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Information about Week 7.2 Taylor Series

Published on March 6, 2014

Author: anhtuantran509

Source: slideshare.net

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Outline 1 Taylor’s Formula for Functions of One Variable 2 “Big O” Notation 3 Taylor’s Formula for Functions of Several Variables 4 Taylor Series Expansions 5 Bond Convexity Kjell Konis (Copyright © 2013) 7.2 Taylor Series 2 / 31

Outline 1 Taylor’s Formula for Functions of One Variable 2 “Big O” Notation 3 Taylor’s Formula for Functions of Several Variables 4 Taylor Series Expansions 5 Bond Convexity Kjell Konis (Copyright © 2013) 7.2 Taylor Series 3 / 31

Taylor’s Formula for Function of One Variable Let f (x ) be at least n times diﬀerentiable and let a be a real number The Taylor polynomial of order n around the point a is Pn (x ) = f (a) + (x − a)f (a) + (x − a)2 (x − a)n (n) f (a) + . . . + f (a) 2 n! n = (x − a)k (k) f (a) k! k=0 Want to use Pn (x ) to approximate f (x ) Questions: Convergence: does Pn (x ) → x as n → ∞? Order: how well does Pn (x ) approximate f (x )? Kjell Konis (Copyright © 2013) 7.2 Taylor Series 4 / 31

Approximation Error Diﬀerence between f (x ) and Pn (x ) is called the nth order Taylor approximation error Taylor approximation error: derivative form Let f (x ) be n + 1 times diﬀerentiable, f (n+1) continuous There is a point c between a and x such that f (x ) − Pn (x ) = (x − a)n+1 (n+1) f (c) (n + 1)! Taylor approximation error: integral form Let f (x ) be n + 1 times diﬀerentiable, f (n+1) continuous x f (x ) − Pn (x ) = a Kjell Konis (Copyright © 2013) 7.2 Taylor Series (x − t)n (n+1) f (t) dt n! 5 / 31

1.0 Example: Linear approximation of log(x ) log(x) −0.5 0.0 log(x) 0.5 P1(x) 0 1 2 x 3 4 Linear approximation of log(x ) around the point a = 1 Taylor polynomial of order 1 for f (x ) = log(x ) (x − a) P1 (x ) = f (a) + f (a) 1! = 0 + (x − 1) 1 1 = x −1 Kjell Konis (Copyright © 2013) 7.2 Taylor Series 6 / 31

Example: Integral Form of Taylor Approximation Error What is the Taylor approximation error at the point x = e? x f (x ) − P1 (x ) = 1 (x − t) f (t) dt 1! e f (e) − P1 (e) = (e − t) 1 e log(e) − (e − 1) = 2−e = = 1 −1 dt t2 1 e − 2 dt t t log(t) + log(e) + e t e 1 e e − log(1) + e 1 = 2 − e ≈ −0.718 Kjell Konis (Copyright © 2013) 7.2 Taylor Series 7 / 31

Example: Derivative Form of Taylor Approximation Error What is the Taylor approximation error at the point x = e? f (x ) − P1 (x ) = (x − 1)(1+1) f (c) (1 + 1)! 2−e = −(e − 1)2 2c 2 c = (e − 1)2 2e − 4 c ∈ [1, e] 1 ≤ c ≈ 1.434 ≤ e ≈ 2.718 Have to know the approximation error to ﬁnd c to ﬁnd the . . . How is this useful? Kjell Konis (Copyright © 2013) 7.2 Taylor Series 8 / 31

Bounding the Taylor Approximation Error Know that the true approximation error occurs at c ∈ [1, e] Follows that |error| ≤ max c∈[1,e] (x − 1)2 f (c) 2! ≤ (e − 1)2 2c 2 c∈[1,e] ≤ 1 (e − 1)2 ≈ 1.476 2 max Thus |f (e) − P1 (e)| < 1.477 Kjell Konis (Copyright © 2013) 7.2 Taylor Series 9 / 31

Outline 1 Taylor’s Formula for Functions of One Variable 2 “Big O” Notation 3 Taylor’s Formula for Functions of Several Variables 4 Taylor Series Expansions 5 Bond Convexity Kjell Konis (Copyright © 2013) 7.2 Taylor Series 10 / 31

“Big O” Notation Consider a degree n polynomial as x → ∞ n ak x k P(x ) = k=0 As n → ∞, the highest order term dominates the others |P(x )| | = lim x →∞ x n x →∞ lim n k k=0 ak x | xn n−1 = lim an + x →∞ ak = |an | x n−k k=0 “Big O” notation provides a compact way to state the same information P(x ) = O(x n ) as x → ∞ Kjell Konis (Copyright © 2013) 7.2 Taylor Series 11 / 31

“Big O” Notation Formally Let f , g : R → R f (x ) = O(g(x )) as x → ∞ means there are C > 0 and M > 0 such that f (x ) ≤ C when x ≥ M g(x ) For ﬁnite points: f (x ) = O(g(x )) as x → a means there are C > 0 and δ > 0 such that f (x ) ≤C g(x ) when |x − a| ≤ δ Example: Taylor polynomial approximation error f (x ) − Pn (x ) = O (x − a)n+1 Kjell Konis (Copyright © 2013) 7.2 Taylor Series as x → a 12 / 31

“Big O” Notation Can also write the Taylor polynomial approximation as f (x ) − Pn (x ) = O (x − a)n+1 f (x ) = Pn (x ) + O (x − a)n+1 f (x ) = f (a) + . . . + (x − a)n (n) f (a) + O (x − a)n+1 n! Linear approximation is second order f (x ) = f (a) + (x − a)f (a) + O (x − a)2 as x → a Quadratic approximation is third order f (x ) = f (a) + (x − a)f (a) + Kjell Konis (Copyright © 2013) (x − a)2 f (a) + O (x − a)3 2 7.2 Taylor Series as x → a 13 / 31

Outline 1 Taylor’s Formula for Functions of One Variable 2 “Big O” Notation 3 Taylor’s Formula for Functions of Several Variables 4 Taylor Series Expansions 5 Bond Convexity Kjell Konis (Copyright © 2013) 7.2 Taylor Series 14 / 31

Taylor’s Formula for Functions of Several Variables Let f be a function of n variables x = (x1 , x2 , . . . , xn ) Linear approximation of f around the point a = (a1 , a2 , . . . , an ) n f (x ) ≈ f (a) + (xi − ai ) i=1 ∂f (a) ∂xi If 2nd order partial derivatives continuous ⇒ 2nd order approximation n (xi − ai ) f (x ) = f (a) + i=1 O x −a 2 n i=1 O = ∂f (a) + O x − a ∂xi n (xi − ai ) i=1 Kjell Konis (Copyright © 2013) as x → a |xi − ai |2 Quadratic approximation around a is O x − a f (x ) ≈ f (a) + 2 3 n n ∂f (xi − ai )(xj − aj ) ∂ 2 f (a) + (a) ∂xi 2! ∂xi ∂xj i=1 j=1 7.2 Taylor Series 15 / 31

Taylor’s Formula in Matrix Notation Let Df (x ) = ∂f ∂f ∂f ... ∂x1 ∂xn ∂xn x1 − a1 x2 − a2 x −a = . . . xn − an ∂2f 2 ∂x1 2 ∂ f D 2 f (x ) = ∂x1.∂x2 . . 2 ∂ f ∂x1 ∂xn ∂2f ∂x2 ∂x1 ... ∂2f 2 ∂x2 ... .. . . . . ∂2f ∂x2 ∂xn ... ∂2f ∂xn ∂x1 ∂2f ∂xn ∂x2 . . . ∂2f 2 ∂xn Linear Taylor approximation f (x ) = f (a) + Df (a)(x − a) + O x − a 2 Quadratic Taylor approximation f (x ) = f (a) + Df (a)(x − a) + Kjell Konis (Copyright © 2013) 1 (x − a)T D 2 f (a)(x − a) + O x − a 2! 7.2 Taylor Series 3 16 / 31

Example: Functions of Two Variables Let f be a function of 2 variables Linear approximation of f around the point (a, b) f (x , y ) ≈ f (a, b) + (x − a) ∂f ∂f (a, b) + (y − b) (a, b) ∂x ∂y Second order approximation since (as (x , y ) → (a, b)) ∂f ∂f f (x , y ) = f (a, b) + (x − a) (a, b) + (y − b) (a, b) ∂x ∂y + O |x − a|2 + O |y − b|2 In matrix notation f (x , y ) = f (a) + Df (a, b) Kjell Konis (Copyright © 2013) x −a + O |x − a|2 + O |y − b|2 y −b 7.2 Taylor Series 17 / 31

Example: Functions of Two Variables (continued) Quadratic approximation of f around the point (a, b) f (x , y ) = f (a, b) + (x − a) + ∂f ∂f (a, b) + (y − b) (a, b) ∂x ∂y (x − a)2 ∂ 2 f ∂2f (a, b) (a, b) + (x − a)(y − b) 2! ∂x 2 ∂x ∂y + (y − b)2 ∂ 2 f (a, b) + O |x − a|3 + O |y − b|3 2! ∂y 2 Matrix notation f (x , y ) = f (a, b) + Df (a, b) + x −a y −b 1 x −a x − a y − b D 2 f (a, b) y −b 2 +O |x − a|3 + O |y − b|3 Kjell Konis (Copyright © 2013) 7.2 Taylor Series 18 / 31

Outline 1 Taylor’s Formula for Functions of One Variable 2 “Big O” Notation 3 Taylor’s Formula for Functions of Several Variables 4 Taylor Series Expansions 5 Bond Convexity Kjell Konis (Copyright © 2013) 7.2 Taylor Series 19 / 31

Taylor Series Expansions If f is inﬁnitely many times diﬀerentiable, can deﬁne Taylor series expansion as n (x − a)k (k) f (a) n→∞ k! k=0 T (x ) = lim Pn (x ) = lim n→∞ A Taylor series expansion is a special case of a power series ∞ n ak (x − a)k = T (x ) = S(x ) ≡ lim n→∞ k=0 Power series coeﬃcients ak = ak (x − a)k k=0 f (k) (a) k! Convergence properties for Taylor series inherited from convergence properties of power series Kjell Konis (Copyright © 2013) 7.2 Taylor Series 20 / 31

Radius of Convergence The radius of convergence is the number R > 0 such that ∞ ak (x − a)k < ∞ S(x ) = ∀ x ∈ (a − R, a + R) k=0 S(x ) inﬁnitely many times diﬀerentiable on the interval (a − R, a + R) S(x ) not deﬁned if x < a − R or if x > a + R If limk→∞ |ak |1/k exists, then R= 1 lim |ak |1/k k→∞ For Taylor series expansions, if limk→∞ R= Kjell Konis (Copyright © 2013) k |f k) (a)|1/k exists, then 1 k lim (k) e k→∞ |f (a)|1/k 7.2 Taylor Series 21 / 31

Radius of Convergence So far, T (x ) < ∞ for x ∈ (a − R, a + R) Want to know if/where T (x ) = f (x ) Theorem: let 0 < r < R, if lim k→∞ rk k! max z∈[a−r ,a+r ] |f (k) (z)| = 0 Then T (x ) = f (x ) ∀ |x − a| ≤ r Kjell Konis (Copyright © 2013) 7.2 Taylor Series 22 / 31

Example Taylor series expansion of f (x ) = log(1 + x ) around the point a = 0 ∞ T (x ) = n (x − 0)k (k) (x − 0)k (k) f (0) = lim f (0) n→∞ k! k! k=0 k=0 f (x ) = (1 + x )−1 , . . . , f (k) (x ) = (−1)(k+1) (k − 1)!(1 + x )−k f (k) (0) = (−1)(k+1) (k − 1)!(1)−k = (−1)(k+1) (k − 1)! Taylor series expansion of f (x ) = log(1 + x ) around a = 0 ∞ T (x ) = ∞ (x − 0)k (k) xk f (0) = (−1)(k+1) k! k k=1 k=1 = x− Kjell Konis (Copyright © 2013) x2 x3 x4 + − + ... 2 3 4 7.2 Taylor Series 23 / 31

Example (continued) Find radius of convergence k k = lim lim (k) 1/k (k+1) (k − 1)!|1/k k→∞ |(−1) k→∞ |f (a)| = k lim k→∞ (k − 1)! 1/k = hmmm . . . Power series deﬁnition 1/k lim |ak | k→∞ (−1)(k+1) = lim k→∞ k = = Kjell Konis (Copyright © 2013) 1/k 1 1/k k→∞ k lim lim u u = 1 u 0 7.2 Taylor Series 24 / 31

Example (continued) Radius of convergence: R = 1 limk→∞ |ak |1/k Where does T (x ) = log(1 + x )? =1 (0 < r < R = 1) z∈[−r ,r ] rn n→∞ n! z∈[−r ,r ] r n (n − 1)! n→∞ n! (1 − r )n = z∈[−r ,r ] rn n→∞ n! = max |f n (z)| = = rn n→∞ n! lim 1 n→∞ n T (x ) = log(1 + x ) for |x | < Kjell Konis (Copyright © 2013) lim lim max max (−1)n+1 (n − 1)! (1 + z)n (n − 1)! |1 + z|n lim lim r 1−r n = 0 for r ≤ 1 2 1 2 7.2 Taylor Series 25 / 31

Outline 1 Taylor’s Formula for Functions of One Variable 2 “Big O” Notation 3 Taylor’s Formula for Functions of Several Variables 4 Taylor Series Expansions 5 Bond Convexity Kjell Konis (Copyright © 2013) 7.2 Taylor Series 26 / 31

Bond Pricing Formula 400 Price 300 200 100 0 0 5% 10% λ 15% 20% The price P of a bond is P = where: n F ck + n [1 + λ] 1 + λk k=1 ck = coupon payment F = face value Kjell Konis (Copyright © 2013) n = # coupon periods remaining λ = yield to maturity 7.2 Taylor Series 27 / 31

Linear Approximation 400 Price 300 200 100 0 0 q 5% 10% λ 15% 20% The tangent line used for approximation L(λ) = P(0.10) + (λ − 0.10) dP (0.10) where dλ dP − −DM P dλ Said last time: approximation can be improved by adding a quadratic term =⇒ approximate using a degree 2 Taylor polynomial Kjell Konis (Copyright © 2013) 7.2 Taylor Series 28 / 31

Convexity Recall: PVk = n ak [1 + λ]k P= k=1 = 1 P dλ2 d2 1 dλ2 P = = = Kjell Konis (Copyright © 2013) ak [1 + λ]k k=1 d 2P = Convexity: C n PVk = 1 P 1 P n ak k=1 n ak [1 + λ]−k k=1 d2 [1 + λ]−k dλ2 n ak k(k + 1)[1 + λ]−(k+2) k=1 1 P[1 + λ]2 n k(k + 1) k=1 7.2 Taylor Series ak [1 + λ]k 29 / 31

Convexity Let P0 and λ0 be the price and yield of a bond Let DM and C be the modiﬁed duration and convexity ∆P ≈ −DM P∆λ + 1 PC (∆λ)2 2 1 P ≈ P0 + −DM P(λ − λ0 ) + PC (λ − λ0 )2 2 ≈ P0 + (λ − λ0 ) dP (λ − λ0 )2 d 2 P (P0 ) + (P0 ) dλ 2 dλ2 400 Price 300 200 100 0 0 Kjell Konis (Copyright © 2013) q 5% 10% λ 7.2 Taylor Series 15% 20% 30 / 31

http://computational-finance.uw.edu Kjell Konis (Copyright © 2013) 7.2 Taylor Series 31 / 31

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