An analytic approximation formula for pricing zero-coupon bonds

This paper presents an analytic approximation formula for pricing zero-coupon bonds, when the dynamics of the short-term interest rate are driven by a one-factor mean-reverting process in which changes in the volatility of the interest rate are a function of the level of the interest rate.     

Pricing American interest rate options under the jump-extended constant-elasticity-of-variance short rate models

This paper demonstrates how to value American interest rate options under the jump-extended constant-elasticity-of-variance (CEV) models. We consider both exponential jumps (see Duffie et al., 2000) and lognormal jumps (see Johannes, 2004) in the short rate process. We show how to superimpose recombining multinomial jump trees on the diffusion trees, creating mixed jump-diffusion trees for the CEV models of short rate extended with exponential and lognormal jumps. Our simulations for the special case of jump-extended Cox, Ingersoll, and Ross (CIR) square root model show a significant computational advantage over the Longstaff and Schwartz’s (2001) least-squares regression method (LSM) for pricing American options on zero-coupon bonds.     

A model for interest rates with clustering effects

We propose a model for short-term rates driven by a self-exciting jump process to reproduce the clustering of shocks on the Euro overnight index average (EONIA). The key element of the model is the feedback effect between the absolute value of jumps and the intensity of their arrival process. In this setting, we obtain a closed-form solution for the characteristic function for interest rates and their integral. We introduce a class of equivalent measures under which the features of the process are preserved. We infer the prices of bonds and their dynamics under a risk-neutral measure. The question of derivatives pricing is developed under a forward measure, and a numerical algorithm is proposed to evaluate caplets and floorlets. The model is fitted to EONIA rates from 2004 to 2014 using a peaks-over-threshold procedure. From observation of swap curves over the same period, we filter the evolution of risk premiums for Brownian and jump components. Finally, we analyse the sensitivity of implied caplet volatility to parameters defining the level of self-excitation.     

The nominal duration of TIPS bonds

This paper studies the price responsiveness (effective duration) of U.S. government issued inflation-indexed bonds, known by the acronym TIPS (Treasury Inflation-Protected Securities), to changes in nominal interest rates, real interest rates, and expected inflation. Using the TIPS pricing formula derived by Laatsch and Klein [Q. Rev. Econ. Finance 43 (2002) 405], we first confirm that TIPS bonds have zero sensitivity to changes solely in expected inflation. By changes solely in expected inflation, we mean that the real rate remains unchanged and the nominal rate changes in accordance with the established Fisher [Publ. Am. Econ. Assoc. 11 (1896)] effect. We show that the first derivative of the TIPS price is zero whenever the real rate is held constant. Thus, the first partial derivative of the TIPS bond pricing formula with respect to expected inflation is zero and the first partial derivative of the TIPS bond price with respect to nominal rates is also zero, given, in each case, that we hold the real rate constant. We then temporarily shift the analysis to zero-coupon TIPS bonds and zero-coupon ordinary Treasury bonds. We prove that the nominal duration of zero-coupon TIPS bonds equals that of zero-coupon ordinary Treasury bonds when the real rate changes but expected inflation is held constant.However, if expected inflation changes and the change in the nominal rate does not yield a constant real rate, zero-coupon TIPS prices will change and they will change by a smaller percentage than will zero-coupon ordinary Treasury bonds. We analyze TIPS responsiveness to changes in nominal rates under such conditions. We derive an approximation to effective duration that demonstrates that the effective durations of various maturity zero-coupon TIPS bonds are approximately linear functions in time to maturity of the effective duration of the one-year zero-coupon TIPS bond, ceteris paribus.Nominal effective duration of TIPS bonds is certainly of interest to fixed income portfolio managers that might have a desire to include such bonds in their portfolio. After all, the greater portion of a typical fixed income portfolio is in traditional, noninflation protected bonds whose major risk exposure is to changes in nominal rates. To properly assess the role of TIPS bonds in the portfolio, portfolio managers need information as to how TIPS bonds respond to the changes in nominal rates that are driving the price behavior of the bulk of the portfolio's assets. Prior to concluding the paper, we demonstrate how portfolio managers can calculate the nominal durations of coupon TIPS bonds using the zero-coupon duration formula we derive.     

The analysis and valuation of interest rate options

This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the Black-Scholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general, and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.     

Jump-diffusion volatility models for variance swaps: An empirical performance analysis

This paper studies a class of tractable jump-diffusion models, including stochastic volatility models with various specifications of jump intensity for stock returns and variance processes. We employ the Markov chain Monte Carlo (MCMC) method to implement model estimation, and investigate the performance of all models in capturing the term structure of variance swap rates and fitting the dynamics of stock returns. It is evident that the stochastic volatility models, equipped with self-exciting jumps in the spot variance and linearly-dependent jumps in the central-tendency variance, can produce consistent model estimates, aptly explain the stylized facts in variance swaps, and boost pricing performance. Moreover, our empirical results show that large self-exciting jumps in the spot variance, as an independent risk source, facilitate term structure modeling for variance swaps, whilst the central-tendency variance may jump with small sizes, but signaling substantial regime changes in the long run. Both types of jumps occur infrequently, and are more related to market turmoils over the period from 2008 to 2021.     

Option pricing under stochastic volatility and tempered stable Lévy jumps

The purpose of this paper is to introduce a stochastic volatility model for option pricing that exhibits Lévy jump behavior. For this model, we derive the general formula for a European call option. A well known particular case of this class of models is the Bates model, for which the jumps are modeled by a compound Poisson process with normally distributed jumps. Alternatively, we turn our attention to infinite activity jumps produced by a tempered stable process. Then we empirically compare the estimated log-return probability density and the option prices produced from this model to both the Bates model and the Black–Scholes model. We find that the tempered stable jumps describe more precisely market prices than compound Poisson jumps assumed in the Bates model.     

The Statistical and Economic Role of Jumps in Continuous-Time Interest Rate Models

This paper analyzes the role of jumps in continuous‐time short rate models. I first develop a test to detect jump‐induced misspecification and, using Treasury bill rates, find evidence for the presence of jumps. Second, I specify and estimate a nonparametric jump‐diffusion model. Results indicate that jumps play an important statistical role. Estimates of jump times and sizes indicate that unexpected news about the macroeconomy generates the jumps. Finally, I investigate the pricing implications of jumps. Jumps generally have a minor impact on yields, but they are important for pricing interest rate options.     

Pricing Bonds and Bond Options with Default Risk

The pricing of bonds and bond options with default risk is analysed in the general equilibrium model of Cox, Ingersoll, and Ross (1985). This model is extended by means of an additional parameter in order to deal with financial and credit risk simultaneously. The estimation of such a parameter, which can be considered as the market equivalent of an agencies' bond rating, allows to extract from current quotes the market perceptions of firm's credit risk. The general pricing model for defaultable zero-coupon bond is first derived in a simple discrete-time setting and then in continuous-time. The availability of an integrated model allows for the pricing of default-free options written on defaultable bonds and of vulnerable options written either on default-free bonds or defaultable bonds. A comparison between our results and those given by Jarrow and Turnbull (1995) is also presented.     

Transform analysis for Hawkes processes with applications in dark pool trading

Hawkes processes are a class of simple point processes that are self-exciting and have a clustering effect, with wide applications in finance, social networks and many other fields. This paper considers a self-exciting Hawkes process where the baseline intensity is time-dependent, the exciting function is a general function and the jump sizes of the intensity process are independent and identically distributed nonnegative random variables. This Hawkes model is non-Markovian in general. We obtain closed-form formulas for the Laplace transform, moments and the distribution of the Hawkes process. To illustrate the applications of our results, we use the Hawkes process to model the clustered arrival of trades in a dark pool and analyse various performance metrics including time-to-first-fill, time-to-complete-fill and the expected fill rate of a resting dark order.     

An equilibrium approach to pricing foreign currency options

The paper presents a modified version of the Garman-Kohlhagen formula for pricing European currency options. The equilibrium approach deviates from the no-arbitrage approach by allowing domestic and foreign interest rates and their dynamics to be determined endogenously in the model. By using the relations between exchange rate dynamics and the dynamics of interest rates, I provide a new characterisation of the relevant volatilities for European currency option pricing, which only depends on parameters describing the variability of the log-exchange rate. The implications of the model for the valuation of American currency options and optimal exercise strategies are examined by applying numerical methods.     

A positive interest rate model with sticky barrier

This paper proposes an efficient model for the term structure of interest rates when the interest rate takes very small values. We make the following choices: (i) we model the short-term interest rate, (ii) we assume that once the interest rate reaches zero, it stays there and we have to wait for a random time until the rate is reinitialized to a (possibly random) strictly positive value. This setting ensures that all term rates are strictly positive.

Our objective is to provide a simple method to price zero-coupon bonds. A basic statistical study of the data at hand indeed suggests a switch to a different mode of behaviour when we get to a low level of interest rates. We introduce a variable for the time already spent at 0 (during the last stay) and derive the pricing equation for the bond. We then solve this partial integro-differential equation (PIDE) on its entire domain using a finite difference method (Cranck–Nicholson scheme), a method of characteristics and a fixed point algorithm. Resulting yield curves can exhibit many different shapes, including the S shape observed on the recent Japanese market.     

Pricing and Hedging Discount Bond Options in the Presence of Model Risk

This paper focuses on pricing and hedging options on a zero-coupon bond in a Heath–Jarrow–Morton (1992) framework when the value and/or functional form of forward interest rates volatility is unknown, but is assumed to lie between two fixed values. Due to the link existing between the drift and the diffusion coefficients of the forward rates in the Heath, Jarrow and Morton framework, this is equivalent to hedging and pricing the option when the underlying interest rate model is unknown. We show that a continuous rangeof option prices consistent with no arbitrage exist. This range is bounded by the smallest upper-hedging strategy and the largest lower-hedging strategy prices, which are characterized as the solutions of two non-linear partial differential equations. We also discuss several pricing and hedging illustrations.     

A slightly depressing jump model: intraday volatility pattern simulation

Hawkes processes have been finding more applications in diverse areas of science, engineering and quantitative finance. In multi-frequency finance various phenomena have been observed, such as shocks, crashes, volatility clustering, turbulent flows and contagion. Hawkes processes have been proposed to model those challenging phenomena appearing across asset prices in various exchanges. The original Hawkes process is an intensity-based model for series of events with path dependence and self-exciting or mutual-exciting mechanisms. This paper introduces a slightly depressing process to model the reverse phenomenon of self-exciting mechanisms. Such a process models the decline in the intensity of jumps observed in market regimes. The proposed birth-immigration-death process captures the decline in jump intensity observed at the start of a daily trading regime while the classical immigration-birth process models an increase in jump intensity towards the close of daily trading. Each of these processes can be expressed as a special case of a simple bivariate Hawkes process.     

Equilibrium positive interest rates: a unified view

This article develops precise connections among two generalapproaches to building interest rate models: a general equilibriumapproach using a pricing kernel and the Heath, Jarrow, and Mortonframework based on specifying forward rate volatilities andthe market price of risk. The connections exploit the observationthat a pricing kernel is uniquely determined by its drift. Throughthese connections we provide, for any arbitrage-free term structuremodel, a representative-consumer real production economy supportingthat term structure model in equilibrium. We put particularemphasis on models in which interest rates remain positive.By modeling the dynamics of the drift of the pricing kernel,we construct a new family of Markovian-positive interest ratemodels.     

Hedging jump risk,expected returns and risk premia in jump-diffusion economies

This article presents a pure exchange economy that extends Rubinstein [Bell J. Econ. Manage. Sci., 1976, 7, 407–425] to show how the jump-diffusion option pricing model of Black and Scholes [J. Political Econ., 1973, 81, 637–654] and Merton [J. Financ. Econ., 1976, 4, 125–144] evolves in gamma jumping economies. From empirical analysis and theoretical study, both the aggregate consumption and the stock price are unknown in determining jumping times. By using the pricing kernel, we determine both the aggregate consumption jump time and the stock price jump time from the equilibrium interest rate and CCAPM (Consumption Capital Asset Pricing Model). Our general jump-diffusion option pricing model gives an explicit formula for how the jump process and the jump times alter the pricing. This innovation with predictable jump times enhances our analysis of the expected stock return in equilibrium and of hedging jump risks for jump-diffusion economies.     

An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston [(1993) The Review of Financial studies, 6, 327–343], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.     

A self-exciting switching jump diffusion: properties,calibration and hitting time

A way to model the clustering of jumps in asset prices consists in combining a diffusion process with a jump Hawkes process in the dynamics of the asset prices. This article proposes a new alternative model based on regime switching processes, referred to as a self-exciting switching jump diffusion (SESJD) model. In this model, jumps in the asset prices are synchronized with changes of states of a hidden Markov chain. The matrix of transition probabilities of this chain is designed in order to approximate the dynamics of a Hawkes process. This model presents several advantages compared to other jump clustering models. Firstly, the SESJD model is easy to fit to time series since estimation can be performed with an enhanced Hamilton filter. Secondly, the model explains various forms of option volatility smiles. Thirdly, several properties about the hitting times of the SESJD model can be inferred by using a fluid embedding technique, which leads to closed form expressions for some financial derivatives, like perpetual binary options.     

Pricing average options under time-changed Lévy processes

This paper presents an approximate formula for pricing average options when the underlying asset price is driven by time-changed Lévy processes. Time-changed Lévy processes are attractive to use for a driving factor of underlying prices because the processes provide a flexible framework for generating jumps, capturing stochastic volatility as the random time change, and introducing the leverage effect. There have been very few studies dealing with pricing problems of exotic derivatives on time-changed Lévy processes in contrast to standard European derivatives. Our pricing formula is based on the Gram–Charlier expansion and the key of the formula is to find analytic treatments for computing the moments of the normalized average asset price. In numerical examples, we demonstrate that our formula give accurate values of average call options when adopting Heston’s stochastic volatility model, VG-CIR, and NIG-CIR models.     

A generalization of reset options pricing formulae with stochastic interest rates

A generalization of reset call options with predetermined dates is derived in the case of time-dependent volatility and time-dependent interest rate by applying martingale method and change of nume?aire or change of probability measure. An analytical pricing formula for the reset call option is also obtained when the interest rate follows an extended Vasicek’s model. Numerical results show that the correlated coefficient between the stock price and interest rate is almost unacted on the price of reset call option with short maturity and Monte Carlo method is inefficient. Monte Carlo method should be only used if there is no closed-formed solution for option pricing.