Financial calculus Introduction to Financial Option Valuation 2 pdf

... the power δt 3 /2 or higher. Exercise 6.4 asks you to show that E  µδt + σ √ δtY i − 1 2 σ 2 δtY 2 i  = µδt − 1 2 σ 2 δt (6.5) and var  µδt + σ √ δtY i − 1 2 σ 2 δtY 2 i  = σ 2 δt + higher ... = S 0 e µt , (6.11) E(S(t) 2 ) = S 2 0 e (2 +σ 2 )t , (6. 12) var(S(t)) = S 2 0 e 2 t (e σ 2 t − 1), (6.13) see Exercise 6.6. 62 Asset price model: Part I is buil...

Ngày tải lên: 21/06/2014, 09:20

... us L−1  i=0 δS 2 i ≈ S(t) 2 L−1  i=0 (µ 2 δt 2 + 2 σ δt 3 2 Y i + σ 2 δtY 2 i ). (8 .2) 8.5 Black–Scholes formulas 81 We may also write d 2 = d 1 − σ √ T − t, (8 .22 ) see Exercise 8 .2. The equation ... 0.1 0 .2 0.3 0.4 0.5 0.6 0.8 1 1 .2 1.4 1.6 d t = 5 × 10 −3 d t = 5 × 10 −4 Asset paths 0 0.1 0 .2 0.3 0.4 0.5 0 0.01 0. 02 0.03 0.04 0.05 Sum-of-square returns σ 2...

Ngày tải lên: 21/06/2014, 09:20

... 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 Asset path E 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 Delta 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 Cash 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 Portfolio Fig. ... 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 Asset path E 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 Delta 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 Cash 0 0.5 1 1.5 2 2.5 3...

Ngày tải lên: 21/06/2014, 09:20

... points to make. (i) Formulas ( 12. 2) and ( 12. 4) were derived without any reference to the idea of hedging to eliminate risk. (ii) Formulas ( 12. 2) and ( 12. 4) were derived without any reference to the ... ( 12. 1) Using (3.8) and the density function (6.10), this may be written e −rT  ∞ 0 (x) xσ √ 2 √ T exp    −  log x − log S 0 − (µ − 1 2 σ 2 )T  2 2σ 2 T   ...

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... [K, 2K], [2K, 3K], Newton’s method takes the form σ n+1 = σ n − F(σ n ) F  (σ n ) , (14.7) 136 Implied volatility Exercise price Option price 5 125 475 522 5 405 5 325 340 5 425 28 0 1 2 5 525 22 6 5 625 ... (10.6) gives ∂ 2 C ∂σ 2 =− S √ T − t √ 2 e − 1 2 d 2 1 d 1 ∂d 1 ∂σ . From (8 .20 ) we have ∂d 1 ∂σ =− log(S/E) +r(T − t) σ 2 √ T − t + 1 2 √ T − t =−  log(S/E) + (...

Ngày tải lên: 21/06/2014, 09:20

... −  B S  1+2r/σ 2 (N( f 2 ) − N(g 2 ))  −Ee −r(T −t)  N (d 2 ) − N(e 2 ) −  B S  −1+2r/σ 2 (N( f 1 ) − N(g 1 ))  . (19.5) Here, d 1 and d 2 are defined in (8 .20 ) and (8 .21 ) and e 1 = log(S/B) +(r + 1 2 σ 2 )(T ... + 1 2 σ 2 )(T − t) σ √ T − t , e 2 = log(S/B) +(r − 1 2 σ 2 )(T − t) σ √ T − t , f 1 = log(S/B) −(r − 1 2 σ 2 )(T − t) σ √ T − t , f 2 = l...

Ngày tải lên: 21/06/2014, 09:20

... Let  V (S, t) := S 1− 2r σ 2 V  X S , t  . Show that ∂  V ∂t + 1 2 σ 2 S 2 ∂ 2  V ∂ S 2 +rS ∂  V ∂ S −r  V = S 1− 2r σ 2  ∂V ∂t  X S , t  + 1 2 σ 2  X S  2 ∂ 2 V ∂ S 2  X S , t  +r X S ∂V ∂ ... = 1 √ 2 σ 2 e −x 2 / (2 2 ) . Our maximum likelihood estimate of σ is then found by maximizing M  i=1 1 √ 2 σ 2 e −  U 2 n+1−i / (2 2 ) , where  U...

Ngày tải lên: 21/06/2014, 09:20

... widths 10 2 [0.0878, 0. 321 9] [0. 123 9, 0.3061] 1.3 10 3 [0 .22 85, 0.3333] [0 .22 38, 0 .29 36] 1.5 10 4 [0 .24 43, 0 .27 64] [0 .23 70, 0 .25 80] 1.5 10 5 [0 .23 59, 0 .24 58] [0 .23 73, 0 .24 40] 1.5 21 .10 Program of Chapter 21 and ... of widths 10 2 [1.8841, 2. 07 52] [1.9875, 2. 00 12] 14.0 10 3 [1.9538, 2. 0087] [1.9976, 2. 0017] 13.4 10 4 [1.9890, 2. 00 62] [1.9997, 2. 0...

Ngày tải lên: 21/06/2014, 09:20

... max(S(0) − E, 0). (24 .2) Similarly, the European put condition (8 .25 ) changes to P(S, 0) = max(E − S(0), 0). (24 .3) 25 7 25 2 Finite difference methods  2 = δ 2 E,  2 = Eδ 2 . 23 .2.  Verify the ... u satisfies the PDE (23 .2) , we have R i j = 1 2 k ∂ 2 u ∂t 2 − 1 12 h 2 ∂ 4 u ∂x 4 + O(k 2 ) + O(h 4 ). (23 .14) 23 .7 Von Neumann stability and convergence 2...

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... software, 26 3 BTCS, 24 0 24 7, 24 9, 25 2, 25 7 26 1, 26 5 convergence, 24 7 24 9, 26 0 Crank–Nicolson, 24 9 25 2, 25 7 26 1, 26 5 for American option, 26 3 for Black–Scholes PDE, 25 7 26 0 FTCS, 24 0 24 9, 25 2, 25 7 26 0, ... 25 7 26 0, 26 5 instability, 24 3 local accuracy, 24 6 24 7, 24 9, 25 1, 25 2 penalty method, 26 3 stencil, 24 2, 24 4, 24 9 upwind, 26 2...

Ngày tải lên: 20/06/2014, 18:20