康奈尔固定收益证券 (16)

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.
t 1 ~ f ( 0 ,v )dv P ( 0 ,t ) e t . P ( 0 ,t 1 )
(16.4)
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Assuming that f ( 0 ,v ) is constant between time t and t+1, we can rewrite this as:
~ P ( 0 ,t ) e f ( 0 ,t )1 P ( 0 ,t 1 )
~ f ( 0 , t ) log( P ( 0 , t ) / P ( 0 , t 1 )) .
(16.6)
This approximates the forward rate curve with a piecewise constant step function.
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EXAMPLE: CONTINUOUSLY COMPOUNDED, PIECEWISE CONSTANT FORWARD RATES This example illustrates the computation of the piecewise constant continuously compounded forward rates. Suppose that from the coupon bond stripping procedure, we obtain the following zero coupon bond prices. T 1 2 3 4 5 P(0,T) .980392 .961168 .942322 .923845 .905730
j B ( 0 ) C P ( 0 , t ) L P ( 0 ,T ) . j j j t 1 j T
(16.1)
For most bonds the index summation dates, t = 1, …, T j 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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Let us represent the price observations at time 0 by the m1 vector
P ( 0 ,1 ) P ( 0 ,2 ) P ( 0 , 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. There are missing zero-coupon bond price observations.
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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B The Initial Forward Rate Curve This section studies estimation of the initial continuously compounded forward rate curve ~ f ( 0 ,T ) 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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A Coupon Bond Stripping This section discusses how to obtain the zerocoupon 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
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Following the formula given in expression (16.6), we generate the following continuously compounded forward rates. T 1 2 3 4 5
~ f ( 0 ,T )
.0198 .0198 .0198 .0198 .0198
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There many methods for getting around the missing zero-coupon bond price observations. We will discuss the simplest (but perhaps most robust) approach. This approach assumes constant forward rates over the missing maturities. To understand this approach, we start with the definition of the continuously compounded forward rates.
B (0 ) j
for j = 1, … , n.
This set could include some Treasury bills or strips.
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From Chapter 10, we know that the arbitrage-free price of the coupon bond B j ( 0 ) with coupons C j principal L j and maturityT j can be written as:
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Instead of piecewise constant forward rate curves, one could use spline techniques instead. In some applications one may want to modify these forward rates so that a particular set of computed prices for interest rate derivatives match market prices. For example, in the pricing of call provisions on Treasury bonds, one may want to adjust the forward rates so that computed noncallable Treasury bond prices match market prices.
~
where
~ ~ f ( 0 ,v ) f ( 0 , t ) for t v t 1 .
(16.5)
Expression (16.5) is easily solved for the forward rates, given the zero-coupon bond price vector in expression (16.3). The solution is:
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The best method to employ, therefore, is to choose an error minimizing procedure.