Arbitrage-free models

In this section, we start from a simple multi-factor HJM term structure model and derive the drift term of the forward rate dynamics required to obtain an arbitrage-free model framework (see HJM [35]). Furthermore, we derive the equivalence between the HJM-firamework and a corresponding extended short rate model. Then, by applying our option pricing technique (see chapter (2)) we are able derive the well known closed-form solution for the price of an option on a discount bond (e.g. caplet or floorlet). 

In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

An arbitrage-free model of the term structure on the other hand can be made to fit precisely with the current, observed term structure, so that observed bond yields are in fact equal to the bond yields calculated by the model. So, an arbitrage-free model is intended to be consistent with the currently observed... 

For these reasons, practitioners may prefer to use an arbitrage-free model if one can be successfully implemented and calibrated. This is not always straightforward, and under certain conditions, it is easier to implement an equilibrium multi-factor model (which we discuss in the next section) than it is to implement a multi-factor arbitrage-free model. Under one particular set of circumstances, however, it is always preferable to use an equilibrium model, and that is when reliable market data is not available. If modelling the term stmcture in a developing or emerging bond market, it will be more efficient to use an equilibrium model. 

Some texts have suggested that equilibrium models can be converted into arbitrage-free models by making the short-rate drift rate time dependent. However, this may change the whole nature of the model, presenting problems in calibration. 

The Ho-Lee (1986) model was one of the first arbitrage-free models and was presented using a binomial lattice approach, with two parameters the standard deviation of the short-rate and the riskpremium of the short-rate. We summarise it here. Following Ho and Lee, let ( ) be the equilibrium price of a zero-coupon bond maturing at time T under state i. That is F( ) is a discount... 

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 problem with some of the models just discussed is that they generate their own term structures that, in the absence of adjustment, do not match the term structure observed in the market. A category of arbitrage-free models proposed by Ho and Lee, Hull and White, and Black, Herman, and Toy seek to eliminate this problem. For example, the Black, Herman and Toy model enjoys a degree of popularity among market practitioners because (1) it takes account of and matches the term structure observed in the market, (2) it eliminates the possibility of... 

Selecting the appropriate term-structure model is more of an art than a science, depending on the particular application involved and the users individual requirements. The Ho-Lee and BDT versions, for example, are arbitrage, or arbitrage-free, models, which means that they are designed to match the current term structure. With such models—assuming, of course, that they specify the evolution of the short rate correctly—the law of noarbitrage can be used to determine the price of interest rate derivatives. 

Real option valuation (ROV) Recently, Gupta and Maranas (2004) revisited a real-option-based concept to project evaluation and risk management. This framework provides an entirely different approach to NPV-based models. The method relies on the arbitrage-free pricing principle and risk neutral valuation. Reconciliation between this approach and the above-described risk definitions is warranted. 

At last, we extend the single-factor framework to a more general multi-faetor USV model. Starting from the arbitrage-free Al-factor bond priee dynamies... 

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

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. 

Zheng, C., 1994. An arbitrage-Free SAINTS Model of Interest Rates. First National Bank of Chicago, Chicago, Working Paper. 

The Vasicek model was the first term-structure model described in the academic literature, in Vasicek (1977). It is a yield-based, one-factor equilibrium model that assumes the short-rate process follows a normal distribution and incorporates mean reversion. The model is popular with many practitioners as well as academics because it is analytically tractable—that is, it is easily implemented to compute yield curves. Although it has a constant volatility element, the mean reversion feature removes the certainty of a negative interest rate over the long term. Nevertheless, some practitioners do not favor the model because it is not necessarily arbitrage-free with respect to the prices of actual bonds in the market. 

What is the significance of this Here we take it as given that because price processes can be described as equivalent martingale measures (which we do not go into here) they enable the practitioner to construct a risk-free hedge of a market instmment. By enabling a no-arbitrage portfolio to be described, a mathematical model can be set up and solved, including risk-free valuation models. 

A short-rate model can be used to derive a complete term structure. We can illustrate this by showing how the model can be used to price discount bonds of any maturity. The derivation is not shown here. Let P t, T) be the price of a risk-free zero-coupon bond at time t maturing at time T that has a maturity value of 1. This price is a random process, although we know that the price at time T will be 1. Assume that an investor holds this bond, which has been financed by borrowing funds of value C,. Therefore, at any time t the value of the short cash position must be C,= —P(t, T) otherwise, there would be an arbitrage position. The value of the short cash position is growing at a rate dictated by the short-term risk-free rate r, and this rate is given by... 

A default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

The fair price of a convertible bond is the one that provides no opportunity for arbitrage profit that is, it precludes a trading strategy of running simultaneous but opposite positions in the convertible and the underlying equity in order to realize a profit. Under this approach we consider now an application of the binomial model to value a convertible security. Following the usual conditions of an option pricing model such as Black-Scholes (1973) or Cox-Ross-Rubinstein (1979), we assume no dividend payments, no transaction costs, a risk-free interest rate, and no bid-offer spreads. 

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