MOMENT EXPLOSIONS AND STATIONARY DISTRIBUTIONS IN AFFINE DIFFUSION MODELS

Many of the most widely used models in finance fall within the affine family of diffusion processes. The affine family combines modeling flexibility with substantial tractability, particularly through transform analysis; these models are used both for econometric modeling and for pricing and hedging of derivative securities. We analyze the tail behavior, the range of finite exponential moments, and the convergence to stationarity in affine models, focusing on the class of canonical models defined by Dai and Singleton (2000) . We show that these models have limiting stationary distributions and characterize these limits. We show that the tails of both the transient and stationary distributions of these models are necessarily exponential or Gaussian; in the non-Gaussian case, we characterize the tail decay rate for any linear combination of factors. We also give necessary and sufficient conditions for a linear combination of factors to be Gaussian. Our results follow from an investigation into the stability properties of the systems of ordinary differential equations associated with affine diffusions.     

Option pricing with orthogonal polynomial expansions

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein–Stein, and Hull–White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier‐transform‐based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.     

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.     

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.     

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.     

STATIC FUND SEPARATION OF LONG‐TERM INVESTMENTS

This paper proves a class of static fund separation theorems, valid for investors with a long horizon and constant relative risk aversion, and with stochastic investment opportunities. An optimal portfolio decomposes as a constant mix of a few preference‐free funds, which are common to all investors. The weight in each fund is a constant that may depend on an investor's risk aversion, but not on the state variable, which changes over time. Vice versa, the composition of each fund may depend on the state, but not on the risk aversion, since a fund appears in the portfolios of different investors. We prove these results for two classes of models with a single state variable, and several assets with constant correlations with the state. In the linear class, the state is an Ornstein–Uhlenbeck process, risk premia are affine in the state, while volatilities and the interest rate are constant. In the square root class, the state follows a square root diffusion, expected returns and the interest rate are affine in the state, while volatilities are linear in the square root of the state.     

Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps

We investigate analytical solvability of models with affine stochastic volatility (SV) and Lévy jumps by deriving a unified formula for the conditional moment generating function of the log-asset price and providing the condition under which this new formula is explicit. The results lay a foundation for a range of valuation, calibration, and econometric problems. We then combine our theoretical results, the Hilbert transform method, various interpolation techniques, with the dimension reduction technique to propose unified simulation schemes for solvable models with affine SV and Lévy jumps. In contrast to traditional exact simulation methods, our approach is applicable to a broad class of models, maintains good accuracy, and enables efficient pricing of discretely monitored path-dependent derivatives. We analyze various sources of errors arising from the simulation approach and present error bounds. Finally, extensive numerical results demonstrate that our method is highly accurate, efficient, simple to implement, and widely applicable.     

SWAPTION PRICING IN AFFINE AND OTHER MODELS

This paper shows that Singleton and Umantsev's method for swaption pricing in affine models can be simplified and extended to other models. Two alternative methods for approximating the option exercise boundary are introduced: one based on the multivariate Taylor series expansion, and the other based on duration‐matched zero‐coupon bond approximation. Applied to affine models and quadratic‐Gaussian models, these methods are found to give accurate swaption prices.     

Consistent recalibration of yield curve models

The analytical tractability of affine (short rate) models, such as the Vasi?ek and the Cox–Ingersoll–Ross (CIR) models, has made them a popular choice for modeling the dynamics of interest rates. However, in order to properly account for the dynamics of real data, these models must exhibit time‐dependent or even stochastic parameters. This breaks their tractability, and modeling and simulating become an arduous task. We introduce a new class of Heath–Jarrow–Morton (HJM) models that both fit the dynamics of real market data and remain tractable. We call these models consistent recalibration (CRC) models. CRC models appear as limits of concatenations of forward rate increments, each belonging to a Hull–White extended affine factor model with possibly different parameters. That is, we construct HJM models from “tangent” affine models. We develop a theory for continuous path versions of such models and discuss their numerical implementations within the Vasi?ek and CIR frameworks.     

A NOTE ON THE DAI–SINGLETON CANONICAL REPRESENTATION OF AFFINE TERM STRUCTURE MODELS*

Dai and Singleton (2000) study a class of term structure models for interest rates that specify the short rate as an affine combination of the components of an N‐dimensional affine diffusion process. Observable quantities in such models are invariant under regular affine transformations of the underlying diffusion process. In their canonical form, the models in Dai and Singleton (2000) are based on diffusion processes with diagonal diffusion matrices. This motivates the following question: Can the diffusion matrix of an affine diffusion process always be diagonalized by means of a regular affine transformation? We show that if the state space of the diffusion is of the form for integers satisfying or , there exists a regular affine transformation of D onto itself that diagonalizes the diffusion matrix. So in this case, the Dai–Singleton canonical representation is exhaustive. On the other hand, we provide examples of affine diffusion processes with state space whose diffusion matrices cannot be diagonalized through regular affine transformation. This shows that for ), the assumption of diagonal diffusion matrices may impose unnecessary restrictions and result in an avoidable loss of generality.     

Option Pricing For Jump Diffusions: Approximations and Their Interpretation

We derive a computable approximation for the value of a European call option when prices satisfy a jump-diffusion model with the coefficients depending explicitly on time. This is achieved by approximating the original coefficients with functions that are piecewise constant in time. We give an interpretation of the approximating option values, in particular in the context of a discrete-time model associated with the approximating continuous-time model.     

TIME‐CHANGED MARKOV PROCESSES IN UNIFIED CREDIT‐EQUITY MODELING

This paper develops a novel class of hybrid credit‐equity models with state‐dependent jumps, local‐stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time‐changed Markov diffusion process with state‐dependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with state‐dependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has local‐stochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state‐dependent jumps, local‐stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time‐changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump‐to‐default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local‐stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.     

Valuation of VIX and target volatility options with affine GARCH models

In this paper we propose semiclosed-form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options under affine GARCH models based on Gaussian and Inverse Gaussian distributions. We illustrate the advantage of the proposed analytic expressions by comparing them with those obtained from benchmark Monte–Carlo simulations. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.     

Noncausal affine processes with applications to derivative pricing

Linear factor models, where the factors are affine processes, play a key role in Finance, since they allow for quasi-closed form expressions of the term structure of risks. We introduce the class of noncausal affine linear factor models by considering factors that are affine in reverse time. These models are especially relevant for pricing sequences of speculative bubbles. We show that they feature nonaffine dynamics in calendar time, while still providing (quasi) closed form term structures and derivative pricing formulas. The framework is illustrated with term structure of interest rates and European call option pricing examples.     

Modeling preference evolution in discrete choice models: A Bayesian state-space approach

We develop discrete choice models that account for parameter driven preference dynamics. Choice model parameters may change over time because of shifting market conditions or due to changes in attribute levels over time or because of consumer learning. In this paper we show how such preference evolution can be modeled using hierarchial Bayesian state space models of discrete choice. The main feature of our approach is that it allows for the simultaneous incorporation of multiple sources of preference and choice dynamics. We show how the state space approach can include state dependence, unobserved heterogeneity, and more importantly, temporal variability in preferences using a correlated sequence of population distributions. The proposed model is very general and nests commonly used choice models in the literature as special cases. We use Markov chain monte carlo methods for estimating model parameters and apply our methodology to a scanner data set containing household brand choices over an eight-year period. Our analysis indicates that preferences exhibit significant variation over the time-span of the data and that incorporating time-variation in parameters is crucial for appropriate inferences regarding the magnitude and evolution of choice elasticities. We also find that models that ignore time variation in parameters can yield misleading inferences about the impact of causal variables. This paper is based on the first author's doctoral dissertation.     

PRICING OPTIONS ON VARIANCE IN AFFINE STOCHASTIC VOLATILITY MODELS

We consider the pricing of options written on the quadratic variation of a given stock price process. Using the Laplace transform approach, we determine semi‐explicit formulas in general affine models allowing for jumps, stochastic volatility, and the leverage effect. Moreover, we show that the joint dynamics of the underlying stock and a corresponding variance swap again are of affine form. Finally, we present a numerical example for the Barndorff‐Nielsen and Shephard model with leverage. In particular, we study the effect of approximating the quadratic variation with its predictable compensator.     

AN EQUILIBRIUM GUIDE TO DESIGNING AFFINE PRICING MODELS

The paper examines equilibrium models based on Epstein–Zin preferences in a framework in which exogenous state variables follow affine jump diffusion processes. A main insight is that the equilibrium asset prices can be computed using a standard machinery of affine asset pricing theory by imposing parametric restrictions on market prices of risk, determined inside the model by preference and model parameters. An appealing characteristic of the general equilibrium setup is that the state variables have an intuitive and testable interpretation as driving the consumption and dividend dynamics. We present a detailed example where large shocks (jumps) in consumption volatility translate into negative jumps in equilibrium prices of the assets as agents demand a higher premium to compensate for higher risks. This endogenous “leverage effect,” which is purely an equilibrium outcome in the economy, leads to significant premiums for out‐of‐the‐money put options. Our model is thus able to produce an equilibrium “volatility smirk,” which realistically mimics that observed for index options.     

A FAMILY OF TERM-STRUCTURE MODELS FOR LONG-TERM RISK MANAGEMENT AND DERIVATIVE PRICING

In this paper we propose a new family of term-structure models based on the Flesaker and Hughston (1996) positive-interest framework. The models are Markov and time homogeneous, with correlated Ornstein-Uhlenbeck processes as state variables. We provide a theoretical analysis of the one-factor model and a thorough emprical analysis of the two-factor model. This allows us to identify the key factors in the model affecting interest-rate dynamics. We conclude that the new family of models should provide a useful tool for use in long-term risk management. Suitably parameterized, they can satisfy a wide range of desirable criteria, including:

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  sustained periods of both high and low interest rates similar to the cycle lengths we have observed over the course of the 20th century in the United Kingdom and the United States

       
  
  

PRICING COUPON-BOND OPTIONS AND SWAPTIONS IN AFFINE TERM STRUCTURE MODELS

This paper provides a numerically accurate and computationally fast approximation to the prices of European options on coupon-bearing instruments that is applicable to the entire family of affine term structure models. Exploiting the typical shapes of the conditional distributions of the risk factors in affine diffusions, we show that one can reliably compute the relevant probabilities needed for pricing options on coupon-bearing instruments by the same Fourier inversion methods used in the pricing of options on zero-coupon bonds. We apply our theoretical results to the pricing of options on coupon bonds and swaptions, and the calculation of "expected exposures" on swap books. As an empirical illustration, we compute the expected exposures implied by several affine term structure models fit to historical swap yields.     

CONSTANT PROPORTION PORTFOLIO INSURANCE IN THE PRESENCE OF JUMPS IN ASSET PRICES

Constant proportion portfolio insurance (CPPI) allows an investor to limit downside risk while retaining some upside potential by maintaining an exposure to risky assets equal to a constant multiple of the cushion , the difference between the current portfolio value and the guaranteed amount. Whereas in diffusion models with continuous trading, this strategy has no downside risk, in real markets this risk is nonnegligible and grows with the multiplier value. We study the behavior of CPPI strategies in models where the price of the underlying portfolio may experience downward jumps. Our framework leads to analytically tractable expressions for the probability of hitting the floor, the expected loss, and the distribution of losses. This allows to measure the gap risk but also leads to a criterion for adjusting the multiplier based on the investor's risk aversion. Finally, we study the problem of hedging the downside risk of a CPPI strategy using options. The results are applied to a jump-diffusion model with parameters estimated from returns series of various assets and indices.