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Damatimi_1 - 3.7 Hef tveggja tmabila CRR lkan ar sem p =...

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3.7 Hef tveggja tímabila CRR líkan þ ar sem p = 0.5, S(0) = 100, u = 1,01, d = 1/ u. Finni ð öll möguleg ver ð eftir 1-tímabil og 2-tímabil. Hverjar eru líkurnar á a ð ver ð i ð a.m.k. 102 eftir tvö tímabil? Lausn: Eftir 1-tímabil: S(0) u = 101; S(0) d = 99.0099 2-tímabil: S(0) u 2 = 102.01; S(0) ud = 100 og S(0) d 2 = 98.0296 Líkurnar á a ð ver ð i ð a.m.k. 102 eftir 2-tímabil er jafnt líkunum á tveimur hækkunum í rö ð , þ .e. p 2 = 0.25. 3.8 Lausn: Sjá Excel skjal á uglunni: dæmi 3.8 3.9 Two diffusion processes X and Y satisfy: dX = (2+ 5t + X)dt + 3dW 1 dY = 4Ydt + 8YdW 1 + 6 dW 2 where W1 and W2 are correlated Brownian motions with ρ = 0.1. Use Ito’s rule to find the dynamics of the processes X 4 , e X , X Y Lausn: G.r.f. a ð ƒ (t,X) diffranlegt fall m.t.t. t og tvídiffranlegt m.t.t. X. Þ á a ð gefnu dreififerlinu dX(t) = μ (t,X(t))dt + σ (t,X(t))dW(t) fullnægir ƒ (t,X(t)) ferlinu: d ƒ (t,X(t)) = [ ƒ t + ½ σ 2 ƒ XX ](t,X(t))dt + ƒ X (t,X(t))dX(t) þ .a. ef vi ð skiptum út fyrir dX(t) og fellum út úr rithætti (t,X(t)) til einföldunar þ á fáum vi ð Ito regluna sem: d ƒ = [ ƒ t + μƒ X + ½ σ 2 ƒ XX ]dt + σ ƒ X dW óformlega skrifa Ito’s regluna sem þ .a. au ð veldara a ð muna: d ƒ = ƒ t dt+ ƒ X dX + ½ ƒ XX dX dX (1) þ ar sem dX dX = ( μ dt + σ dW) × ( μ dt + σ dW) = σ 2 dt (2) Til a ð átta sig á hagn ý ti Ito’s reglunar þ á ímynda sér a ð ferli ð X(t) l ý si heg ð un hlutabréfaver ð s og ƒ (t,X(t)) l ý si þ á heg ð un á aflei ð u, valrétti, af þ ví hlutabréfi. Ito gefur
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