PRICING SWAPTIONS AND COUPON BOND OPTIONS IN AFFINE TERM STRUCTURE MODELS

We propose an approach to find an approximate price of a swaption in affine term structure models. Our approach is based on the derivation of approximate swap rate dynamics in which the volatility of the forward swap rate is itself an affine function of the factors. Hence, we remain in the affine framework and well-known results on transforms and transform inversion can be used to obtain swaption prices in similar fashion to zero bond options (i.e., caplets). The method can easily be generalized to price options on coupon bonds. Computational times compare favorably with other approximation methods. Numerical results on the quality of the approximation are excellent. Our results show that in affine models, analogously to the LIBOR market model, LIBOR and swap rates are driven by approximately the same type of (in this case affine) dynamics.     

Pricing executive stock options with averaging features under the Heston–Nandi GARCH model

In this paper, we focus on the pricing issue of four types of executive stock options (ESOs) in the Heston–Nandi generalized autoregressive conditional heteroskedasticity option pricing model. Based on the derived benchmark strike prices in the proposed framework, we obtain the closed-form pricing formulae for four types of ESOs. In the numerical part, we investigate the sensitivity and cost efficiency of ESOs and suggest that systematic risk (stock β) and the fraction of wealth invested in restricted stock could impede the cost efficiency of ESOs.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

Black's Model of Interest Rates as Options, Eigenfunction Expansions and Japanese Interest Rates

Black's (1995) model of interest rates as options assumes that there is a shadow instantaneous interest rate that can become negative, while the nominal instantaneous interest rate is a positive part of the shadow rate due to the option to convert to currency. As a result of this currency option, all term rates are strictly positive. A similar model was independently discussed by Rogers (1995) . When the shadow rate is modeled as a diffusion, we interpret the zero-coupon bond as a Laplace transform of the area functional of the underlying shadow rate diffusion (evaluated at the unit value of the transform parameter). Using the method of eigenfunction expansions, we derive analytical solutions for zero-coupon bonds and bond options under the Vasicek and shifted CIR processes for the shadow rate. This class of models can be used to model low interest rate regimes. As an illustration, we calibrate the model with the Vasicek shadow rate to the Japanese Government Bond data and show that the model provides an excellent fit to the Japanese term structure. The current implied value of the instantaneous shadow rate in Japan is negative.     

MultiFactor Valuation of Floating Range Notes

Under a one-factor Gaussian Heath-Jarrow-Morton model, Turnbull (1995) as well as Navatte and Quittard-Pinon (1999) have provided explicit pricing solutions for range notes contracts. The present paper generalizes such closed-form solutions for the context of a multifactor Gaussian HJM framework.     

THE AFFINE LIBOR MODELS

We provide a general and flexible approach to LIBOR modeling based on the class of affine factor processes. Our approach respects the basic economic requirement that LIBOR rates are nonnegative, and the basic requirement from mathematical finance that LIBOR rates are analytically tractable martingales with respect to their own forward measure. Additionally, and most importantly, our approach also leads to analytically tractable expressions of multi‐LIBOR payoffs. This approach unifies therefore the advantages of well‐known forward price models with those of classical LIBOR rate models. Several examples are added and prototypical volatility smiles are shown. We believe that the CIR process‐based LIBOR model might be of particular interest for applications, since closed form valuation formulas for caps and swaptions are derived.     

LINEAR-QUADRATIC JUMP-DIFFUSION MODELING

We aim at accommodating the existing affine jump-diffusion and quadratic models under the same roof, namely the linear-quadratic jump-diffusion (LQJD) class. We give a complete characterization of the dynamics of this class by stating explicitly the structural constraints, as well as the admissibility conditions. This allows us to carry out a specification analysis for the three-factor LQJD models. We compute the standard transform of the state vector relevant to asset pricing up to a system of ordinary differential equations. We show that the LQJD class can be embedded into the affine class using an augmented state vector. This establishes a one-to-one equivalence relationship between both classes in terms of transform analysis.     

The Market Model of Interest Rate Dynamics

A class of term structure models with volatility of lognormal type is analyzed in the general HJM framework. The corresponding market forward rates do not explode, and are positive and mean reverting. Pricing of caps and floors is consistent with the Black formulas used in the market. Swaptions are priced with closed formulas that reduce (with an extra assumption) to exactly the Black swaption formulas when yield and volatility are flat. A two–factor version of the model is calibrated to the U.K. market price of caps and swaptions and to the historically estimated correlation between the forward rates.     

ISOLATING THE WILD CARD OPTION

Many embedded options are difficult to value the wild card option in the Treasury bond futures contract is one of these embedded options. We illustrate how narrow theoretical bounds on the value of this option, relative to the price of the contract, may be obtained in the presence of other embedded options. Simulations suggest that the value of the wild card option is close to zero. This implies that, in this economy, a simpler pricing model of the Treasury bond futures contract, which ignores the wild card option, will result in only a small loss of accuracy.     

GENERALIZATION OF THE DYBVIG–INGERSOLL–ROSS THEOREM AND ASYMPTOTIC MINIMALITY

The long‐term limit of zero‐coupon rates with respect to the maturity does not always exist. In this case we use the limit superior and prove corresponding versions of the Dybvig–Ingersoll–Ross theorem, which says that long‐term spot and forward rates can never fall in an arbitrage‐free model. Extensions of popular interest rate models needing this generalization are presented. In addition, we discuss several definitions of arbitrage, prove asymptotic minimality of the limit superior of the spot rates, and illustrate our results by several continuous‐time short‐rate models.     

Characterizing Gaussian Models of the Term Structure of Interest Rates

Models of the term structure of interest rates are considered for which, under the martingale measure, instantaneous forward rates are Gaussian. The possible forms of the covariance structure are characterized under appropriate formulations of the Markov property. It is demonstrated that imposing Markovian assumptions limits severely the covariances that may be obtained and that the strongest such formulation together with stationarity implies that the whole forward rate surface is necessarily a Gaussian random field described by just three parameters.     

LIFTING QUADRATIC TERM STRUCTURE MODELS TO INFINITE DIMENSION

We introduce an infinite dimensional generalization of quadratic term structure models of interest rates, aiming that the lift will give us a deeper understanding of the classical models. We show that it preserves some of the favorable properties of the classical quadratic models.     

BILINEAR TERM STRUCTURE MODEL

The Gaussian Affine Term Structure Model (ATSM) introduced by Duffie and Kan is often used in finance to price derivatives written on interest rates or to compute the reserve to hedge a portfolio of credits (CreditVaR), and in macroeconomic applications to study the links between real activity and financial variables. However, a standard three‐factor ATSM, for instance, implies a deterministic affine relationship between any set of four rates, with different times‐to‐maturity, and these relationships are not observed in practice. In this paper, we introduce a new class of affine term structure models, called Bilinear Term Structure Model (BTSM). This extension breaks down the deterministic relationships between rates in structural factor models by introducing lagged factor values, and the linear dependence by considering quadratic effects of the factors.