# old7ans - 13 Let P(r t T denote the price at time t of\$l to...

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Unformatted text preview: 13. Let P(r, t, T) denote the price at time t of\$l to be paid with certainty at time T, ts T, ifthe short rate at time t is equal to r. For a Vasicek model you are given: Calculate r* . i (A) 0.04 (B) 0.05 (C) 0.06 i (D) 0.07 6%)) 0.08 Dwam‘ t, g l v i , i g 2 § Q l i i i l g g EXAM MFE: Spring 2007 - l3 - GO TO NEXT PAGE Actuarial Models — Financial Economics Segment 7. You are given the following information: Bond maturit ( ears _ 2 1 0.9434 0.8817 g E E E i i ‘1 1 i g A European call option, that expires in 1 year, gives you the right to purchase a 1-year bond for 0.9259. T: I K The bond forward price is lognormally distributed with volatility a = 0.05 . Using the Black formula, calculate the price of the call option. Magi .rm _ F5 Fa}. £00011” : P[o") = .‘HJ‘I “’3‘”; (A) 0.011 (B) 0.014 I”: F/K) + ‘31: (‘7. (C) 0.017 41: (ft-3' " ‘z'u’ (D) 0020 A; = .1111: ad? a .1520 @ 0.022 1010.31: .5332— M(¢l,\f«.\$’€36 i i E g E g g i E EXAM MFE: Spring 2007 - 7 — GO TO NEXT PAGE Actuarial Models —— Financial Economics Segment 30. You are given the following market data for zero-coupon bonds with a maturity payoff of\$100. 2 A 2-period Black-Derman-Toy interest tree is calibrated using the data from above: Bond Price \$ Volatili in Yearl 94.34 88.50 Year 0 ru r0 <: rd Calculate rd, the effective annual rate in year 1 in the “down” state. 5.94% (B) 6.60% (C) 7.00% (D) 7.27% (E) 7.33% 10% Year 1 too : -""'"'"" " = , 06 r“ 443‘! I | Ice mfg a? T06 “T w" ‘ 75 29. The following is a Black-Derman-Toy binomial tree for effective annual interest rates. Year 0 Year 1 Year 2 6% [be ’- .4433“ 5% .2074” ’14—- : .qcu‘l‘t r0 “"1 1.03%“ 3% ’ I -- ~"‘“'°" 2% f; " ﬁfe”:— Compute the volatility in year 1 of the 3—year zero-coupon bond generated by the tree. 1 (A) 140/ £54}- : y; :5 KW; =.o€(.ozi 0 ‘0; m" m aorta“ (B) 18% (C) 22% .L 26% W “ (E) 30% rm 4- 9/0341 ‘IL. '9“ l ' := ‘QqJ‘IOL .03 l . qt/rIQ'L—I : .02)633 00%)5’IS 20' t W t 9’ P633 M— , t J, 3 :02- aﬁoQ‘f’J ‘ ’0‘], f 73 5 i l l l i i 14. You are to use a Black-Derman-Toy model to determine Fo,2[P(2, 3)], the forward price for time—2 delivery of a zero—coupon bond that pays 1 at time 3. In the Black-Derman-Toy model, each period is one year. The following effective annual interest rates are given: I’d = 30% ru = 60% Fad = 20% ruu : Determine 1000 >< Fo,2[P(2, 3)]. (A) 667 (B) 678 i (C) 690 (D) 709 @712 i: AquM" [479, t i .....L_ “Ly 49.4. .__. low)” : ('EMlH/‘o [.60 "h (IA-r, M°3 mo ,1... .2... ,L. .4." a.) + Hr. 1.36 1.46 + ((1%, 7‘5; 1.7.6 '; JPN-632, Hf. ‘ ' l + -'L_._J_ __ NW') ‘ '73 Lee '17}, («30 Wu (m, =3 -°(‘16°37'/(l+¢,) q M vi 0 1"” “0‘” 0w . 'é'm‘f/(W‘el gm % Exam MFE: Spring 2009 - 14 _ GO ON To NEXT PAGE @ M“ /,J"0 \ 1-f- /.‘u‘0 é— ): '.2 \ 137\$;. .Lo P(o,3§ 14. You are using the Vasicek one-factor interest—rate model with the short-rate process calibrated as ‘ dr(t) = 0.6 b — r(t)]dt + odZ(t). For t S T, let P(r, t, T) be the price at time t of a zero-coupon bond that pays \$1 at time T, if the short—rate at time t is r. The price of each zero—coupon bond in the Vasicek model follows an lto process, dP[r(t), t, T] P[r(t),t,T] : 64”“); t, T] dt — q[r([)’ t, T] You are given that a(0.04, 0, 2) = 004139761. Find a(0.05, l, 4). - “(mmwlwl Wm) ' t i P, '1 M' f P VAJIMK: f! U “‘W‘M‘ﬂ: Fiﬁ-0M (14,”)! 11mph: .. .o‘HS‘MH-.o‘l “(-0335 #3 -.0J' 6’0 " g“<ﬂ)/6 34 ...
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## This note was uploaded on 03/11/2011 for the course MATH 476 taught by Professor Staff during the Winter '08 term at University of Illinois, Urbana Champaign.

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old7ans - 13 Let P(r t T denote the price at time t of\$l to...

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