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We examine the shapes attainable by the forward- and yield-curve in the widely-used Svensson family, including the Nelson-Siegel and Bliss subfamilies. We provide a complete classification of all attainable shapes and partition the parameter space of each family according to these shapes. Building upon these results, we then examine the consistent dynamic evolution of the Svensson family under absence of arbitrage. Our analysis shows that consistent dynamics further restrict the set of attainable shapes, and we demonstrate that certain complex shapes can no longer appear after a deterministic time horizon. Moreover a single shape (either inverse of normal curves) must dominate in the long-run.

The paper is devoted to the study of the short rate equation of the form $$ dR(t)=F(R(t)) dt +\sum_{i=1}^{d}G(R(t-))dZ_i(t)$$ with deterministic functions $F,G_1,...,G_d$ and a multivariate L\'evy process $Z=(Z_1,...,Z_d)$ with possibly dependent coordinates. The equation is supposed to have a nonnegative solution which generates an affine term structure model. The L\'evy measure $\nu$ of $Z$ is assumed to admit a spherical decomposition based on the representation $\mathbb{R}^d=S^{d-1}\times (0,+\infty)$, where $S^{d-1}$ stands for the unit sphere. Then $\nu(dy)=\lambda(d\xi)\times \gamma_{\xi}(dr)$, where $\lambda$ is a measure on $S^{d-1}$ and $\gamma_{\xi}$ on $(0,+\infty)$. Under some assumptions on spherical decomposition, a precise form of the generator of $R$ is determined and it is shown that the resulted term structure model is identical to that generated by the equation $$ d R(t)=(a R(t)+b) dt+C\cdot (R(t-))^{1/\alpha} dZ^{\alpha}(t), \quad R(0)=x, $$ with some constants $a,b,C$ and a one dimensional $\alpha$-stable L\'evy process $Z^{\alpha}$, where $\alpha\in(1,2)$. The case when $\nu$ has a density is considered as a special case. The paper generalizes the classical results on the Cox-Ingersoll-Ross (CIR) model, \cite{CIR}, as well as on its extended version from \cite{BarskiZabczykCIR} and \cite{BarskiZabczyk} where $Z$ is a one-dimensional L\'evy process. It is the starting point for the classification in the spirit of \cite{DaiSingleton} and \cite{BarskiLochowski} for the affine models with dependent L\'evy processes., Comment: arXiv admin note: text overlap with arXiv:2204.07245

We characterize affine term structure models of non-negative short rate $R$ which may be obtained as solutions of autonomous SDEs driven by independent, one-dimensional L\'evy martingales, that is equations of the form $$ dR(r)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=r_0\geq 0, \quad t>0, \quad (1)$$ with deterministic real functions $F,G_1,...,G_d$ and independent one-dimensional L\'evy martingales $Z_1,...,Z_d$. Using a general result on the form of the generators of affine term structure models due to Filipovi\'c, it is shown, under the assumption that the Laplace transforms of the driving noises are regularly varying, that all possible solutions $R$ of (1) may be obtained also as solutions of autonomous SDEs driven by independent stable processes with stability indices in the range $(1,2]$. The obtained models include in particular the $\alpha$-CIR model, introduced by Jiao et al., which proved to be still simple yet more reliable than the classical CIR model. Results on heavy tails of $R$ and its limit distribution in terms of the stability indices are proven. Finally, results of numerical calibration of the obtained models to the market term structure of interest rates are presented and compared with the CIR and $\alpha$-CIR models., Comment: arXiv admin note: substantial text overlap with arXiv:2303.08477, arXiv:2204.07245

The collateral choice option allows a collateral-posting party the opportunity to change the type of security in which the collateral is deposited. Due to non-zero collateral basis spreads, this optionality significantly impacts asset valuation. Because of the complexity of valuing the option, many practitioners resort to deterministic assumptions on the collateral rates. In this article, we focus on a valuation model of the collateral choice option based on stochastic dynamics. Intrinsic differences in the resulting collateral choice option valuation and its implications for collateral management are presented. We obtain sensitivities of the collateral choice option price under both the deterministic and the stochastic model, and we show that the stochastic model attributes risks to all involved collateral currencies. Besides an inability to capture volatility effects, the deterministic model exhibits a digital structure in which only the cheapest-to-deliver currency influences the valuation at a given time. We further consider hedging an asset with the collateral choice option by a portfolio of domestic and foreign zero-coupon bonds that do not carry the collateral choice option. We propose static hedging strategies based on the crossing times of the deterministic model and based on variance-minimization under the stochastic model. We show how the weights of this model can be explicitly determined with the semi-analytical common factor approach and we show in numerical experiments that this strategy offers good hedging performance under minimized variance.

We consider a method of lines (MOL) approach to determine prices of European and American exchange options when underlying asset prices are modelled with stochastic volatility and jump-diffusion dynamics. As the MOL, as with any other numerical scheme for PDEs, becomes increasingly complex when higher dimensions are involved, we first simplify the problem by transforming the exchange option into a call option written on the ratio of the yield processes of the two assets. This is achieved by taking the second asset yield process as the numeraire. We also characterize the near-maturity behavior of the early exercise boundary of the American exchange option and analyze how model parameters affect this behavior. Using the MOL scheme, we conduct a numerical comparative static analysis of exchange option prices with respect to the model parameters and investigate the impact of stochastic volatility and jumps to option prices. We also consider the effect of boundary conditions at far-but-finite limits of the computational domain on the overall efficiency of the MOL scheme. Toward these objectives, a brief exposition of the MOL and how it can be implemented on computing software are provided., Comment: 38 pages with 19 figures. To appear in Quantitative Finance. Sections 2 to 4 of this paper present a condensed version of similar arguments made in our earlier arXiv preprint (arXiv:2002.10194). arXiv admin note: text overlap with arXiv:2002.10194

The collateral choice option gives the collateral posting party the opportunity to switch between different collateral currencies which is well-known to impact the asset price. Quantification of the option's value is of practical importance but remains challenging under the assumption of stochastic rates, as it is determined by an intractable distribution which requires involved approximations. Indeed, many practitioners still rely on deterministic spreads between the rates for valuation. We develop a scalable and stable stochastic model of the collateral spreads under the assumption of conditional independence. This allows for a common factor approximation which admits analytical results from which further estimators are obtained. We show that in modelling the spreads between collateral rates, a second order model yields accurate results for the value of the collateral choice option. The model remains precise for a wide range of model parameters and is numerically efficient even for a large number of collateral currencies., Comment: 25 pages, 5 figures, 4 tables

In this paper, we introduce a model that adds a non-linearity to discounting: the discounting factor may depend on the notional (i.e., discounted values are no longer linear in the notional). In the first part of the paper, we provide a discounting when discount factors cannot be derived from market products. That is, a risk-neutralising trading strategy cannot be performed. This is the case when one needs a risk-free (default-free) discounting, but default protection on funding providers is not traded. For this case, we derive a default compensation factor that describes the present value of a strategy to compensate for default (like buying default protection would do). In a second part of the paper, we introduce a model where the survival probability, and hence the discount factor, depends on the notional. This model introduces an effect not present in the classical modelling of a time-dependent survival probability. Our model allows that large liquidity requirements are more likely to default instantly than small ones. Combined, the two models build a framework where discounting (and hence valuation) is non-linear: discount factors depend on the amount to be discounted. Our approach builds on top of the classical theory of discounting (which may either be given as market-implied or be derived from a model of utility, consumption and production). In that sense, it is rather a generalisation than an alternative. The modelling approach has specific relevance for climate models, where discounting is an important aspect in assessing the severity of future events. Our model may result in non-decaying discount factors (negative discount rates) for certain scenarios., Comment: 40 pages, 8 figures

In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton (1976), Heston (1993), and Bates (1996). A Radon-Nikodym derivative process is also introduced to facilitate the shift from the objective market measure to other equivalent probability measures, including the equivalent martingale measure. Under the equivalent martingale measure, we derive the integro-partial differential equation that characterizes the exchange option prices. We also derive representations of the European exchange option price using the change-of-numeraire technique proposed by Geman et al. (1995) and the Fourier inversion formula derived by Caldana and Fusai (2013), and show that these two representations are comparable. Lastly, we show that the American exchange option price can be decomposed into the price of the European exchange option and an early exercise premium., Comment: 45 pages, 1 figure

We price European and American exchange options where the underlying asset prices are modelled using a Merton (1976) jump-diffusion with a common Heston (1993) stochastic volatility process. Pricing is performed under an equivalent martingale measure obtained by setting the second asset yield process as the numeraire asset, as suggested by Bjerskund and Stensland (1993). Such a choice for the numeraire reduces the exchange option pricing problem, a two-dimensional problem, to pricing a call option written on the ratio of the yield processes of the two assets, a one-dimensional problem. The joint transition density function of the asset yield ratio process and the instantaneous variance process is then determined from the corresponding Kolmogorov backward equation via integral transforms. We then determine integral representations for the European exchange option price and the early exercise premium and state a linked system of integral equations that characterizes the American exchange option price and the associated early exercise boundary. Properties of the early exercise boundary near maturity are also discussed., Comment: 42 pages, 2 figures, 1 table

In this paper, we study term structure movements in the spirit of Heath, Jarrow, and Morton [Econometrica 60(1), 77-105] under volatility uncertainty. We model the instantaneous forward rate as a diffusion process driven by a G-Brownian motion. The G-Brownian motion represents the uncertainty about the volatility. Within this framework, we derive a sufficient condition for the absence of arbitrage, known as the drift condition. In contrast to the traditional model, the drift condition consists of several equations and several market prices, termed market price of risk and market prices of uncertainty, respectively. The drift condition is still consistent with the classical one if there is no volatility uncertainty. Similar to the traditional model, the risk-neutral dynamics of the forward rate are completely determined by its diffusion term. The drift condition allows to construct arbitrage-free term structure models that are completely robust with respect to the volatility. In particular, we obtain robust versions of classical term structure models., Comment: rewrote/restructured the paper while main results remain unchanged

Yield curve modeling is an essential problem in finance. In this work, we explore the use of Bayesian statistical methods in conjunction with Nelson-Siegel model. We present the hierarchical Bayesian model for the parameters of the Nelson-Siegel yield function. We implement the MAP estimates via BFGS algorithm in rstan. The Bayesian analysis relies on the Monte Carlo simulation method. We perform the Hamiltonian Monte Carlo (HMC), using the rstan package. As a by-product of the HMC, we can simulate the Monte Carlo price of a Bond, and it helps us to identify if the bond is over-valued or under-valued. We demonstrate the process with an experiment and US Treasury's yield curve data. One of the interesting observation of the experiment is that there is a strong negative correlation between the price and long-term effect of yield. However, the relationship between the short-term interest rate effect and the value of the bond is weakly positive. This is because posterior analysis shows that the short-term effect and the long-term effect are negatively correlated., Comment: Book chapter; 14 pages, 5 tables, 7 figures

We study the Hull-White model for the term structure of interest rates in the presence of volatility uncertainty. The uncertainty about the volatility is represented by a set of beliefs, which naturally leads to a sublinear expectation and a G-Brownian motion. The main question in this setting is how to find an arbitrage-free term structure. This question is crucial, since we can show that the classical approach, martingale modeling, does not work in the presence of volatility uncertainty. Therefore, we need to adjust the model in order to find an arbitrage-free term structure. The resulting term structure is affine with respect to the short rate and the adjustment factor. Although the adjustment changes the structure of the model, it is still consistent with the traditional Hull-White model after fitting the yield curve. In addition, we extend the model and the results to a multifactor version, driven by multiple risk factors., Comment: rewrote/restructured the paper while main results remain unchanged; left out the part about drift uncertainty; included a section about yield curve fitting and a multifactor extension

Nelson and Siegel curves are widely used to fit the observed term structure of interest rates in a particular date. By the other hand, several interest rate models have been developed such their initial forward rate curve can be adjusted to any observed data, as the Ho-Lee and the Hull and White one factor models. In this work we study the evolution of the forward curve process for each of this models assuming that the initial curve is of Nelson-Siegel type. We conclude that the forward curve process produces curves belonging to a parametric family of curves that can be seen as extended Nelson and Siegel curves., Comment: 13 pages

We model bond's price curves corresponding to the sovereign uruguayan debt nominated in USD, as an alternative to the official bond prices publication released by the Central Bank of Uruguay (CBU). Four different gaussian models are fitted, based on historical data issued by the CBU, corresponding to some of the more frequently traded bonds. The main difficulty we approach is the absence of liquidity in the bond market. Nevertheless the adjustment is relatively good, giving the possibility of non-arbitrage pricing of the whole family of non traded instruments, and also the possibility of pricing derivative securities., Comment: 18 pages, 9 figures

In the LIBOR market model, forward interest rates are log-normal under their respective forward measures. This note shows that their distributions under the other forward measures of the tenor structure have approximately log-normal tails.