Lecture19-07

Lecture19-07 - Lecture 19: Volatiltiy ! "# ' ( ) ) ) i...

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Unformatted text preview: Lecture 19: Volatiltiy ! "# ' ( ) ) ) i σn n Si ui Si/Si-1 σ n2 = u= 1 m −1 m i =1 m i =1 n n ( un −i − u ) 2 1 m un −i $ %& ! "# + ) ) ) ui m -1 (Si-Si-1)/Si-1 ui m σ2 = n 1 m m 2 i =1 n − i u $ %& * ! "# ! ( ( σ2 = n w here m i =1 m i =1 2 α i un − i αi = 1 $ %& , ! "# ' / 0 1 2+ "# $ % L& V m i =1 2 α iu n −i σ 2 = γV L + n where γ+ m i =1 αi = 1 $ %& . ! "# -+'+ ) ' ) 2 σ 2 = λσ 2 −1 + (1 − λ ) u n −1 n n ' % u( $ %& 3 ! "# ' ) )) ) ) -+' * -/ λ +,. $ %& 4 ! "# 5'/ 0 1 0 $ 2 σ 2 = γ V L + α u n −1 + βσ 2 −1 n n 2 "# % 1 =1 γ + α + β =1 $ %& ! "# 5'/ 0 1 1 ω = γV 0 "# % 2 2 2 2 σ n = ω + α u n −1 + βσ n −1 VL = ω 1− α − β $ %& % ! "# 7 )1 2 σ n2 = 0.000002 + 0.13un −1 + 0.86σ n2−1 ) $ $ ,, , -, ( -2 / $ %& 6 ! "# 7 )1 -2 3 2) 0.000002 + 0.13 × 0.0001 + 0.86 × 0.000256 = 0.00023336 -5 42 $ %& ! "# 5 ' / 0 1 82 σ n2 = ω + p i =1 2 α i u n−i + q j =1 β j σ n2− j $ %& ! "# +7 + ) ' ' $9 ( ( $ %& * ! "# 7 )) ) %% ) 6 p(1− p)9 )) ' ' &p, $ %& , ! "# 7 7 Maximize: ∏ i =1 m − ui2 1 exp 2v 2πv Taking logarithms this is equivalent to maximizing: m u2 − ln(v) − i v i =1 1m 2 Result : v= ui m i =1 $ %& . ! "# ' ) ' 5'/ 0 ∏ i =1 m u 1 exp − 2 vi 2 πvi 2 i or m i =1 − ln( vi ) − u i2 vi $ %& 3 ! "# 7 )1 )8 )" )8 9 ) & ω% % α m i =1 ' β − ln( v i ) − u i2 vi ω% % α β ' $ %& 4 ! "# : ): )) ): $ 0 "# % ! $ %& ! "# 0 ;5 (+ < ) )) ui ;9 $< ' u (=σi2 $ %& % ! "# E[σ 2 + k ] = VL + (α + β) k (σ 2 − VL ) n n ' 1 m m −1 k =0 2 E σ n+k m [ $ %& 6 ! "# Define a = ln 1 α+β per annum for a T - day option is The volatility 252 V L + 1 − e − aT [V ( 0 ) − V L ] aT $ %& ! "# : ) 0 "# % % % ) ' : A 2 & ; 2 , ,@ -/ 5 , ,3 4 , ,/ -3 > , , ,(? 4, , ,, -3 $ %& ! "# xi=(Xi-Xi-1)/Xi-1 σx,n: σy,n& n& yi=(Yi-Yi-1)/Yi-1 n$ n$ n$ = σu,n σv,n) $ %& * X Y ! "# )) )8 7) * n +λ n$ B $λ xn- yn$ $ %& , ! "# = $ ' Ω, % $ w TΩ w ≥ 0 ; $ %& . ! "# 7 ' 1 0 0.9 0 1 0.9 0.9 0.9 1 $ %& 3 ...
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This note was uploaded on 12/14/2011 for the course FIN 5515 taught by Professor Staff during the Spring '10 term at FSU.

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