The asymptotic expansion of the regular discretization error of Itô integrals

We study an Edgeworth‐type refinement of the central limit theorem for the discretization error of Itô integrals. Toward this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula. Two applications to finance are given; the asymptotics of discrete hedging error under the Black–Scholes model and the difference between continuously and discretely monitored variance swap payoffs under stochastic volatility models.     

The dynamics of long forward rate term structures

In this article, we look at study the dynamics of forward rates with maturities longer than 14 years. We re‐document the phenomenon of the downward sloping long forward rate term structure using U.S. Treasury STRIPS data over the period 1988 to 2007. By calibrating Diebold F. X. and Li C.‐L.'s ( 2006 ) dynamic Nelson C. R. and Siegel A. F. ( 1987 ) and Christensen J. H. E., Diebold F. X., and Rudebusch G. D.'s ( 2007 ) arbitrage‐free Nelson‐Siegel models, we find that both models explain the empirical phenomenon very well. Out‐of‐sample comparison shows that imposing no‐arbitrage restriction indeed improves the forecasting performance. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 30:957–982, 2010     

Interest Rate Dynamics and Consistent Forward Rate Curves

We consider as given an arbitrage‐free interest rate model M, and a parametrized family of forward rate curves G. We study the question as to when the given family G is consistent with the dynamics of the interest rate model M, in the sense that M actually will produce forward rate curves belonging to G. We allow the interest rate model to be driven by a multidimensional Wiener process, as well as by a marked point process, and we give necessary and sufficient conditions for consistency. As test cases, we study some popular models, obtaining both positive and negative results about consistency. We also introduce a natural exponential‐polynomial family of forward rate curves, and for this family we give necessary and sufficient conditions for the existence of consistent interest rate models with deterministic volatility functions.     

Linkages in the term structure of interest rates across sovereign bond markets

This paper investigates the linkages in the sovereign bond yields across different maturity spectrums among both developed and Asian countries. Term structure of interest rate is estimated using the Dynamic Nelson Siegel model and Kalman filter. The degrees of integration and transmission of shocks from one country to another are measured using forecast error variance decomposition in the generalized vector autoregression (VAR) model. The level factor showed higher spillover index across the countries. Regional influence is found to be higher in slope and curvature factors among the Asian countries. The linkages are high during periods of crisis.     

Short Rate Dynamics and Regime Shifts*

We characterize the dynamics of the US short‐term interest rate using a Markov regime‐switching model. Using a test developed by Garcia, we show that there are two regimes in the data: In one regime, the short rate behaves like a random walk with low volatility; in another regime, it exhibits strong mean reversion and high volatility. In our model, the sensitivity of interest rate volatility to the level of interest rate is much lower than what is commonly found in the literature. We also show that the findings of nonlinear drift in Aït‐Sahalia and Stanton, using nonparametric methods, are consistent with our regime‐switching model.     

THE RANGE OF TRADED OPTION PRICES

Suppose we are given a set of prices of European call options over a finite range of strike prices and exercise times, written on a financial asset with deterministic dividends which is traded in a frictionless market with no interest rate volatility. We ask: when is there an arbitrage opportunity? We give conditions for the prices to be consistent with an arbitrage-free model (in which case the model can be realized on a finite probability space). We also give conditions for there to exist an arbitrage opportunity which can be locked in at time zero. There is also a third boundary case in which prices are recognizably misspecified, but the ability to take advantage of an arbitrage opportunity depends upon knowledge of the null sets of the model.     

On the Existence of Finite-Dimensional Realizations for Nonlinear Forward Rate Models

We consider interest rate models of the Heath–Jarrow–Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite-dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite-dimensional realization. A number of concrete applications are given, and all previously known realization results (as far as existence is concerned) for Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov process. In particular we give a short proof of a result by Jeffrey showing that the only forward rate models with short rate dependent volatility structures which generically possess a short rate realization are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.     

THE MULTIVARIATE supOU STOCHASTIC VOLATILITY MODEL

Using positive semidefinite supOU (superposition of Ornstein–Uhlenbeck type) processes to describe the volatility, we introduce a multivariate stochastic volatility model for financial data which is capable of modeling long range dependence effects. The finiteness of moments and the second‐order structure of the volatility, the log‐ returns, as well as their “squares” are discussed in detail. Moreover, we give several examples in which long memory effects occur and study how the model as well as the simple Ornstein–Uhlenbeck type stochastic volatility model behave under linear transformations. In particular, the models are shown to be preserved under invertible linear transformations. Finally, we discuss how (sup)OU stochastic volatility models can be combined with a factor modeling approach.     

Accounting for stochastic interest rates,stochastic volatility and a general correlation structure in the valuation of forward starting options

A quantitative analysis on the pricing of forward starting options under stochastic volatility and stochastic interest rates is performed. The main finding is that forward starting options not only depend on future smiles, but also directly on the evolution of the interest rates as well as the dependency structures among the underlying asset, the interest rates, and the stochastic volatility: compared to vanilla options, dynamic structures such as forward starting options are much more sensitive to model specifications such as volatility, interest rate, and correlation movements. We conclude that it is of crucial importance to take all these factors explicitly into account for a proper valuation and risk management of these securities. The performed analysis is facilitated by deriving closed‐form formulas for the valuation of forward starting options, hereby taking the stochastic volatility, stochastic interest rates as well the dependency structure between all these processes explicitly into account. The valuation framework is derived using a probabilistic approach, enabling a fast and efficient evaluation of the option price by Fourier inverting the forward starting characteristic functions. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:103–125, 2011     

BESSEL PROCESSES,STOCHASTIC VOLATILITY,AND TIMER OPTIONS

Motivated by analytical valuation of timer options (an important innovation in realized variance‐based derivatives), we explore their novel mathematical connection with stochastic volatility and Bessel processes (with constant drift). Under the Heston (1993) stochastic volatility model, we formulate the problem through a first‐passage time problem on realized variance, and generalize the standard risk‐neutral valuation theory for fixed maturity options to a case involving random maturity. By time change and the general theory of Markov diffusions, we characterize the joint distribution of the first‐passage time of the realized variance and the corresponding variance using Bessel processes with drift. Thus, explicit formulas for a useful joint density related to Bessel processes are derived via Laplace transform inversion. Based on these theoretical findings, we obtain a Black–Scholes–Merton‐type formula for pricing timer options, and thus extend the analytical tractability of the Heston model. Several issues regarding the numerical implementation are briefly discussed.     

An empirical comparison of continuous time models of the short term interest rate

This article tests the performance of a wide variety of well-known continuous time models—with particular emphasis on the Black, Derman, and Toy (1990; henceforth BDT) term structure model—in capturing the stochastic behavior of the short term interest rate volatility. Many popular interest rate models are nested within a more flexible time-varying BDT framework that allows us to compare the models and find the proper specification of the dynamics of short rates. The empirical results indicate that the equilibrium models that do not allow the drift and diffusion parameters to vary over time and parameterize the volatility only as a function of interest rate levels overemphasize the sensitivity of volatility to the level of interest rate and fail to model adequately the serial correlation in conditional variances. On the other hand, the GARCH-based arbitrage-free models with time-dependent parameters in the drift and diffusion functions define the volatility only as a function of unexpected information shocks and fail to capture adequately the relationship between interest rate levels and volatility. This study shows that the most successful models in capturing the dynamics of short term interest rates are those that introduce time-dependent parameters to the short rate process and define the conditional volatility as a function of both the interest rate levels and the last period's unexpected news. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 777–797, 1999     

A note on the long rate in factor models of the term structure

In this paper, we consider factor models of the term structure based on a Brownian filtration. We show that the existence of a nondeterministic long rate in a factor model of the term structure implies, as a consequence of the Dybvig–Ingersoll–Ross theorem, that the model has an equivalent representation in which one of the state variables is nondecreasing. For two‐dimensional factor models, we prove moreover that if the long rate is nondeterministic, the yield curve flattens out, and the factor process is asymptotically nondeterministic, then the term structure is unbounded. Finally, we provide an explicit example of a three‐dimensional affine factor model with a nondeterministic yet finite long rate in which the volatility of the factor process does not vanish over time.     

PORTFOLIO OPTIMIZATION AND STOCHASTIC VOLATILITY ASYMPTOTICS

We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean‐reverting, this is a singular perturbation problem for a nonlinear Hamilton–Jacobi–Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a “practical” strategy that does not require tracking the fast‐moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single‐factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution.     

APPROXIMATING GARCH-JUMP MODELS, JUMP-DIFFUSION PROCESSES, AND OPTION PRICING

This paper considers the pricing of options when there are jumps in the pricing kernel and correlated jumps in asset prices and volatilities. We extend theory developed by Nelson (1990) and Duan (1997) by considering the limiting models for our approximating GARCH Jump process. Limiting cases of our processes consist of models where both asset price and local volatility follow jump diffusion processes with correlated jump sizes. Convergence of a few GARCH models to their continuous time limits is evaluated and the benefits of the models explored.     

FAST SWAPTION PRICING IN GAUSSIAN TERM STRUCTURE MODELS

We propose a fast and accurate numerical method for pricing European swaptions in multifactor Gaussian term structure models. Our method can be used to accelerate the calibration of such models to the volatility surface. The pricing of an interest rate option in such a model involves evaluating a multidimensional integral of the payoff of the claim on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state space by a hyperplane tangent to the maximum probability point on the boundary and simplify the multidimensional integration into an analytical form. The maximum probability point can be determined using the gradient descent method. We demonstrate that our method is superior to previous methods by comparing the results to the price obtained by numerical integration.     

Consistent calibration of HJM models to cap implied volatilities

This article proposes a calibration algorithm that fits multifactor Gaussian models to the implied volatilities of caps with the use of the respective minimal consistent family to infer the forward‐rate curve. The algorithm is applied to three forward‐rate volatility structures and their combination to form two‐factor models. The efficiency of the consistent calibration is evaluated through comparisons with nonconsistent methods. The selection of the number of factors and of the volatility functions is supported by a principal‐component analysis. Models are evaluated in terms of in‐sample and out‐of‐sample data fitting as well as stability of parameter estimates. The results are analyzed mainly by focusing on the capability of fitting the market‐implied volatility curve and, in particular, reproducing its characteristic humped shape. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:1093–1120, 2005     

The drift factor in biased futures index pricing models: A new look

The presence of bias in index futures prices has been investigated in various research studies. Redfield ( 11 ) asserted that the U.S. Dollar Index (USDX) futures contract traded on the U.S. Cotton Exchange (now the FINEX division of the New York Board of Trade) could be systematically arbitraged for nontrivial returns because it is expressed in so‐called “European terms” (foreign currency units/U.S. dollar). Eytan, Harpaz, and Krull ( 4 ) (EHK) developed a theoretical factor using Brownian motion to correct for the European terms and the bias due to the USDX index being expressed as a geometric average. Harpaz, Krull, and Yagil ( 5 ) empirically tested the EHK index. They used the historical volatility to proxy the EHK volatility specification. Since 1990, it has become more commonplace to use option‐implied volatility for forecasting future volatility. Therefore, we have substituted option implied volatilities into EHK's correction factor and hypothesized that the correction factor is “better” ex ante and therefore should lead to better futures model pricing. We tested this conjecture using twelve contracts from 1995 through 1997 and found that the use of implied volatility did not improve the bias correction over the use of historical volatility. Furthermore, no matter which volatility specification we used, the model futures price appeared to be mis‐specified. To investigate further, we added a simple naïve δ based on a modification of the adaptive expectations model. Repeating the tests using this naïve “drift” factor, it performed substantially better than the other two specifications. Our conclusion is that there may be a need to take a new look at the drift‐factor specification currently in use. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:579–598, 2002     

HEATH–JARROW–MORTON INTEREST RATE DYNAMICS AND APPROXIMATELY CONSISTENT FORWARD RATE CURVES

We study a finite-dimensional approach to the Heath–Jarrow–Morton model for interest rate and introduce a notion of approximate consistency for a family of functions in a deterministic and stochastic framework. This amounts to asking the decrease of the minimum distance in least squares sense. We start from a general linearly parameterized set of functions and extend the theory to a nonlinear Nelson–Siegel family. Necessary and sufficient condition to have approximately consistency are given as well as a criterion of stability for the approximation.     

Modeling the conditional mean and variance of the short rate using diffusion,GARCH, and moving average models

This article introduces a two‐factor‐discrete‐time‐stochastic‐volatility model that allows for departures from linearity in the conditional mean and incorporates serially correlated unexpected news, asymmetry, and level effects into the definition of conditional volatility of the short rate. The new class of econometric specifications nests many popular existing symmetric and asymmetric GARCH as well as diffusion models of the short‐term interest rate. This study attempts to determine the correct specification of conditional mean and variance of the short rate by developing a more general econometric framework that allows for nonlinear effects in the drift of the short rate, and that defines the conditional volatility as a nonlinear function of unexpected information shocks and interest rate levels. The existing and alternative models are compared in terms of their ability to capture the stochastic behavior of the short‐term riskless rate. The empirical results indicate that the relative performance of the two‐factor models in predicting the future level and variance of interest‐rate changes is superior to the nested models. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:717–751, 2000     

A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES

We propose a model which can be jointly calibrated to the corporate bond term structure and equity option volatility surface of the same company. Our purpose is to obtain explicit bond and equity option pricing formulas that can be calibrated to find a risk neutral model that matches a set of observed market prices. This risk neutral model can then be used to price more exotic, illiquid, or over‐the‐counter derivatives. We observe that our model matches the equity option implied volatility surface well since we properly account for the default risk in the implied volatility surface. We demonstrate the importance of accounting for the default risk and stochastic interest rate in equity option pricing by comparing our results to Fouque et al., which only accounts for stochastic volatility.