Lecture_16 - 16 Parameter Estimation The previous chapters...

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1 16 Parameter Estimation The previous chapters take the input parameters, the initial forward rate curve and the volatility function(s) as given. This chapter studies how to obtain these inputs from observable market prices of zero-coupon bonds, coupon bonds and various other interest rate options. This chapter does not exhaust the possible approaches to this problem.
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2 A Coupon Bond Stripping This section discusses how to obtain the zero- coupon bond prices implicit in observed coupon bond prices. Usually, the on-the-run Treasury securities have the most liquid markets and provide the most accurate prices. In theory, looking at market prices, we can observe a collection of coupon-bond prices, denoted by ) 0 ( j B for j = 1, … , n. This set could include some Treasury bills or strips.
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3 From Chapter 10, we know that the arbitrage-free price of the coupon bond ) 0 ( j B with coupons j C principal j L and maturity j T can be written as: = + = j T 1 t ) j T , 0 ( P j L ) t , 0 ( P j C ) 0 ( j B . (16.1) For most bonds the index summation dates, t = 1, …, j T correspond to six month intervals (the time between coupon payments). To “strip out the zero-coupon bonds” means to solve this set of linear equations [expression (16.1) for all j] for the zero-coupon bond prices ( P(0,t) for all t ) . Depending upon the set of coupon bonds included, this system could have no solutions, one solution or many solutions.
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4 The best method to employ, therefore, is to choose an error minimizing procedure. . 2 j T 1 t ) j T , 0 ( P j L ) t , 0 ( P j C ) 0 ( j n 1 j imize min to } n , , 1 j : j T max{ t 0 for ) t , 0 ( P Choose = + = = B K (16.2)
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5 EXAMPLE: COUPON BOND STRIPPING This example shows how to strip zero-coupon bond prices from a set of coupon bond prices. Suppose that we go to our fixed income broker and he gives us the following prices for five different coupon bonds. Bond Price B(0) Coupon C Maturity T Face Value L 100.2451 2.25 1 100 101.9415 3 2 100 100 2 3 100 101.9038 2.5 4 100 98.8215 1.75 5 100 The coupon bonds have different coupon payments and maturity dates, but equal face values.
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6 One can solve expression (16.2) for the underlying zero-coupon bond prices. The solution is: T P(0,T) 1 .980392 2 .961168 3 .942322 4 .923845 5 .905730 In this special case, the sum of squared errors is zero as there is an exact set of zero-coupon bond prices that underlie the coupon bonds. ±
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7 B The Initial Forward Rate Curve This section studies estimation of the initial continuously compounded forward rate curve ) T , 0 ( f ~ for all T measured on a per-year basis. These forward rates are determined from the set of zero-coupon bond prices P(0,T) for all T determined as the solution to expression (16.2). These zero-coupon bond prices will have maturities with discrete spacings (perhaps 6 months apart).
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8 Let us represent the price observations at time 0 by the m × 1 vector ) m , 0 ( P ) 2 , 0 ( P ) 1 , 0 ( P M . (16.3) The first difficulty encountered is that the number of observed zero-coupon bond prices each day (m) are insufficient to determine the continuously compounded forward rates of all maturities.
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This note was uploaded on 10/31/2010 for the course NBA 5550 taught by Professor Jarrow,robert during the Fall '08 term at Cornell University (Engineering School).

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Lecture_16 - 16 Parameter Estimation The previous chapters...

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