Motivated by the behavior of short-term interest rates since the 2007-2008 Financial Crisis, we consider short-rate models with lesser-known but perhaps compatible boundary behaviors at r = 0. Specifically, we explore (i) slowly-reflecting boundaries, also known as sticky boundaries, and (ii) jump-returns from a boundary.

I discuss spectral expansions for jump-diffusions, stressing exit problems, correct boundary conditions, and numerics. The linear operators involved (the generators and their adjoints) are generally non-self-adjoint operators with a discrete, but complex-valued spectrum. I discuss how to prove discreteness, treating the jumps as a perturbation. For numerics, I show how to employ NDSolve in addition to the eigensystem solvers.

We consider time series modelling of SPX under jump-diffusions. When volatility is stochastic, parameter inference can be difficult if the instantaneous volatility is considered latent (unobservable): a situation considered elsewhere in this volume. However, if the SPX series is combined with the VIX series, the latency can often be removed and maximum likelihood estimation of popular financial models can proceed directly. We develop the case of a jump-diffusion where the jumps are induced by a stochastic time-change of a diffusion.

My approach uses affine jump-diffusions with volatility jumps and some time inhomogeneity. Stability analysis of autonomous ODE systems plays a key role. General VIX option formulas under this class of processes are found, where the market prices for VIX forwards are state variables and matched exactly. A basic one-factor version is shown capable of good matches to VIX option skew patterns, both under typical markets and under high stress environments like the Financial Crisis.

This is the first of two more chapters on statistical inference for continuous-time stochastic processes. In this one, we show how to discretize generic stochastic volatility models and turn them into Hidden Markov Models. Inference for the latter is well-developed and a natural way to deal with the latent nature of stochastic volatility. The net result is a tractable inference program based upon a discretized process which converges to the target continuum process.

We discuss parameter inference when diffusions are continuously observed. Although highly idealized, this setup raises interesting theoretical issues. The issues are discovered and resolved within worked examples:

• inference with boundary processes present (slow reflection);
• a critical case of the Ornstein-Uhlenbeck process; and
• inference under Feller’s square-root process.

While we review standard theory, our emphasis is on non-standard behavior in elementary examples. These involve boundary behavior and/or lack of ergodicity.

We review the essentials of Feller’s (1951) square-root process, including two fundamental solutions and the role of flux at the origin. Applications include the joint (stock and volatility) transition densities for both the square-root and 3/2 model, and the CIR bond process with absorption.

We explore the mathematics and numerics of the SABR model, mostly in equity language. The motivation: it’s a fascinating exercise in mathematical finance. The model was developed for interest rate derivatives. Applications tend to rely on small-time asymptotics. There are plausible equity option applications if various extensions are made. But, since the extensions break most of the tractability (and the result could hardly be called `SABR’), that topic is mentioned only briefly at the end.

How to best handle discrete dividends is still a contentious issue. In a 2003 article co-authored with E.G. Haug and J. Haug, we espoused the piecewise GBM class of models as the preferred solution. Here, I take the same approach, updating that discussion with the significant and subsequent advances in both numerics and theory.

This chapter provides additional background for the many invocations of NDSolve with little elaboration in other chapters. We concentrate on two tricky issues common to PDE problems in quantitative finance: semi-infinite domains and singular boundaries.

Two-sided exit problems have few exact results, but there is a class of processes which are analytically tractable. Consider a one-dimensional Levy process where the jumps are compound-Poisson and, in at least one direction, have a relatively simple structure. This structure needs to be simple exponential or a natural generalization called here “rational-transform-type”. For these, we show how a Wiener-Hopf method leads to closed-form solutions, up to a (numerical) Laplace inversion. This extends an earlier solution due to J. Kemperman. The “double-exponential jump-diffusion” (DEJD) developed by S. Kou and H. Wang is a nice special case that is solved in detail. The numerical results were used earlier in the book to double-check the spectral theory codes developed in Ch. 2

In Vol. I, we discussed the implied volatility smile near expiration under stochastic volatility models. Our results were extensive — yet incomplete. Subsequent work by several authors has clarified the theory. Complete solutions under diffusion systems are found using ideas from Riemannian geometry and related theory.

To illustrate the geometrical theory, we develop in detail the T \rightarrow 0 asymptotic smile for the CEV(p)-vol class of stochastic volatility models. This class encompasses all volatility SDEs of the form: dV = \beta(V) \, dt + \xi \, V^p dB_t, with a stock price-volatility correlation. Here p is any real number. The drift \beta(V) plays no role in the leading asymptotics. Previously for this model, only the asymptotic smiles for the special cases p=\frac{1}{2} and p=1 were available in the literature.