The Basic Assumptions Al The market is frictionless. There are no taxes and no transaction costs, and all securities are perfectly divisible. A2 The market clears at discrete points in time, which are separated in regularintervals. For simplicity, we use each period as a unit of time. A3 The bond market is complete. A4 At each time n, there are a finite numberof states of the world. For state i, we denote the equilibriumprice of the discount bond of maturity T by p n T.
Note that p n. This function is called the discountfunction. Within the context of the model, the discount function completely describesthe term structureof interest rates of the ith state at time n. The discount function p n. It must be positive since the function represents assets' values. Equation 2 says that a discount bond with maturity in the distant future must have a negligible value. Assumptions Al through A4 are the standardperfect capital market assumptions in a discrete state-time framework.
The Binomial Lattice Now we describethe evolution of the term structure. Initially, we observe the discount function P. At the initial time, by convention, we have the 0-state. So we have P. The superscript denotes the time, and the subscript denotes the state. Therefore, there are only two states of the world at time 1.
When p l. We confine each discount function, in either upstate or downstate, to also attain only one of two possible functions. Specifically,conditional on the discount function P. At this point, we do not specify any particularfunctional forms for PMY.
Similarly, conditional on the discount function P. Note that the binomial lattice assumptionrequiresthe discount function attained by an upstate followed by a downstate to be equal to the discount function reachedby a downstate followed by an upstate.
The stochastic process of the discount function in subsequent periods is describedanalogously. We requirethe discount function to depend only on the numberof upstate movements and not on the sequence in which they occur. A discount function is defined for each time n and state i. This set of discount functions is said to form a binomial lattice. A vertex of this lattice is specified by n, i. The term structurecan evolve from one vertex to another by different paths, but this will not affect the value of the discount function at the vertex at the end of the path.
That is, the discount functions are path independent. Often, it is more convenient to represent a term structure by the yield curve as opposedto the discount function. The Binomial Process of Bond Prices When the term structureevolves in a binomiallattice, the price of each discount bond must follow a binomialprocess with time-dependentstep size.
In particular, consider the discount bond with maturity N. Initially, the bond price is, by definition, P N. After the first period,the bond shortens its maturityto N - 1 , and therefore, given the discount functions in the upstate and downstate, we can determinethe bond prices;they are P 1 N - 1 and P ' N - 1 in an upstate and downstate, respectively.
All subsequent prices are determined analogously. The stochastic price process of a discount bond is depicted in Figure 1. The discount function is depicted in each state and time in a binomial lattice. Figure 1 shows that the discount function always originates from unity. It increases in value in an upstate but drops in value in a downstate.
Now consider the three- period bond. Initially, its value is P 3. At time 1, it becomes a two-periodbond, and its value can be either Pl1 2 or p 1 2.
At time 2, this bond becomes a one- period bond, and its value cannot deviate too much from unity in any state of the world and must convergeto unity at maturity.
This model of a bond price stochastic process is similar to the binomialprocess proposedby Cox, Ross, and Rubinstein CRR [8] and Rendleman and Bartter [18] for stocks. However, there are two main differences. First, when pricing interest rate contingent claims, in most cases we are concerned with how the prices of discount bonds with different maturities move relative to each other.
That is why we focus on the binomial lattice of a term structure rather than a binomial process of a particularbond. Second, in our model, the step size is time dependentto ensure that the bond value convergesto unity at maturity.
The bond price uncertainty is small at the two extreme time points: for the time horizon in the immediate future and near bond maturity. The price uncertainty is large for time horizons away from these two endpoints. We achieve this by isolating the two effects. The first effect is the resolution of uncertainty of the term structure.
As the time horizon lengthens, we are more uncertain about the term structureconfiguration. There are more variations of the term structure when n is large. The second effect is that the bond price uncertaintymust decreasewhen the time horizonapproaches maturity since the bond price cannot significantly deviate from unity when the maturityis short.
Now consider a particularbond. As the time horizon increases, uncertaintyof the term structureconfigurationincreases, leading to largerbond price variance. However,at the same time, the bond has shorter maturity in the future,and the latter effect the maturityeffect prevails.
When the time horizon is sufficiently distant in the future, the latter effect may dominate the former, leading to a decrease in bond price uncertainty. We can now compareour model with that of Schaefer and Schwartz [20].
In their approach,they seek to model the stochastic process of a bond price. In their case, they must specify a process such that its variance is time dependent. This section introducesthe necessary constraints on the term structuremovement such that the movement is consistent with an arbitrage-freeenvironment.
We also intro- ducesome simplifyingrestrictionsso that we can developa procedureto construct a "desirable"term structuremovement. Furthermore, the discount function must be the implied forwarddiscount function F n T to avoid any arbitrageopportunities. This observationwas made in Merton [14]. Schaefer and Schwartz [20] analyzea particularprocesswherethe instantaneousvarianceis proportionalto the underlying bond duration. See Ho and Lee [ Pricing Interest Rate ContingentClaims In a certainty world, if the next period discount function differs from Fin T , then investors can realizearbitrageprofits.
Therefore,in modelingterm structure uncertainty,we are concernedwith how the discount function is perturbedfrom the implied forwardfunction in the following period. Thus, roughly, they specify the difference between the upstate and downstate prices over the next period. When h T is significantly greaterthan unity for all values of T, then all the bond prices will rise substantially in the upstate.
The ImpliedBinomialProbabilityr Given a binomial lattice of a term structuremovement,we also need to ensure that there is no arbitrageprofit to be made in forming arbitraryportfolios of the discount bonds. Specifically, if we take any two discount bonds with different maturities and construct a portfolio of these two bonds such that the portfolio realizes a risk-free return over the next period, then the risk-free rate must be the return of a one-perioddiscount bond.
This arbitrage-freecondition imposes a restriction on the perturbationfunctions at each vertex n, i. The method to calculate the risk-free hedge is similar to that of CRR, and the details of the argumentsare given in AppendixA. The result shows that, when the bond price upwardmovement is significant, the bond price downwardmovement must also be sizable such that the weighted average of the movements is the same across all bonds.
The implied binomial probability can be understood in the Cox, Ross, and Rubinstein context of binomial option pricing. For this reason, the implied binomial probability wris the "risk-neutral"probability of CRR within their model's context. For large 7r,the model says that the price change for the next periodis mainly a price decrease. Similarly,when r is small approximatelyzero , the price change is dominated by a price rise.
Equation 10 shows that, if one cannot arbitrageusing the discount bond, then this ratio must be the same for all bonds. The Path-IndependentCondition In constructing the binomial lattice, we assume a discount function evolving from one state to another depending only on the number of the upwardmove- ments and not on the sequencein which they occur.
To investigatethe implicationof this constraint,considerthe discount function p n T at time n and state i. Then, the aboveexpression,derivedalso in CRR, followsin a straightforward mannerfromequation We compare it with other models of interest rate movements. Equations 19 and 20 show that the AR model is uniquely determined by two constants 7r and 6.
Section IIB has given the intuitive explanation of 7r. The larger the spread, the greater the interest rate variability. For this model, at any state-time, consider a bond with maturity T. The bond's upstate price relative to the implied forward price is h T ; therefore, the longer the maturity, the larger the price change. For short-term bonds, the price change is negatively related to 6. Accordingto equation 17 , we see that 0 c 6 c 1. Vasicek [22], Dothan [9], Richard [19], and Cox, Ingersoll, and Ross [7] have proposed equilibrium models of the term structure.
In these models, all the discount bonds are priced relative to the stochastic short rate in such a way that there are no arbitrageopportunitiesin trading the discount bonds. Since the AR model also requires all bonds to be priced relative to a bond, and hence to a specific interest rate, it would provide valuable insight to show how bonds are priced relative to the short rate and how the model may be identified with these single-state-variablemodels.
The short rate, within the context of our model, is the rate of a one-period discount bond. Now we wish to determine the stochastic process of the short rate. Our approach can be used to price… Expand. View via Publisher. Save to Library Save. Create Alert Alert. Share This Paper. Background Citations. Methods Citations. Results Citations.
Figures from this paper. Citation Type. Has PDF. Publication Type. More Filters. Enormous progress has been made by academics and finance practitioners alike in modeling the dynamics of the term structure of interest rates in the past 35 years.
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We then show that the AR model can be used to price interest rate contingent claims relative to the observed complete term structure of interest rates. This paper. The crux of the problem in pricing interest rate contingent claims is to model the term structure movements and to relate the movements to the assets' prices. Yield Curve Modeling Python Using the duration measures defined by Bierwag , we derive the formulae of duration far zero-coupon bonds, coupon bonds and bond portfolios under the Heath, Jarrow and Morton HJM term structure framework.
Yield Curve Analysis Pdf In the absence of any prices between the 6m — 2y tenor period, the choice became a function of a macroeconomic view. Account Options Default Risk.
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