V S S d + ρs S + σ S V S 4 d + σs S dz

5 Since Π is a replicaing porfolio, in order o mach dπ = dv, i is necessary o choose = S This leads o + σ V S rm = 0 S Noe ha M = V S = V S, he Black-Scholes equaion for V is hen given by S + σ V S + rs S S 9 Consider c = Nd 1 and p = N d 1, where ln S + r + σ τ d 1 = σ τ We have d 1 σ = τ ln S + rτ σ, τ rv = 0 S d ln + 1 τ = στ 3 r + σ Financial inerpreaion: When he opion is sufficienly ou-of-he-money currenly, a higher volailiy of he asse price or a longer ime o expiry implies a greaer value of dela The opion becomes more likely for he opion o expire in-he-money When he opion is currenly in-he-money, ln S changes in sign; so higher σ and longer τ lead o a small value of dela 10 The convexiy of he European call price wih respec o he asse price implies cs, τ; cs, τ; S S Le S S, we obain c c, hence e S S c 1 For a European pu opion, we have e p = p S S p = cs, τ; S SN d 1 e rτ N d SN d 1 In order o have e p < 1, his is equivalen o observe SN d 1 < e rτ N d This occurs when S is sufficienly low or he pu is sufficienly in-he-money 11 a By Io s Lemma and observing = 0, i follows ha f df = + σ S f d + f S S ds = Θ + σ S Γ d 5 τ

6 By virue of no arbirage, we se df = rfd and obain Θ + σ S Γ = rf b When he asse value is sufficienly high, he call will always be exercised wih erminal payoff S T Therefore, he call price ends asympoically o S e rτ By differeniaing S e rτ wih respec o τ, i is seen ha he hea ends asympoically o re τ from below 1 Using he risk neural valuaion principle, he value of he European call opion is given by c M S, τ;, M = e rτ E Q [c M S T, 0;, M = e rτ E Q [maxs T, 0 + e rτ E Q [maxs T M, 0 = cs, τ; cs, τ; + M Alernaively, we can derive he price relaion wihou he assumpion of exisence of Q The capped European call and he porfolio of long one uni of European call wih srike and shor one uni of European call wih srike + M have he same payoff a T under all scenarios of S T, so heir curren prices have he same relaion This relaion is model free, ha is, i is independen of he underlying assumpion of he underlying asse price disribuion 13 The erminal payoff funcion of his call is given by c L S, τ;, α = minmaxs T, 0, αs T Using he risk neural valuaion principle, i follows ha c L S, τ;, α = e rτ E Q [c L S T, 0;, αs T = e rτ E Q [maxs T, 0 + αs T maxmaxs T, 0, αs T = e rτ E Q [maxs T, 0 1 αe rτ E Q [max S T = cs, τ; 1 αc S, τ; 1 α Noe ha he dela of c L is an increasing funcion of S This is because cl = Nd 1 1 αn d 1 1 α and cl S = nd 1 1 Sσ τ 1 αn d 1 1 α 1 Sσ τ > 0 1 α, 0 As S, c L becomes α unis of forward wih zero forward price, so cl α as S Here, cl is always less han α and ends o α from below when he asse price becomes exceedingly large 6