General Stochastic Clocks

This is the first post in a series on asset price dynamics that are subject to a stochastic clock. The main idea is to not model the volatility as a stochastic process but instead make the rate at which time progresses randomly. In this post, we outline the basic idea and define the terminology.

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Asset Price Model

Let X = \left\{ X_t: t \in \mathbb{R}_+ \right\} be a Lévy process and Y = \left\{ Y_t: t \in \mathbb{R}_+ \right\} be a differentiable finite-variation time-change process on a filtered probability space \left( \Omega, \mathbb{P}, \mathbb{F} \right). We recall that as a time-change, Y has initial value Y_0 = 0 and is a right-continuous, strictly increasing process, see e.g. Chapter IV.3 in Protter (2004). We interpret \mathbb{P} to be the physical probability measure and model the stock price S = \left\{ S_t: t \in \mathbb{R}_+ \right\} as

    \[ S_t = \exp \left\{ \gamma t + X \left( Y_t \right) \right\}. \]

Here, \gamma \in \mathbb{R} is some constant, not to be confused with the real-world drift of the stock price. We will elaborate on this in a later post, when discussing measure changes for stochastic clock models. Since the time-change Y is differentiable and of finite-variation, it can be represented as

    \[ Y_t = \int_0^t y_u \mathrm{d}u \]

for some process y = \left\{ y_t: t \in \mathbb{R}_+ \right\}. Thus, y is the instantaneous rate of activity process. From the definition of Y is follows that y has to be strictly positive. In the following post, we discuss the case where y follows a non-Gaussian Ornstein-Uhlenbeck process.

The economic interpretation of this setup is that y models the rate of information arrival. It captures the effect that asset prices change as new information becomes available. The volatility of these changes is the higher, the more information or the more significant information is released.

Characteristic Function

We are now interested in finding the characteristic function \phi_{\ln \left( S_t \right)}(\omega) of the logarithmic asset price \ln \left( S_t \right). By definition,

    \begin{align*} \phi_{\ln \left( S_t \right)}(\omega) & = \mathbb{E} \left[ \exp \left\{ \mathrm{i} \omega \left( \gamma t + X \left( Y_t \right) \right) \right\} \right] \\ & = e^{\mathrm{i} \omega \gamma t} \mathbb{E} \left[ \mathbb{E} \left[ \left. \exp \left\{ \mathrm{i} \omega X \left( Y_t \right) \right\} \right| \sigma \left( Y_t \right) \right] \right]. \end{align*}

Here, we conditioned on the value of the time-change in the second step. Now let \phi_{X_t}(\omega) be the characteristic function of X_t. Since X is a Lévy process, it is given in terms of its characteristic triplet \left( \sigma^2, \nu, \gamma \right) by the Lévy-Khinchin representation, see e.g. Theorem 3.1 in Cont and Tankov (2004) or Theorems 1.2.14 and 1.3.3 in Applebaum (2004). Furthermore, letting \psi_{X_1}(\omega) be the corresponding characteristic exponent, it takes the form

    \[ \phi_{X_t}(\omega) = \exp \left\{ t \psi_{X_1}(\omega) \right\}, \]

see e.g. Proposition 3.2 in Cont and Tankov (2004) or Theorem 1.3.3 Applebaum (2004). Thus,

    \begin{align*} \phi_{\ln \left( S_t \right)}(\omega) & = e^{\mathrm{i} \omega \gamma t} \mathbb{E} \left[ \phi_{X \left( Y_t \right)}(\omega) \right]\\ & = e^{\mathrm{i} \omega \gamma t} \mathbb{E} \left[ \exp \left\{ Y_t \psi_{X_1}(\omega) \right\} \right]. \end{align*}

Now let \phi_{Y_t}(\omega) be the characteristic function of the time-change Y_t, then

    \[ \phi_{\ln \left( S_t \right)}(\omega) = e^{\mathrm{i} \omega \gamma t} \phi_{Y_t} \left( -\mathrm{i} \psi_{X_1}(\omega) \right). \]

In the next post, we will find a general expression for \phi_{Y_t}(\omega).


Applebaum, David (2004) Lévy Processes and Stochastic Calculus: Cambridge University Press

Cont, Rama and Tankov, Peter (2004) Financial Modelling with Jump Processes: Chapman & Hall

Protter, Philip E. (2004) Stochastic Integration and Differential Equations: Springer

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