Equation (30) describes the process for the propagation of the spot rate and is given by
(B.1) Comparing the above with the general equation (A.3), it is seen that (B.2) (B.3) Equation (A.15) becomes
(B.4)
To apply the principal of an arbitrage-free term structure, consider equation (A.16).
(B.5) Any security Wi with maturity si is subject to the same relationship (B.5) such that
(B.6) Consider a portfolio W consisting of owning an amount of W2 and shorting an amount W1 such that
(B.7) where
(B.8)
and
dr = k(θ–r)dt+σ rdz
a = k(θ–r) b = σ r
dP k(θ–r)∂P
∂r
--- ∂P
∂t
--- 1
2---σ2r∂2P
∂r2 ---
+ + dt σ r∂P
∂r ---dz +
=
dP = Pàdt P– ρdz
dWi = Wiàidt W– iρidz
W = W2–W1
W2 ρ1
ρ1–ρ2
---
W
=
Term Structure Modeling 133
(B.9)
Then
Applying equation (B.6), the above becomes
(B.10)
Since the stochastic element dz is not present in equation (B.10), the rate of return on the portfolio W is equal to the riskless rate r. Therefore equation (B.10) may be written as
Thus
This gives the following relationship
and
(B.11)
Since the maturities s1 and s2 were chosen arbitrarily, the above is true for any maturity s. Therefore, the term
W1 ρ2
ρ1–ρ2
---
W
=
dW = dW2–dW1
dW à2ρ1
ρ1–ρ2
---
Wdt ρ1ρ2 ρ1–ρ2
---
Wdz
– à1ρ2
ρ1–ρ2
---
Wdt ρ1ρ2 ρ1–ρ2
---
Wdz +
–
=
à2ρ1–à1ρ2
ρ1–ρ2
---
Wdt
=
dW = rWdt
r à2ρ1–à1ρ2
ρ1–ρ2
---
=
rρ1–rρ2 = à2ρ1–à1ρ2
à2–r ρ2
--- à1–r
ρ1
---
=
5-Audley/Chin-TermStructModel Page 133 Thursday, August 29, 2002 10:00 AM
is not a function of maturity and may be written as
(B.12) whereq(t,r) is the market price of risk.
Applying separation of variables, choose q(t,r) to be the following (B.13) whereλ(t) is the risk premium which can be shown to be
(B.14) (As the time, t, approaches the maturity, s, the risk premium decreases toward zero, which reflects the decreasing risk associated with shorter- term instruments.) Equation (B.12) is rewritten as
(B.15) This states that the expected return of a bond is equal to the riskless rate plus another term related to the risk premium.
With equations (A.18) and (B.3), the above becomes
(B.16) Substituting the above into equation (B.5), (B.5) becomes
Equating the coefficients of dt between the above and equation (B.4), (B.17) à–r
---ρ
à–r
---ρ = q t r( , )
q t r( , ) = λ( )t r
λ( )t 1 2---σ
---κ{1 exp– [–κ(s t– )]}
=
à = r+λ rρ
à r λ r σ r∂P
∂r ---
– 1
P---
+
=
dP P r λσr∂P
∂r ---1
P---
–
dt P– ρdz
=
∂P
∂t
--- rP [k(θ–r) λσ+ r]∂P
∂r ---
– 1
2---σ2r∂2P
∂r2 --- –
=
Term Structure Modeling 135
subject to the boundary condition
P(r,s) = 1 (B.18)
This completes the derivation of equation (31).
Finally, if we assume a separation of variables for P(r,t) of the form (B.19) it can be derived that the “target” spot rate, θ, of the form
(B.20)
(B.21) will provide a solution to equation (31) that will exactly reprice the ref- erence set where the discount function d(t0,T) and the forward rates F(t0, t0 + T) are derived from the reference set as described in the body of this chapter using spline functions. Furthermore, this property is true for all volatilities when the risk premium of equation (B.14) is used.
P r t( , ) = exp[C t( )–B t( )r]
θ(t0+T) d dT---
– lnd t( 0,T) 1 k--- d2
dT2
---lnd t( 0,T) –
=
θ(t0+T) F t( 0,t0+T) 1 k--- d
dT---F t( 0,t0+T) +
=
5-Audley/Chin-TermStructModel Page 135 Thursday, August 29, 2002 10:00 AM
CHAPTER 6
137
A Practical Guide to Swap Curve Construction
Uri Ron Senior Trader Bank of Canada
waps are increasingly used by governments, financial intermediaries, corporations, and investors for hedging, arbitrage, and to a lesser extent, speculation. Swaps are also used as benchmarks for evaluating the performance of other fixed-income markets, and as reference rates for forecasting.
Swaps offer an operationally efficient and flexible means of trans- forming cash flow streams. The swap market has little or no government regulation, and provides a high degree of privacy. The swap market’s liquidity, depth, and high correlation with fixed-income products, other than plain vanilla government bonds, render its derived term structure a fundamental pricing mechanism for these products and a relevant benchmark for measuring the relative value of different fixed-income products.1
The role of the swap term structure as a relevant benchmark for pricing and hedging purposes is expected to increase as government fis- cal situations improve. An improved fiscal situation reduces the size of government debt programs, in effect decreasing the liquidity and effi- ciency of government debt markets. Furthermore, the financial markets
1For correlations of swap rates and other fixed-income rates for the U.S. market, see M. Fleming, “The Benchmark U.S. Treasury Market: Recent Performance and Pos- sible Alternatives,” FRBNY Economic Policy Review (April 2000).
S
6-Ron-PracticalGuide Page 137 Thursday, August 29, 2002 10:04 AM
crisis in the fall of 1998 reinforced the “flight to quality” phenomenon, where spreads between governments’ issues and other fixed-income securities widened substantially under adverse market conditions, thereby calling into question the role of the government market as a rel- evant benchmark for nongovernment issues. The swap term structure again emerges as a potential substitute.
With the increased importance of the swap market, practitioners recognize the importance of a consistent and computationally efficient swap term structure for marking to market financial transactions; mark- ing to market is the practice of valuing an instrument to reflect current market conditions. While the general framework for the construction of the swap term structure is widely known, the derivation details are vague and not well documented. This chapter attempts to bridge this gap by carefully covering all angles of the swap term-structure deriva- tion procedure while leaving enough flexibility to adjust the constructed term structure to the specific micro requirements and constraints of each primary swap market.
Marking to market fixed-income portfolios is instrumental for trad- ing, accounting, performance valuation, and satisfying inter-institution collateralization requirements. The current methodology in capital mar- kets for marking to market fixed-income securities is to estimate and discount future cash flows using rates derived from the appropriate term structure. The swap term structure is increasingly used as the founda- tion for deriving relative term structures and as a benchmark for pricing and hedging.
The first section describes the motivation for using the swap term structure as a benchmark for pricing and hedging fixed-income securi- ties. A swap term-structure derivation technique designed to mark to market fixed-income products is then described in detail. Finally, differ- ent aspects of the derived swap term structure are discussed.