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Models to Describe Interest Rate Uncertainty

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Published: 6th Dec 2019

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Tagged: Finance


An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. Interest’s rates are fundamental to a capitalist society. Interest rates are normally expressed as a percentage rate over the period of one year. Interest rates are also a tool of monetary policy and are taken into account when dealing with variables like investment, inflation, and unemployment. In traditional actuarial investigations, the interest rate is assumed to be deterministic and hence there is only one source of uncertainty, the mortality uncertainty, to be considered. Concerns about the effects of including a stochastic interest rate in the model have been growing during the last decade. The literature has tended to focus on annuities and the model adopted to describe the interest rate uncertainty, in a continuous framework, has usually involved the use of a Brownian motion (Beekman and Fuelling,1990- 1991), (Dufresne, 1990), (De Schepper et al, 1992), (Parker, 1994), (Perry and Stadje, 2001), (Perry et al, 2003). When the market rates are high, volatility is expected to be high or when interest rates are low, volatility will be low. (Brennan & Schwartz, 1980; Chan et al., 1992; Cox et al., 1985; Nowman, 1997; Nowman&Staikouras, 1998)

A derivative which has as an underlie the ability to pay or receive a given amount of money at a given interest rate. Interest rate derivatives are the most popular kind of derivative, and include interest rate swaps and forex swaps. Features of interest rate swaps and forex swaps are swap of fixed-for-floating interest rate, a master agreement for fixed rate interest, a floating or variable rate which is reset periodically, a set-off exercise at every reset time to swap a fixed-for-floating interest rate and floating interest rate is to based on a certain benchmark (Dr. Mohd Daud Bakar, 1971)

Interest rate swaps is an exchange of interest payments on a specific principal amount. This is a counterparty agreement, and so can be standardized to the requirements of the parties involved. An interest rate swap usually involves just two parties, but occasionally involves more. Often, an interest rate swap involves exchanging a fixed amount per payment period for a payment that is not fixed (the floating side of the swap would usually be linked to another interest rate, often the LIBOR) an interest rate swap, the principal amount is never exchanged; it is just a notional principal amount (D. K. MalhotraIn, 1998).

Background research

Richard J. Rendleman, Jr. (1949) is a composer whose works have been performed by The North Carolina Symphony, The South Carolina Philharmonic Orchestra, The United States Navy Band and a number of other orchestras, chamber groups and choral ensembles. His compositions have been recorded by the Seattle Symphony, The Warsaw National Philharmonic, The Czech National Symphony, The Slovak Radio Orchestra and the St. Stephens Chamber Orchestra. Rendleman is also Professor of Finance at the Kenan-Flagler Business School of the University of North Carolina at Chapel Hill.

Rendleman married Nancy Sherwin in 1974, received his Ph.D. from UNC in 1976, and taught at Northwestern University before moving back to his home state to take a teaching position at Duke’s Fuqua School of Business. Although his academic career was moving along Rendleman still couldn’t shake his desire to make something happen with his music. (Melinda S. Stubbee, 1996)

Rendleman began composition studies with Robert Ward, and for the first few years thereafter composed mainly for piano in 1981. Later, his compositions expanded into chamber, vocal and orchestral works. Prior to studying with Ward, Rendleman’s music training consisted of five years of piano lessons as a young child followed by four years of junior high and high school band. Some of Rendleman’s earliest compositions were instrumental pieces written in 1966-1967 for Rhythm and Blues band, Soul Inc., known by many in the Salisbury, North Carolina area as “Carolina’s Finest Show and Dance Ensemble.” Rendleman is also Professor of Finance at the Kenan-Flagler Business School of the University of North Carolina at Chapel Hill. He is best known for his work in derivative securities markets, particularly implied volatility and two-state (or binomial) option pricing and for his work on the relationship between stock returns and quarterly earnings surprises. For better or worse, his work on earnings surprises in the early to mid 1980s has contributed significantly to Wall Street’s sensitivity to quarterly earnings during the 1990s ( Richard Rendleman, 2009)

Literature Review

The literature review is based on a Rendleman Bartter model developed by Yeliz Yolcu [2005]. Rendleman bartter models are the first short rate models being proposed in the financial literature. As they said before, drift and diffusion terms contain constant parameters. Moreover, they produce an endogenous term structure (current term structure of rates is an output rather than an input of the model) of interest rates in that initial term structure of rates need not to match the observed market data no matter how the parameters are chosen. Rendleman-Bartter model (1980) formulate as drt = rtdt + rtdWt; where; are nonnegative constants.When they look at roughly, we think that Dothan and Rendleman-Bartter have the same formulation. However, in the original paper of Dothan he aimed to present a valuation formula for default free bonds for a certain class of tastes when the rate follows a geometric Wiener process and so he started with his work by the above formulation (3.1.10) under the objective probability. Since, they are dealing with the risk neutral valuation, they change directly this formulation to risk neutral framework. Moreover, without doubt the analytic expression of this model is the same as that Dothan Model. Indeed, they will give the intuition of constructing this kind of model. Rendleman and Bartter assume that the short-term interest rate behaves like a stock price less than ideal. One important difference between the stock price and and interest rate is interest rate appeared to be pulled back to some long run average level over time (mean reversion) due to economic facts. (Rogers, 1995) Unlike Vasicek, they do not corporate with the mean reversion.

Michael D. Goldberg [2000] – The simplest models of the short-term interest rate are those in which the interest rate follows arithmetic or geometric Brownian motion. For example, they could write dr = adt + σdZ (24.23).In this specification, the short-rate is normally distributed with mean r0 + at and variance σ2t. There are several objections to this model. First, the short-rate can be negative. It is not reasonable to think the nominal short rate can be negative, since if it were, investors would prefer holding cash under a mattress to holding bonds. Second, the drift in the short-rate is constant. If a > 0, for example, the short-rate will drift up over time forever. In practice if the short-rate rises, they expect it to fall; example, it is mean-reverting. Thirdly, the volatility of the short-rate is the same whether the rate is high or low. In practice, they expect the short-rate to be more volatile if rates are high.

Rendleman and Bartter model, by contrast assumes that the short-rate follows geometric Brownian motion: dr = ardt + σrdz (24.24) While interest rates can never be negative in this model, one objection to equation (24.24) is that interest rates can be arbitrarily high. In practice they would expect rates to exhibit mean reversion; if rates are high, they expect them on average to decrease. The Rendleman-Bartter model, on the other hand, says that the probability of rates going up or down is the same whether rates are 100% or 1%. In the Rendleman-Bartter model, interest rates could not be negative because both the mean and variance in that model are proportional to the level of the interest rate. Thus, as the short-rate approaches zero, both the mean and variance also approach zero and it is never possible for the rate to fall below zero (Richard J. Rendleman, Jr., 1980). In the Vasicek model, by contrast, rates can become negative because the variance does not vanish as r approaches zero (Oldrich Vasicek, 1977).

Professor Landsman reviews research on both the relevance and reliability of reporting fair values for loans and other financial instruments (Landsman, 2005). Accounting standard setters define fair value as the amount that would be paid or received for the item being valued in an arm’s length transaction between knowledgeable parties (FASB, 2004a). This is a market value definition and the standard setters have indicated that, if available, a current market price for the item is said to be the best estimate of its fair value. Relevance means that the fair value is capable of making a difference to financial statement users’ decisions. Reliability means that the reported fair value represents what it is purported to represent (Barth et al, 2001). Professor Landsman concludes that the evidence on fair value reporting supports its relevance. On reliability, he suggests there is some uncertainty, using evidence from Barth, Landsman, and Rendleman (1998) based on testing a pricing model for corporate bonds. He further discusses banks’ use of their private information in determining loan fair values and consequences of model valuation errors on earnings volatility.

Short Title: Rendleman-Bartter Model

In the context of finance the The Rendleman-Bartter model is a very important theory. The Rendleman-Bartter Model could be called a short rate model. This model deals with rates of interest. The Rendleman-Bartter Model tries to explain the growth of rates of interest and is among the earliest models that dealt with rates of interest for a shorter period of time. It applied the random process that had been used to explain the movements of the basic prices of stock options. According to the model, the instantaneous rate of interest changes in accordance with the geometric Brownian motion, which is also called the exponential Brownian motion. The geometric Brownian motion is an uninterrupted random process. In the Rendleman-Bartter Model, market risk is the sole factor that is responsible for the changes in the rates of interest. This is why the Rendleman-Bartter Model could also be called a kind of “one factor model”

In this model Wt is nothing but a Wiener process. It models the risk factor involved in random markets. σ is the drift parameter of the Rendleman-Bartter model. θ is the standard deviation parameter of the Rendleman-Bartter model. The drift parameter of the Rendleman-Bartter model stands for the extent of fluctuation in the rate of interest. This rate is normally fixed, anticipated and instantaneous. The standard deviation parameter of the Rendleman-Bartter model ascertains the unpredictability of rates of interest. (Richard J Rendleman and Brit J Bartter,1979)

Market risk is the risk that the value of a portfolio, either an investment portfolio or a trading portfolio, will decrease due to the change in value of the market risk factors. The four standard market risk factors are stock prices, interest rates, foreign exchange rates, and commodity prices. The associated market risk are equity risk, interest rate risk, currency risk and commodity risk (Y Amihud, 1992).

Equity risk is the risk that one’s investments will depreciate because of stock market dynamics causing one to lose money. The measuring of risk used in the equity markets is typically the standard deviation of a security’s price over a number of periods. The standard deviation will delineate the normal fluctuations one can expect in that particular security above and below the mean, or average. However, since most investors would not consider fluctuations above the average return as “risk”, some economists prefer other means of measuring it (E Dimson, 2002).

Interest rate risk is the risk borne by an interest-bearing asset, such as a loan or a bond, due to variability of interest rates. In general, as rates rise, the price of a fixed rate bond will fall, and vice versa. Interest rate risk is commonly measured by the bond’s duration. Asset liability management is a common name for the complete set of techniques used to manage risk within a general enterprise risk management framework (George J. Hall, Thomas J. Sargent, 2010).

Currency risk is a kind of risk that arises from the change in price of one currency against another. Whenever investors or companies have assets or business operations across national borders, they face currency risk if their positions are not hedged. Transaction risk is the risk that exchange rates will change unfavorably over time. It can be hedged against using forward currency contracts. Translation risk is an accounting risk, proportional to the amount of assets held in foreign currencies. Changes in the exchange rate over time will render a report inaccurate, and so assets are usually balanced by borrowings in that currency. The exchange risk associated with a foreign denominated instrument is a key element in foreign investment. This risk flows from differential monetary policy and growth in real productivity, which results in differential inflation rates (Gunther Schnabl, 2006).

Commodity risk refers to the uncertainties of future market values and of the size of the future income, caused by the fluctuation in the prices of commodities. These commodities may be grains, metals, gas, electricity etc. A commodity enterprise needs to deal with price risk, quantity risk, cost risk, political risk (DF Larson, 1998).

The Wiener process in mathematics a continuous-time stochastic process named in honor of Norbert Wiener. It is often called Brownian motion, after Robert Brown. It is one of the best known Levy processes (cadlag stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and physics. The Wiener process plays an important role both in pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm-Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a Gaussian white noise process, and so is useful as a model of noise in electronics engineering, instrument errors in filtering theory and unknown forces in control theory

(N Dohi, 1993).

The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker-Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman-Kac formula, a solution to the Schrodinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model (MJ Hinich, 2010).

The limitation of Rendleman-Bartter Model

The assumption that the short-term interest rate behaves like a stock price is a natural starting point but it is less than ideal. One important difference between interest rates and stock price is that interest rates appear to be pulled back to some long-run level over time. This phenomenon is known as mean reversion. When r is high, mean reversion tends to cause it to have a positive drift. The Rendleman and Bartter model does not incorporate mean reversion. There are compelling economic arguments in favour of mean reversion. Hen rates are high, the economy tends to slow down and there is low demand for funds from borrowers. As a result, rates decline. When rates are low, there tends to be a high demands for funds on the part for borrowers and rates tend to rise (Y YOLCU, 2005).

Other examples of short-rate model

1. Particular short-rate models

Vasicek model

Vasicek model is a mathematical model describing the evolution of interest rates. It is a one type of “one-factor model” as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives, and has also been adapted for credit markets, although its use in the credit market is in principle wrong, implying negative probabilities (Oldrich Vasicek, 1977).

Ho–Lee model

Ho–Lee model is a short rate model to predict future interest rates. It is the simplest model that can be calibrated to market data, by implying the form of θt from market prices. Ho and Lee does not allow for mean reversion (T.S.Y. Ho, S.B. Lee, 1986).

Hull–White model

In financial mathematics, Hull–White model,which also called the ‘extended Vasicek model’ is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today’s term structure of interest rates. It is relatively straight-forward to translate the mathematical description of the evolution of future interest rates onto a tree or lattice and so interest rate derivatives such as bermudan swaptions can be valued in the model. The model is still popular in the market today (John C. Hull, Alan White, 1990).

Cox–Ingersoll–Ross model

In mathematical finance, the Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. It is a kind of “one factor model” and short rate model as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was as an extension of the Vasicek model (John C. Cox, Jonathan E. Ingersoll, Stephen A. Ross, 1985).

Black–Karasinski model

In financial mathematics, the Black–Karasinski model is a mathematical model of the term structure of interest rates. It is a one-factor model as it describes interest rate movements as driven by a single source of randomness. It belongs to the class of no-arbitrage models, i.e. it can fit today’s zero coupon bond prices, and in its most general form, today’s prices for a set of caps, floors or European swaptions ( Fischer Black, Piotr Karasinski, 1991).

Black–Derman–Toy model

Black-Derman-Toy model (BDT) is a popular short rate model. It is a one-factor model, that is, a single stochastic factor – the short rate – determines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the lognormal distribution, and is still widely used. It was first developed for in-house use by Goldman Sachs in the 1980s and was published in the Financial Analysts Journal in 1990. A personal account of the development of the model is provided in one of the chapters in Emanuel Derman’s memoir “My Life as a Quant.” Under BDT, using a binomial lattice one calibrates the model parameters to fit both the current term structure of interest rates (yield curve), and the volatility structure for interest rate caps (usually as implied by the Black-76-prices for each component caplet). Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and IRDs ( Fischer Black, Emanuel Derman, Bill Toy, 1980).

2. Multi-factor short-rate models

Longstaff–Schwartz model

Longstaff–Schwartz model is a two-factor model of the term structure of interest rates. It produces a closed-form solution for the price of zero coupon bonds and a quasi-closed-form solution for options on zero coupon bonds. The model is developed in a Cox-Ingersoll-Ross framework with short interest rates and their volatility as the two sources of uncertainty in the equation (Longstaff, Francis and Eduardo Schwartz , 1992).

Chen model

The Chen model is a mathematical model describing the evolution of interest rates. It is a type of “three-factor model” of short rate model as it describes interest rate movements as driven by three sources of market risk. It was the first stochastic mean and stochastic volatility. In an authoritative review of modern finance, Chen model is listed as a major term structure model. Different variants of Chen model are still being used in financial institutions worldwide. James and Webber devote a section to discuss Chen model in their book; Gibson et al. devote a section to cover Chen model in their review article. Andersen et al. devote a paper to study and extend Chen model. Gallant et al. devote a paper to test Chen model and other models; Cai devotes her PhD dissertation to test Chen model and other competing models (Lin Chen, 1994).


The Rendleman-Bartter model for pricing interest rate options makes the assumption that the probability distribution of an interest rate, a bond price, or some other variable at a future point in time is lognormal. They are widely used for valuing instruments such as caps, European bond option, and European swap option. The Rendleman-Bartter Model is analytically tractable and relatively simple to implement, however this model has limitation as it does not provide a description of how interest rates evolve through time. Consequently, it cannot be used for valuing interest rate derivatives such as American-style swap option, callable bonds, and structured notes. To overcome the limitation of this model, alternative approaches are build ,such as building of what is known as term structure model. This model describing the evolution of all zero-coupon interest rates. In conclusion, each interest rate modeling approach has its own advantages and disadvantages, selecting the suitable model to be used in each interest rate derivatives in very important as it will bring out different outcome.


1) Charlotte Christiansen[2008], Level-ARCH short rate models with regime switching: Bivariate modeling of US and European short rates, International Review of Financial Analysis, 17(5), 925-948

2) Dwight Grant & Gautam Vora[2003], Analytical implementation of the Ho and Lee model for the short interest rate, Global Finance Journal, 14(1), 19-47

3) James J. Kung, Lung-Sheng Lee[2009], Option pricing under the Merton model of the short rate, Mathematics and Computers in Simulation, 80(2), 378-386

4) Robert Faff, Philip Gray[2006], On the estimation and comparison of short-rate models using the generalised method of moments, Journal of Banking & Finance, 30(11), 3131-3146

5) Zhenyu Cui, Don Mcleish[2010], Comment on ‘Option pricing under the Merton model of the short rate’ by Kung and Lee, Mathematics and Computers in Simulation, In Press, Accepted Manuscript, Available online

6) Rosario Dell’Aquila, Elvezio Ronchetti & Fabio Trojani[2003], Robust GMM analysis of models for the short rate process, Journal of Empirical Finance, 10(3), 373-397

7) Garland B. Durham[2003], Likelihood-based specification analysis of continuous-time models of the short-term interest rate, Journal of Financial Economics, 70(3), 463-487

8) Torben G. Andersen & Jesper Lund[1997], Estimating continuous-time stochastic volatility models of the short-term interest rate, Journal of Econometrics, 77(2), 343-377

9) Sotiris K. Staikouras[2006], Testing the stabilization hypothesis in the UK short-term interest rates: Evidence from a GARCH-X model, The Quarterly Review of Economics and Finance, 46(2), 169-189

10) Sandy Suardi[2008], Are levels effects important in out-of-sample performance of short rate models?, Economics Letters, 99(1), 181-184

11) Michael D. Goldberg[2000], On empirical exchange rate models: what does a rejection of the symmetry restriction on short-run interest rates mean?, Journal of International Money and Finance, 19(5), 673-688

12) Haitham A. Al-Zoubi[2009], Short-term spot rate models with nonparametric deterministic drift, The Quarterly Review of Economics and Finance, 49(3), 731-747

13) Turan G. Bali & Liuren Wu[2006], A comprehensive analysis of the short-term interest-rate dynamics, Journal of Banking & Finance, 30(4), 1269-1290

14) Massimo Guidolin, Allan Timmermann[2009], Forecasts of US short-term interest rates: A flexible forecast combination approach, Journal of Econometrics, 150(2), 297-311

15) Rajna Gibson, François-Serge Lhabitant and Denis Talay (2001). Modeling the Term Structure of Interest Rates, The Journal of Risk, 1(3), 37–62

16) Riccardo Rebonato (2002), Modern Pricing of Interest-Rate Derivatives. Princeton University Press, Journal of Financial Economics, 18(2), 133-167

17) Gunther Schnabl (2006), Capital Markets and Exchange Rate Stabilization in East Asia – Diversifying Risk Based on Currency Baskets, Hamburg Institute of International Economics (HWWI),

18) Elroy Dimson, Paul Marsh and Mike Staunton (2002), Global Evidence on the equity risk premium, Journal of Applied Corporate Finance.

19) George J. Hall and Thomas J. Sargent (2010), Interest rate risk and other determinants of post-WWII U.S. Government Debt/GDP dynamics, National Bureau of economic research.

20) Yakov Amihud, Bent Jesper Christensen & Haim Mendelson (1992), Further Evidence on the Risk-Return Relationship ,Research papaer no. 1248.

21) Richard J.Rendleman, Jr, Brit J.Bartter (1980), The pricing of options on debt securities, Journal of Financial and Quantitative Analysis, 15(1), pg 11-24

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Long Title:

How can we use “Rendleman Bartter Model” as one of the short-rate model in valuation of interest rate derivatives?

Short title:

Rendleman Bartter Model






Choo En Leh



Lee Pei Fen


Madam Dr. Devinaga A/P K. Rasiah

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Background research



Literature review



Sub theory and theory






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