Spot rates curve

Calibration to the current spot rate curve, using the volatilities implied by the prices of exchange-traded optimis therefore, the model would be implemented using volatility parameters that were exactly similar to those implied by the traded option prices. In practice, this can be a lengthy process ... 

Calibrating the model to the current spot rate curve, using volatility parameters that are approximately close enough to result in prices that are near to those of observed exchange-traded options. This is usually the method that is adopted. 

We can compare fitted yield curves to an actual spot rate curve wherever there is an active government (risk-free) zero-couprai market in operation. In the United Kingdom, a zero-couprai bmid market was introduced in December... 

To value a nongovernment bond, the arbitrage-free value is found by adding a suitable spread to the government spot rates. A spot rate curve can be created using any benchmark such as LIBOR. 

YIELD SPREAD MEASURES RELATIVE TO A SPOT RATE CURVE... 

The nominal spread measure has several drawbacks. For now, the most important is that the nominal spread fails to account for the term strnctnre of spot rates for both bonds. We will pose an alternative spread measnre that incorporates the spot rate curve. 

The zero-volatility spread, also referred to as the Z-spread or static spread, is a measure of the spread that the investor would realize over the entire benchmark spot rate curve if the bond were held to maturity. Unlike the nominal spread, it is not a spread at one point on the yield curve. The Z-spread is the spread that will make the present value of the cash flows from the nongovernment bond, when discounted at the benchmark rate plus the spread, equal to the nongovernment bond s market price plus accrued interest. A trial-and-error procedure is used to compute the Z-spread. 

The nominal spread for the nongovernment bond is 148.09 basis points. Let s use the information presented in Exhibit 3.15. The second column in Exhibit 3.15 shows the cash flows for the 7%, 5-year nongovernment issue. The third column is a hypothetical benchmark spot rate curve that we will employ in this example. The goal is to determine the spread that, when added to all the Treasury spot rates, will produce a present value for the non-government bond equal to its market price of 101.9141. 

What does the Z-spread represent for this nongovernment security Since the Z-spread is relative to the benchmark euro spot rate curve, it represents a spread required by the market to compensate for all the risks of holding the nongovernment bond versus a government bond... 

A Z-spread can be calculated relative to any benchmark spot rate curve in the same manner. The question arises what does the Z-spread mean when the benchmark is not the euro benchmark spot rate curve (i.e., default-free spot rate curve) This is especially true in Europe where swaps curves are commonly used as a benchmark for pricing. When the government spot rate curve is the benchmark, we indicated that the Z-spread for nongovernment issues captured credit risk, liquidity risk, and any option risks. When the benchmark is the spot rate curve for the issuer, for example, the Z-spread reflects the spread attributable to the issue s liquidity risk and any option risks. Accordingly, when a Z-spread is cited, it must be cited relative to some benchmark spot rate curve. This is essential because it indicates the credit and sector risks that are being considered when the Z-spread is calculated. Vendors of analytical systems such Bloomberg commonly allow the user to select a benchmark. 

The zero-volatility or static spread is the spread that when added to the government spot rate curve will make the present value of the cash flows equal to the bond s price plus accrued interest. When spread is defined in this way, spread dnration is the approximate percentage change in price for a 100 basis point change in the zero-volatility spread holding the government spot rate curve constant. 

The most popular version of this approach was developed by Thomas Ho in 1992. This approach examines how changes in US Treasury yields at different points on the spot curve affect the value of a bond portfolio. Ho s methodology has three basic steps. The first step is to select several key maturities or key rates of the spot rate curve. Ho s approach focuses on 11 key maturities on the spot rate curve. These rate durations are called key rate durations. The specific maturities on the spot rate curve for which a key rate duration is measured are 3 months, 1 year, 2 years, 3 years, 5 years, 7 years, 10 years, 15 years, 20 years, 25 years, and 30 years. However, in order to illustrate Ho s methodology, we will select only three key rates 1 year, 10 years, and 30 years. 

This section describes the relationships among spot interest rates and the actual market yields on zero-coupon and coupon bonds. It explains how an implied spot-rate curve can be derived from the redemption yields and prices observed on coupon bonds, and discusses how this curve may be used to compare bond yields. Note that, in contrast with the common practice, spot rates here refer only to rates derived from coupon-bond prices and are distinguished from zero-coupon rates, which denote rates actually observed on zero-coupon bonds trading in the market. 

Calibration to the current spot rate yield curve, using a pre-specified volatility level and not the volatility values given by the prices of exchange-traded optiOTis. This may result in mispriced bonds and options if the selected volatilities are not accurate ... 

So, in this case the forward rate is decreasing at the point T when the spot rate is at its maximum value. This is illustrated hypothetically in Figure 3.1 and it is common to observe the forward rate curve decreasing as the spot rate is increasing. 

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

A multi-factor model of the whole yield curve has been presented by Heath et al. (1992). This is a seminal work and a ground-breaking piece of research. The approach models the forward curve as a process arising from the entire initial yield curve, rather than the short-rate only. The spot rate is a stochastic process and the derived yield curve is a function of a number of stochastic factors. The HJM model uses the current yield curve and forward rate curve, and then specifies a continuous time stochastic process to describe the evolution of the yield curve over a specified time period. 

The expression describes a stochastic process composed of n independent Wiener processes, from which the whole forward rate curve, from the initial curve at time 0, is derived. Each individual forward rate maturity is a function of a specific volatility coefficient. The volatility values ( t, t, T, w)) are not specified in the model and are dependent on historical Wiener processes. From Equation (4.28) following the HJM model, the spot rate stochastic process is given by Equation (4.29) ... 

In the HIM model, the processes for the bond price and the spot rate are not independent of each other. As an arbitrage-free pricing model, it differs in crucial respects from the equilibrium models presented in the previous chapter. The core of the HIM model is that given a current forward rate curve, and a function capturing the dynamics of the forward rate process, it models the entire term structure. 

The discount function relates the current cash bond yield curve with the spot yield curve and the implied forward rate yield curve. From Equation (5.3) we can set ... 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

Consider the case of Tesla Motors Inc., the bond has semi-annual coupon payments. In this case, the present value is determined by using the spot rate for each payment date in which the international yield curve has been implemented. 

In chapter 2 of the companion volume to this book in the boxed-set library, Corporate Bonds and Structured Financial Products, we introduced the concept of the yield curve, and reviewed some preliminary issues concerning both the shape of the curve and to what extent the curve could be used to infer the shape and level of the yield curve in the future. We do not know what interest rates will be in the future, but given a set of zero-coupon (spot) rates today we can estimate the future level of forward rates using a yield curve model. In many cases however we do not have a zero-coupon curve to begin with, so it then becomes necessary to derive the spot yield curve from the yields of coupon bonds, which one can observe readily in the market. If a market only trades short-dated debt instruments, then it will be possible to construct a short-dated spot curve. 

It is important for a zero-coupon yield curve to be constructed as accurately as possible. This because the curve is used in the valuation of a wide range of instruments, not only conventional cash market coupon bonds, which we can value using the appropriate spot rate for each cash flow, but other interest-rate products such as swaps. 

In an arbitrage-free model, the initial term structure described by spot rates today is an input to the model. In fact such models could be described not as models per se, but essentially a description of an arbitrary process that governs changes in the yield curve, and projects a forward curve that results from the mean and volatility of the current short-term rate. An equilibrium term structure model is rather more a true model of the term structure process in an equilibrium model the current term structure is an output from the model. An equilibrium model employs a statistical approach, assuming that market prices are observed with some statistical error, so that the term structure must be estimated, rather than taken as given. 

The first column in Exhibit 3.5 simply lists the quarterly periods. Next, Column (2) lists the number of days in each quarterly coupon period assumed to be 91 days. Column (3) indicates the assumed current value of 3-month LIBOR. In period 0, 3-month LIBOR is the current 3-month spot rate. In periods 1 through 16, these rates are implied 3-month LIBOR forward rates derived from the current LIBOR yield curve. For ease of exposition, we will call these rates forward rates. Recall for a floater, the coupon rate is set at the beginning of the period and paid at the end. For example, the coupon rate in the first period depends on the value of 3-month LIBOR at period 0 plus the quoted margin. In this first illustration, 3-month LIBOR is assumed to remain constant at 5%. Column (4) is the quoted margin of 15 basis points and remains fixed to maturity. 

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