EQUILIBRIUM SHORT-RATE MODELS VS NO-ARBITRAGE MODELS: LITERATURE REVIEW AND COMPUTATIONAL EXAMPLES

Equilibrium short-rate models vs no-arbitrage models: Literature review and computational examples. Econometrics. Ekonometria. Advances in Applied Data Analysis , 25 (3). Abstract: In this paper equilibrium short-rate models are compared against no-arbitrage short-rate models. This article is composed of the introduction to this literature and a review, followed by numerical examples of one-factor short-rate models; the Cox-Ingersoll-Ross (CIR) model and the Vasicek model. No-arbitrage models were presented through the Hull-White (HW) model, the Binomial lattice model for bond pricing and interest rate modelling, the Black-Karasinski (BK) model, and the Heath-Jarrow-Morton (HJM) model. The results prove that no single interest rate model exists that can be used for all purposes. These models were compared in terms of volatility, mean reversion process and convergence. The end results confirm the dependence of volatility on the level rate as a determinant of the predictive success of these models.


Introduction
Short-rate models are a mathematical model that are used in the evaluation of interest rate derivatives to illustrate the evolution of interest rates over time by determination of the evolution of the short rate ( ) over time. The free short-term interest rate is a key economic variable, since it affects the short end of the term structure and has implications in the pricing of the full range of fixed income securities and derivatives, see (Andersen and Lund, 1997). In the literature, most of the models for option or discount bond pricing, or at least several of them, are developed in a continuous time framework, with the arithmetic Brownian motion 1 , e.g. (Black and Scholes, 1973;Merton, 1973). Other models such as the Vasicek (1977) model use the Ornstein--Uhlenbeck process 2 which is a Gauss-Markov process, that over time tends to drift towards its mean function; such a process is termed as mean-reverting, see (Björk, 2009). Mean-reversion is understood here as the change of the market return in the direction of a reversion level as a reaction to a prior change in the market return, see (Hillebrand, 2003). Empirical evidence on "on the extent to which stock prices exhibit mean-reverting behaviour" can be found in (Poterba and Summers, 1988). The two major classes of interest rate models can be described as: "no-arbitrage term structure models" and "short rate models", see also (Cairns (2004;Treepongkaruna and Gray, 2003).The first equilibrium models of short rate were built on the assumption of how the economy works. Equilibrium models or short-rate models such as those developed by Vasicek (1977), Dothan (1978), Cox, Ingersoll, Ross (1980), Cox, Ingersoll, and Ross (CIR) (1985), the Ho-Lee Model (1986) was the first paper that developed this idea prominently. Longstaff (1989Longstaff ( , 1992, Longstaff and Schwartz (1992), Schwartz (1979, 1982) and Chan, Karolyi, Longstaff, and Sanders (hereafter CKLS) (1992) all started with an stochastic differential equation (SDE) model and developed pricing mechanisms for bonds under an equilibrium framework, in these models drift and diffusion parameters are not allowed to vary over time, see also (Buetow, Fabozzi and Sochacki, 2012). The inverse of the CIR model 3 is discussed in 1 Definition of Brownian motion: Let ( , ℱ ) ∈(0,∞ ) be an ℝ-valued continuous stochastic process in probability space (Ω, ℱ, ), then ( , ℱ ) ∈(0,∞ ) is called standard Brownian motion if: = 0; − ∼ (0, − ); − ⊥ ℱ . An ℝ valued process is called -dimensional Brownian motion with initial value ∈ ℝ if = + ( 1 , … . , ), ∀ ∈ (0, ∞), where are standard Brownian motions, see (Ewald, 2007). is a mean-reversion rate. Here ( ) follows normal distribution with ( (0) − 2 ; 1 − − ).
The squared radius of the vector ( ) is ( ) = ∑ ( ) 2 → ( ) = ∑ (2 ( ) ( ) + (Ahn and Gao, 1999) and then (Aït-Sahalia, 1999). No-arbitrage models (no-arbitrage bounds represent mathematical relations that are specifying limits on portfolio prices. These price bounds are a specific example of good-deal bound, see (Birge, 2008)). The general approach to option pricing is first to assume that prices do not provide arbitrage opportunities. For a more general explanation on good-deal bounds see Björk, Tomas, Slinko, Irina. (2006). Black and Karasinski, (1991), Black, Derman, and Toy (1990) 4 , Ho and Lee (1986), Heath, Jarrow, and Morton (1992), and Hull andWhite (1990, 1993) begin with the same or similar SDE models as the equilibrium approach, but use market prices to generate an interest rate lattice. No-arbitrage models are the preferred framework to evaluate the value of interest rate derivatives. The derivation of the option prices is obtained by replicating the payoffs provided by the option using the underlying asset and risk-free borrowing/lending. The simplest form of no-arbitrage model is put-call parity to obtain values market prices of bonds that are exact. Equilibrium models will not price bonds exactly. In this paper the authors took into consideration: the Hull-White one factor model to simulate the price of a bond and the total return of a bond portfolio, followed by the CIR model to simulate the daily short rates. The tree models are presented through the Black-Karasinski (BK) model (the (BK) Black-Karasinski model is a single-factor, log-normal version of the HW model), and the Heath-Jarrow-Morton (HJM) model. The (HJM) Heath-Jarrow-Morton model considers a given initial term structure of interest rates and a specification of the volatility of forward rates to build a tree representing the evolution of the interest rates, based on a statistical process (see also MathWorks® help center). These models use sets of zero-coupon bonds 5 to predict changes in interest rates 6 . This paper compared two types of models and drew conclusions as to which models are best suited to be used (the authors tested their volatility with respect to the level rate, as a determinant of the success of the model).

Fundamental theorem of asset pricing
Theorem 1. Suppose that bond prices evolve in a way that is stochastic, if ∃ ≡ , under which for ∀ , the discounted price process ( , )/ℬ( ) is martingale 7 for all 4 Empirical investigation of the Black-Derman-Toy model (BDT) (Bali, 1999), uses the data set originally constructed by Fama (1984) and then updated by the Center for Research in Security Prices (CRSP). ( ) 2( , ) , where ( , ): = ∫ − ∫ . 6 A zero-coupon bond is a bond that, instead of carrying a coupon, is sold at a discount from its face value, pays no interest during its life, and pays the principal only at maturity. 7 Martingale is a sequence of variables: 0 , 1 , …. With finite means such that the conditional expectation of +1 is given as: ( +1 | 0 , … , ) = . One dimensional random walk is an example of martingale, see (Doob, 1953). ∀ : 0 < < . If this holds then the market is complete if and only if is a unique measure under which ( , )/ℬ( ) are martingales.
Where ℱ is a algebra 8 generated by the price histories up to the time, implies expectations with respect to the equivalent martingale measure. Fair value ( ) at time under which is some ℱ measurable derivative payment payable, then the discounted price process (1) ( , + 2) is martingale under from → + 1, by the martingale Representation theorem ∃ ( , ) which is a predictable process such that:   The Vasicek model (1977) is a one-factor short-rate model, that describes interest rate movements driven by only one factor of market risk. Vasicek (1977) proposed the following model:

The Vasicek model
where is a risk-free interest rate; represents a rate at which ( ) reverts to the long--term mean; represents volatility of short-term interest rates; ̃( ) is the standard Brownian movement under risk neutral measure , and > 0; > 0; > 0, ( + ) given ( ) is normally distributed under risk-neutral measure with mean and

The Cox-Ingersoll-Ross (CIR) model (1985)
In the Cox-Ingersoll-Ross model (1985), the short-rate is assumed to satisfy the following differential equation: where , , > 0 with 2 > 2 and is a Brownian motion under risk-free measure. Now, let 0 ≤ ≤ ≤ , the short rate in the CIR model satisfies: and also: In the CIR model, the price of a zero-coupon bond with maturity at the time ∈ [0, ] is given as: , here: ℎ = √ 2 + 2 2 .
Bond price dynamics in the CIR model is given as: ( , ) = ( ) ( , ) − √ ( , ) ( , ) ( ). And: Under T-forward measure 10 ℚ , the short rate in the CIR model satisfies: In the CIR model the instantaneous forward rate with maturity is given as: ( , ) = ( , ) + ( ) + ( , ), and satisfies the following stochastic differential equation 11 : ( , ) = √ ( ) ( , ) ( ). Extensions of the models were provided most notably in (Maghsoodi, 1996). There coefficients were replaced with time-varying functions in order to make it consistent with a pre-assigned term structure of interest rates and possibly volatilities, while one extension is given in Chen (1996), which is a three-factor model. The unique solution to ( ) = ( − ( )) + √ ( ) is given as: For zero-coupon bonds by taking = 1; = 0, and we have ( Hull and White (1990) explored extensions of the Vasicek model (1977) that provide an exact fit to the initial term structure 12 . One version of the extended Vasicek model that they considered was: .

The Hull-White model (1990)
The Ornstein-Uhlenbeck process provides the following: Function ( ) can be calculated from the initial term structure: ( ) = (0, ) + (0, ) + 2 2 (1 − −2 ). The last term of the equation is small so if ignored it: (0, ) + [ (0, ) − ]. This shows that, on average, follows the slope of the initial instantaneous forward rate curve. When it deviates from that curve, it reverts back to it at rate . Or, when a current term structure is matched, Hull and White (1993): (1 − −2 ). It turns out that the time-s value of the T-maturity discount bond has distribution, see (Hull and White, 1996): where . These equations define the price of a zero-coupon bond at a future time in terms of the short rate at time and the prices of bonds today. The latter can be calculated from today's term structure. ( , ) is log-normally distributed. The solution for here is given as in (Gurrieri, Nakabayashi, and Wong, 2009 Then for 1 > we have: . From the previous one can write: 13 Here ℱ is the information set or price history as defined before. The probability law in the Markov models requires prices as input and the probability law of ( , )|ℱ , where < < which is equal to that of ( , )| ( ) in some finite dimensional Ito process.
is also a Markov process, that is the law of

The Black-Karasinski model (1991)
Black and Karasinski (BK) developed a model, within a discrete time framework, where the target rate, mean reversion rate and local volatility are deterministic functions of , see (Svoboda, 2004). The Black-Karasinski model (1991) assumes that evolves to: Black-Karasinski in a similar way to Hull-White introduced the "target interest rate" ( ): ( ) = [( − )] + , thus the Black-Karasinski model 14 in discrete time, where ( ) or is: where = +1 − . Black, Derman, and Toy (1990) proposed a binomial model with the volatility of a short rate proportional to This is also a single stochastic model, and the interest rate determines the future evolution of all interest rates. Buetow, Hanke, and Fabozzi (2001) suggested that the BDT model follows the following stochastic differential equation: where is a standard Brownian motion under a risk-neutral probability measure, is an instant short rate volatility. In the specification of the binomial tree BK model, the mean reversion may be equated to the rate of change of local volatility, as the short--term interest rate gets closer to the target rate:

The Heath-Jarrow-Morton model (HJM) (1992)
This model 15 is generated through Heath, Jarrow, and Morton, 1990, 1991, 1992, who formulated the dynamics of the forward rate curve ( , ) where is a time of maturity, starting from a given initial curve (0, ). They showed that the curve's arbitrage free risk-neutral dynamics is determined by the forward-rate volatilities ( , ): This is effectively infinite dimensional SDE (Jamshidian, 2010), driven by the finite dimensional Brownian motion. Regarding the volatility and the Gaussian interest rate (Jamshidian, 1991)  (iii) (0, ) is deterministic and satisfies,∫ | (0, )| < ∞; 0 (iv) [|∫ ( , ) | 0 |] < ∞, see (Cairns, 2004). 16 A Bermuda option is a type of exotic options contract that can only be exercised on predetermined dates, often on one day every month. Exotic options are hybrid securities that are often customizable to the needs of the investor.

Simulate the price of a bond using a Hull-White one-factor model until the bond's maturity
Here the authors used MATLAB to simulate the price of a bond using the Hull-White one-factor model 19 until the bond's maturity. First, they defined the zero-curve data, then the CouponRate = 0, followed by the Hull-White parameters: = 0 .1; and = 0.01. . 19 The Hull-White one-factor model is specified using the zero curve, alpha, and sigma parameters. Zero curve is used to evolve the path of the future interest rates. HW1F = HullWhite1F (RateSpec, alpha, sigma). Then the simulation parameters are defined: nTrials = 100; nPeriods = 12*5; ∆ = 1/12. The simulation gives the following result: Source: authors' own calculation.
Simulated prices of the bond are converging in approximately five years, since the start of the simulation.

Simulate the total return of a bond portfolio until maturity using the Hull-White one-factor model
Now there is a simulation of total return of a bond portfolio until maturity by using the As can be seen in the following graph, the simulated portfolio bond returns first are converging around zero returns but not zero, and afterwards they are diverging. References to these models in the Matlab financial toolbox are in (Brigo and Mercurio, 2006) and (Hull, 2011). Source: authors' own calculation. Source: authors' own calculation.
When = 0 the Hull-White model becomes the Ho-Lee model and the simulated bond price is depicted in the following graph 20 .

The Cox-Ingersoll-Ross process
The Cox-Ingersoll-Ross model 21 is a continuation in the equilibrium short-rate models. This exercise of the CIR model uses the following parameters: = [0 0.25]; observation times, = 0.2; mean-reversion parameter, = 0.05; long-term mean, = 0.1; volatility 0 = 0.04; starting value. The resulting empirical distribution is similar to normal: is the change in the short-term interest rate over a small interval, ( ) is a function of time determining the average direction in which moves, chosen such that movements in are consistent with today's zero coupon yield curve, is the mean reversion rate, is a small change in time, is the annual standard deviation of the short rate, is the Brownian motion. 21 The CIR process satisfied the following equation: ( ) = ( − ( )) + √ ( ).  From the previous graph one concludes that for higher volatility, the parameter sample path of the CIR processes has a higher drift.

The Vasicek model
The Vasicek function is given as: = ( , , , ) 22 and the model is: From the previous graph it can be seen that on the first graph, the left amplitude is positive when calculating from trough 23 to peak or a bullish retracement, which is a term to describe a minor pullback (a moderate drop in a stock or commodities pricing chart in an otherwise trend of recent peaks, this can be also called a consolidation, or an asset oscillating between well-defined patterns of trade; i.e. market indecisiveness). 22 Here is the speed of the mean reversion, is the risk-neutral long-term mean of the short rate, is the volatility or std. of the short rate, r is the current short rate at time t. 23 A trough is the stage of the economy's business cycle that marks the end of a period of declining business activity and the transition to expansionor as in this examplea trough is the low end of price decline and at the same time a point of transition to a high peak. From the above graphs it can be seen that a convergence takes more steps when the security's price is going through a positive shift, which is not the case in a minor negative pullback. Pricing drops are relatively short in duration. The amplitude also allowed to estimate the volatility (level) or the amount of risk present in a particular investment. In the previous example the arguments in the function were: = 0.07; = 0.3; = 0.08; = 0.01, and the maturity matrix is given as = [0: : 10].
26 Forward interest rate stochastic integral will be: From the previous graph one can see the evolution of the short rate in one years' time by using the CIR (1985) model.

The Binomial lattice model for bond price and interest rate modelling
These models can be viewed as approximations to the more sophisticated models. The study used martingale 27 , see (Feller, 1971) pricing on this lattice to compute the bond prices, which means using no-arbitrage models. The value process associated with the trading strategy is defined as: By definition, the price of a discount bond 28 with maturity date is given as, see (Li, Ritchken, and Sankarasubramanian, 1995): ( , ) = − ∫ ( , ) . Ritchken and Sankarasubramanian (1995) identify the class of volatility structures that permit the terms structure to be presented by a two-state Markovian model 29 . To provide an illustration of the binomial lattice no-arbitrage models, the study used the Matlab code by Krishna (2021). Hence now the price of a bond is given as:     28 The forward rate diffusion process is given as:   In the next two figures the Black-Karasinski model is presented as a plot and path diagram presentation.  The conclusion in this 'no-arbitrage' model is that the starting price does not differ much from the closing price, and the ratio between the two is close to 1, i.e.

The Heath-Jarrow-Morton interest-rate tree
In this model 31 the following parameters are given in order to estimate the path of bond prices: Rates .or = ( , ) + ( , ) , is not correlated to due to no-arbitrage.

Conclusion
The Cox et al. (1985) model, andthe Vasicek (1977) model are similar, they assume that all interest rate contingent claims are based on short-term interest rates. The mean--reversion term in this model assumes that the yield curve on a bond or security cannot follow random walk as in a stock movement. The mean reversion term meant that short-term rate is higher than the long-term , so that the short-term rate would fall adjusting gradually to the long-term interest rate. Long-term interest rates will follow random walk and a martingale sequence with finite means such that the conditional expectation of +1 is given as: ( +1 | 0 , … , ) = , see (Mishkin, 1978;Pesando, 1979). Conversely the short-term rate is lower than the long-term interest rate, and it will rise to long-term rate, see (Ho and Lee (2004) 32 . Now the arbitrage-free condition − = ( , ), where is the market price of risk. No-arbitrage models are making a departure from economic theory and they assume that the yield curve follows a random movement, just like the model used to describe a stock price movement. One 32 Dybvig (1989) shows that the one-factor model offers an appropriate first-order approximation for modelling the yield curve movement, i.e. that the yield curve can be confined to exhibit one-factor movement where that factor is only movement in .
shortcoming of the no-arbitrage models such as the Ho-Lee (1986) and Hull and White (1990) model, the Black-Karasinski model (1991), the Heath, Jarrow and Morton (1992) model and binomial tree models is that interest rate movement does not exhibit mean reversion process, and the volatility of the one-period rate is constant at each node point.The assumption in the no-arbitrage models seems to be more unrealistic, namely: first assuming a perfect capital market 33 , the yield curve can move only up and down, one period interest rate volatility is the same in all of the states of the model, there exists no-arbitrage opportunity in any node point on the binomial lattice. The distribution used in the models is normal or log-normal namely: Vasicek (1977), Hull and White (1990) used normal distribution, while Cox, et al. (1985), Black-Karasinski model (1991) used log-normal distributions. The Heath, Jarrow, and Morton (1992) model has become a popular term structure model in the interest rate derivatives pricing theory. In the HJM (1992) model, the only inputs needed to construct the term structure are the initial yield curve and the volatility structure for all forward rates. No--arbitrage models such as the HJM (1992) model are in general non-Markovian, and contingent models are considered difficult to price with the lattice method. To solve this problem the HJM (1992) model version used in this paper is a single-factor model with a forward volatility structure that depends on the time and the time to maturity, the instantaneous spot rate and forward rate to a fixed maturity. The results showed that convergence in the Hull and White (1990) model was achieved in five periods (years) since the start. Regarding the simulation of the total return on bond portfolio until maturity using the Hull-White one-factor model, the results are that the bond returns first converged around zero returns and afterwards diverged, and negative returns are allowedmeaning that the premium exceeds the income the investor will receive during their holding period. In this model, returns converged in four periods (years) of time. In the simulated CIR model, the conclusion was that volatility was not constant as in the no-arbitrage models, and that the higher volatility parameter sample path of CIR processes has a higher drift. The simulation of the Vasicek model proved that convergence takes more steps when the interest rate is going through a positive shift.When a positive shift of price occurs in the Vasicek model, convergence is achieved in five steps, In the case of a negative pullback of interest rate, convergence is achieved in one step. Binomial lattice models were the first of the no-arbitrage models simulated in this paper. From the example it can be seen that they are more useful since they provide information about short-rate dynamics,underlying asset price and the call price and put option, for binomial lattice models see (Benninga and Wiener, 1999). In the Black-Karasinski model with constant volatility, the conclusion was that when the starting price does not differ much from the closing price, the ratio between the two is close to 1.The HJM model simulated in this paper gives the solution of the closing price with a lower and upper bound. In conclusion it can be confirmed that there is no single interest rate model that can be used for all the purposes. The authors concluded that since interest rate volatility is fundamental to the valuing of the contingent claims and hedging interest risk, the models based on the rate structure should focus on the dependence on the volatility of the level of the rate. This notion was also put forward in the empirical literature, see: (Chan, Karolyi, Longstaff, and Sanders, 1992).