Occupied Processes: Going with the Flow

TL;DR

The paper develops an Itô calculus for occupied processes by augmenting a Markov process $X$ with its occupation flow $\mathcal{O}$ to form a Markovian pair $(\mathcal{O},X)$, and derives a simple chain rule $df(\mathcal{O}_t,X_t) = (\partial_{\mathfrak{o}} + \tfrac{1}{2}\partial_{xx})f(\mathcal{O}_t,X_t) d\langle X \rangle_t + \partial_x f(\mathcal{O}_t,X_t) dX_t$, with calendar-time variants that link to functional Itô calculus. Through a Feynman-Kac framework for occupied SDEs, the authors obtain a broad class of path-dependent PDEs where the occupation flow plays the role of time, while the state variable remains finite-dimensional and amenable to standard elliptic PDE techniques. They demonstrate a unified Markovian lift for pricing exotic options and variance derivatives, and introduce the Local Occupied Volatility (LOV) model to achieve vanilla-calibration fidelity alongside path-dependent volatility features, plus forward-occupation models to capture the evolution of forward occupation surfaces. The work also studies optimal stopping for spot local time, providing both theoretical insights and practical numerical methods (LSMC, corridor-based approximations) for occupation-based stopping problems, with implications for pricing and hedging complex derivatives. Overall, the framework offers a versatile, numerically tractable approach to model, price, and manage a wide class of path-dependent instruments in finance.

Abstract

A stochastic process $X$ becomes \textit{occupied} when it is enlarged with its occupation flow $\mathcal{O}$, which tracks the time spent by the path at each level. Crucially, the occupied process $(\mathcal{O},X)$ enjoys a Markov structure when $X$ is Markov. We develop a novel Itô calculus for occupied processes that lies midway between Dupire's functional Itô calculus and the classical version. We derive a surprisingly simple Itô formula and, through Feynman-Kac, unveil a broad class of path-dependent PDEs where $\mathcal{O}$ plays the role of time. The space variable, given by the current value of $X$, remains finite-dimensional, thereby paving the way for standard elliptic PDE techniques and numerical methods. In the financial applications, we demonstrate that occupation flows provide unified Markovian lifts for exotic options and variance instruments, allowing financial institutions to price and manage derivatives books with a single numerical solver. We then explore avenues in financial modeling where volatility is driven by the occupied process. In particular, we propose the local occupied volatility (LOV) model which not only calibrates to European vanilla options but also offers the flexibility to capture stylized facts of volatility or fit other instruments. We also present an extension of forward variance models that leverages the entire forward occupation surface.

Figures (17)

• Figure 1: Occupation flow $\mathop{\mathrm{\mathcal{O}}}\nolimits$ and its dynamics $d\mathop{\mathrm{\mathcal{O}}}\nolimits_t = \delta_{X_t}d\langle X\rangle_t$.
• Figure 2: The occupation derivative $\partial_{\mathfrak{o}}f(\mathop{\mathrm{\mathcal{O}}}\nolimits_t,X_t)$ gives the sensitivity of $f$ with respect to a Dirac impulse at the spot $X_t$.
• Figure 3: Unified Markovian lift for exotic options and structured products.
• Figure 4: Unified Markovian lift for variance derivatives.
• Figure 5: Occupation measure using calendar time ($\tilde{\mathop{\mathrm{\mathcal{O}}}\nolimits}_t$, left) compared with exponential time ($\mathop{\mathrm{\mathcal{O}}}\nolimits^{\kappa}$, right). As can be seen, $\mathop{\mathrm{\mathcal{O}}}\nolimits^{\kappa}$ puts more emphasis on recent levels of the path.

Theorems & Definitions (68)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1
• Definition 2
• Definition 3
• Proposition 3
• proof