Pricing and hedging bond options and sinking-fund bonds under the CIR model

<abstract><p>This article derives simple closed-form solutions for computing Greeks of zero-coupon and coupon-bearing bond options under the CIR interest rate model, which are shown to be accurate, easy to implement, and computationally highly efficient. These novel analytical solutions allow us to extend the literature in two other directions. First, the static hedging portfolio approach is used for pricing and hedging American-style plain-vanilla zero-coupon bond options under the CIR model. Second, we derive analytically the comparative static properties of sinking-fund bonds under the same interest rate modeling setup.</p></abstract>

Lévy Interest Rate Models with a Long Memory

This article proposes an interest rate model ruled by mean reverting Lévy processes with a sub-exponential memory of their sample path. This feature is achieved by considering an Ornstein–Uhlenbeck process in which the exponential decaying kernel is replaced by a Mittag–Leffler function. Based on a representation in term of an infinite dimensional Markov processes, we present the main characteristics of bonds and short-term rates in this setting. Their dynamics under risk neutral and forward measures are studied. Finally, bond options are valued with a discretization scheme and a discrete Fourier’s transform.

Site-Selective Allylic C-H Amination with Aliphatic Amines

The direct coupling of olefins and alkyl amines represents the most efficient and atom-economical approach to prepare aliphatic allylamines which are fundamental building blocks. However, the method that achieves this goal while exhibiting exquisite control over the site at which the amine is introduced remains elusive. Herein, we report that the combination of a photocatalyst and a cobaloxime enables site-selective allylic C–H amination of olefins with secondary alkyl amines to afford allylic amines, eliminating the need for oxidants. This reaction proceeds by a radical-based mechanism distinct from those of existing allylic amination reactions. It affords the product resulting from cleavage of the stronger, primary allylic C–H bonds over other weaker allylic C–H bond options. DFT calculations reveal that this selectivity originates from a cobaloxime-promoted hydrogen atom transfer (HAT) process. Our method is compatible with a broad scope of alkenes, and can be extended to achieve a site- and diastereoselective amination of natural terpenes.

On the Convergence of a Crank–Nicolson Fitted Finite Volume Method for Pricing American Bond Options

This paper develops and analyses a Crank–Nicolson fitted finite volume method to price American options on a zero-coupon bond under the Cox–Ingersoll–Ross (CIR) model governed by a partial differential complementarity problem (PDCP). Based on a penalty approach, the PDCP results in a nonlinear partial differential equation (PDE). We then apply a fitted finite volume method for the spatial discretization along with a Crank–Nicolson time-stepping scheme for the PDE, which results in a nonlinear algebraic equation. We show that this scheme is consistent, stable, and monotone, and hence, the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. To solve the system of nonlinear equations effectively, an iterative algorithm is established and its convergence is proved. Numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the new numerical method.

Near Exact Calculation of American Options

A new practical approach for the analysis of American (bond) options is developed which is a combination of the closed form solutions and binomial lattice models. The model is calibrated to the observed term structure of rates and traded volatilities and is arbitrage free. The convergence is very fast, but numerically intensive. By extrapolation the near exact premium of an American (bond) option can be calculated.

Martingale Models for the Short Rate

This chapter is devoted to an overview and analysis of the most common short rate models, such as the Vasiček, Dothan, Hull–White, and CIR models. These models are analyzed and classified from the point of view of positive short rates, normal distribution, mean reversion, and computability. In particular we present the theory of affine term structures, and discuss the inversion of the yield curve. Analytical results for bond prices and bond options are presented for all the affine models.

Valuing and Analyzing Fixed Income Derivatives

Derivatives valuation is based on the key principle of no-arbitrage pricing. This chapter presents valuation models for various types of fixed income derivatives, including forward rate agreements (FRAs), interest rate swaps, Eurodollar and Treasury bond futures, bond options, caps and floors, swaptions, and options on interest rate futures. Following the financial crisis that began in the summer of 2007, major changes occurred in the practice of fixed income derivatives valuation, particularly regarding the adoption of overnight indexed swaps (OIS) as a source of the risk-free rate. This chapter shows how OIS discounting is implemented in FRA pricing and swap valuation. Traditional approaches such as cost of carry valuation in futures pricing are illustrated. With respect to option valuation, this chapter explains the risk-neutral pricing approach as well as closed-form solutions such as the Black model. The chapter also provides numeric examples to illustrate the practical use of the presented models and formulas.