a3_sol.pdf

# This is just a straight line s x 1 x x 2 1 6 20 marks

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This is just a straight line S ( x ) = 1 + x, x [ - 2 , 1]. 6. ( 20 marks ) A cubic spline with ( n + 1) knot points t 1 < t 2 < · · · < t n < t n +1 can be expressed as f ( t ) = n X i =1 P i ( t ) I i ( t ) (5) where P i ( t ) = a i + b i ( t - t i ) + c i ( t - t i ) 2 + d i ( t - t i ) 3 and I i ( t ) = ( 1 , if t t i , 0 , otherwise. That is, f ( t ) = P 1 ( t ) , for t [ t 1 , t 2 ) P 1 ( t ) + P 2 ( t ) , for t [ t 2 , t 3 ) P 1 ( t ) + P 2 ( t ) + P 3 ( t ) , for t [ t 3 , t 4 ) . . . n X i =1 P i ( t ) , for t [ t n , t n +1 ) (a) Imposing the following continuity and smoothness conditions on ( 5 ), f ( t - i ) = f ( t + i ) f 0 ( t - i ) = f 0 ( t + i ) f 00 ( t - i ) = f 00 ( t + i ) for i = 2 , ..., n , results in an expression with ( n + 3) unknown parameters for any knot points t i . Determine the expression for f ( t ) with continuity and smoothness conditions imposed. 9

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(b) A discount curve is a function f defined on all future times t such that f ( t ) represents the value today of receiving one unit of currency t years in the future. We are interested in interpolating a discount curve from a set of listed US treasury bonds, which pay a coupon semi-annually. Let τ 1 , τ 2 , ..., τ K denote the set of all payment times of all bonds in the data set. Let C j,k denote the payment of bond j , at time τ k . From a no-arbitrage argument the price of a bond is the present value of discounted cash flows, P j = K X k =1 C j,k f ( τ k ) (6) where P j is the current market price of bond j . A set of eight bonds are shown below, including a dummy bond with maturity 0. All bonds have a principal or face value of 100. US Treasury Bonds ( B j ) B 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 maturity 0 years 1 year 2 years 4 years 5 years 8 years 9 years 10 years price 100 99.821 98.3203 98.0313 97.1172 91.3438 95.0234 99.0000 coupon 0 0.6875 0.6875 1.0000 1.0000 0.8125 1.1250 1.3725 Note: You can download q6.m from uw learn, which provides a set up of this data. Use the cubic spline, f ( t ), discussed in part (a) to interpolate the discount curve. Impose two additional boundary constraints: i) Use a left-end clamped boundary, f 0 ( t 1 ) = 0, to eliminate one more parameter in the expression for f ( t ) that you obtained in part (a). ii) Make the second last maturity an inactive knot (i.e ‘not-a-knot’). Thus use seven knot points, not eight: t 1 t 2 t 3 t 4 t 5 t 6 t 7 time (years) 0 1 2 4 5 8 10 You should have a total of eight unknown parameters in f ( t ). Substitute the expression of f ( t ) into ( 6 ) to obtain an equation for each of the eight bonds shown above. Solve the resulting 8 × 8 linear system using the matlab \ command. Print the solution obtained, and evaluate and plot f ( t ) for values t = 0 : 0 . 1 : 10. (c) The zero-coupon yield curve is a function y , such that y ( t ) is the annualized interest rate on a t -year investment with no intermediate payments, i.e. all principle and interest paid after t years. Assuming continuous compounding, the discount curve is related to the zero-coupon yield curve as follows, f ( t ) = exp {- ty ( t ) } 10
Hence, the zero-coupon yield curve can be obtained as a simple transformation of the discount curve, y ( t ) = - ln ( f ( t )) t Use the evaluations of f ( t ) you obtained in part (b) to plot y ( t ) for t = 0 . 1 : 0 . 1 : 10.

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