On the Deterministic-Shift Extended CIR Model in a Negative Interest Rate Framework

In the framework of a multifactor LIBOR market model (LMM), this paper presents a new approach for finding an approximate distribution of constant maturity swap (CMS) rates under the terminal martingale measure. With this approach, we derive an analytical pricing formula for CMS range notes, which is both intuitive and tractable. Many exotic CMS rate derivatives are widely traded in the marketplace or embedded in structure notes.

Valuation of Swaps and Options on Constant Maturity CDS Spreads

A Unified Market Model for Swaptions and Constant Maturity Swaps

A Review of CMS Swap Pricing Approaches

In this work, we present our findings of the so‐called CIR#, which is a modified version of the Cox, Ingersoll, and Ross (CIR) model, turned into a forecasting tool for any term structure. The main feature of the CIR# model is its ability to cope with negative interest rates, cluster volatility, and jumps. By considering a dataset composed of money market interest rates during turmoil and calmer periods, we show how the CIR# performs in terms of directionality of rates and forecasting error. Comparison is carried out with a revamped version of the CIR model (denoted ), the Hull and White model, and the exponentially weighted moving average (EWMA) which is often adopted whenever no structure in data is assumed. To confirm the analysis, testing and validation is performed on both historical and ad hoc data with different metrics and clustering criteria.

The Cox–Ingersoll–Ross (CIR) model and the Vasicek model are two well-known single factor models of the interest spot rate. In this paper, we construct a mapping by means of which the price of a zero-coupon bond in the CIR model may be obtained from a corresponding price in the Vasicek model. We use symmetry analysis to construct this mapping and verify it by transforming three arbitrary solutions of the pricing equation in the Vasicek model into solutions of the corresponding equation in the CIR model. Copyright © 2010 John Wiley & Sons, Ltd.

Futures-Style Options on Euro-Deposit Futures: Nihil Sub Sole Novi?

A lookback option is a path-dependent option, offering a payoff that depends on the maximum or minimum value of the underlying asset price over the life of the option. This paper presents a new mean-reverting uncertain stock model with a floating interest rate to study the lookback option price, in which the processing of the interest rate is assumed to be the uncertain counterpart of the Cox–Ingersoll–Ross (CIR) model. The CIR model can reflect the fluctuations in the interest rate and ensure that such rate is positive. Subsequently, lookback option pricing formulas are derived through the α-path method and some mathematical properties of the uncertain option pricing formulas are discussed. In addition, several numerical examples are given to illustrate the effectiveness of the proposed model.

Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate products, there is an inherent “tug of war” between the pursuit of calibration and pricing precision, tractability, and implementation efficiency. After reviewing the main ideas and implementation techniques underlying both single- and multi-factor models, we offer our own approach based on dimension reduction via Markovian projection. Specifically, on the theoretical side, we provide a reinterpretation and extension of the classic result due to Gyöngy which covers non-probabilistic, discounted, distributions relevant in option pricing. Thus, we show that for purposes of swaption pricing, a potentially complex and multidimensional process for the underlying swap rate can be collapsed to a one-dimensional one. The empirical contribution of the paper consists in demonstrating that even though we only match the marginal distributions of the two processes, Bermudan swaptions prices calculated using such an approach appear well-behaved and closely aligned to counterparts from more sophisticated models.

We test the theory of the term structure of indexed-bond prices due to Cox, Ingersoll, and Ross (CIR). The econometric method uses Hansen's generalized method of moments and exploits the probability distribution of the single-state variable in CIR's model, thus avoiding the use of aggregate consumption data. It enables us to estimate a continuous-time model based on discretely sampled data. The tests indicate that CIR's model for index bonds performs reasonably well when confronted with short-term Treasury-bill returns. The estimates indicate that term premiums are positive and that yield curves can take several shapes. However, the fitted model does poorly in explaining the serial correlation in real Treasury-bill returns.

We consider the exact path sampling of the squared Bessel process and other continuous-time Markov processes, such as the Cox–Ingersoll–Ross model, constant elasticity of variance diffusion model, and confluent hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale transformation, and change of measure. All these diffusions are broadly used in mathematical finance for modeling asset prices, market indices, and interest rates. We show how the probability distributions of a squared Bessel bridge and a squared Bessel process with or without absorption at zero are reduced to randomized gamma distributions. Moreover, for absorbing stochastic processes, we develop a new bridge sampling technique based on conditioning on the first hitting time at the boundary of the state space. Such an approach allows us to simplify simulation schemes. New methods are illustrated with pricing path-dependent options.

Using interest rate derivative market prices, this paper derives the term structure of the LIBOR-overnight index swap (OIS) spread, which is considered as the funding liquidity risk premium, following the Cox–Ingersoll–Ross model. The probability density functions of the LIBOR-OIS spread constructed from its dynamics were fat-tailed distributions during the crisis of 2008 and the tails further extended after the Lehman failure reflecting deepened uncertainty about the funding liquidity risk. The dynamics provides information to estimate the probability of systemic funding liquidity shocks using a first-passage-time approach. The probability deviated from zero on 18 September 2008 to a material level that provided an early warning signal of the aggregate liquidity shock on 29 September 2008 when the interbank market was totally paralysed.

In the models of the two preceding chapters, the derived prices and yields are functions of the state variables and the few parameters entering the dynamics of the state variables. Consequently, the resulting term structure of interest rates will typically not fit the currently observed term structure perfectly. This seems inappropriate when the models are to be applied to the pricing of derivative securities — if the model cannot price the underlying bonds correctly, why trust the model prices of derivatives? This chapter shows how to calibrate the model to the observed yield curve by replacing one of the constant parameters with a carefully chosen time-dependent. Special attention is given to the one-factor Vasicek and Cox–Ingersoll–Ross models. The models can also be calibrated to additional market information such as the term structure of interest rate volatilities by replacing other parameters with time-dependent functions. However, a direct dependence on calendar time in the drift and volatility of the short rate or other state variables is unrealistic, and care has to be taken not to introduce unrealistic time dependencies that are likely to affect the prices of the derivative securities.

This chapter aims presents the deep relationships between the Cox-Ingersoll-Ross (CIR) type-processes, the Squared Bessel (BESQ) processes and the family of affine processes, according to specific dynamics for the dividend structure behind the market scenarios, aiming at deriving pricing formulas in individual markets as well as analytical solvable or numerical tractable, schemes for dividend processes in volatility stabilized markets.

In this paper, an effective numerical method is proposed for a linear complementarity problem (LCP) arising in the valuation of American bond options under the Cox–Ingersoll–Ross (CIR) model. Firstly, a variable substitution is used to simplify the linear complementary model. Secondly, the finite difference method is adopted to discretize the simplified model, and an equivalent variational form is obtained. Based on the positive definiteness of the discretized matrix, a projection and contraction method (PCM) is adopted for the resulting discretized variational problem. Finally, numerical experiments highlight the effectiveness and performance of the proposed algorithm.