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.
- Ornstein-Uhlenbeck Stochastic Clocks
- Gamma Ornstein-Uhlenbeck Stochastic Clocks
- Cox-Ingersoll-Ross Stochastic Clocks
- Inverse Gaussian Ornstein-Uhlenbeck Stochastic Clocks
Asset Price Model
Let be a Lévy process and be a differentiable finite-variation time-change process on a filtered probability space . We recall that as a time-change, has initial value and is a right-continuous, strictly increasing process, see e.g. Chapter IV.3 in Protter (2004). We interpret to be the physical probability measure and model the stock price as
Here, 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 is differentiable and of finite-variation, it can be represented as
for some process . Thus, is the instantaneous rate of activity process. From the definition of is follows that has to be strictly positive. In the following post, we discuss the case where follows a non-Gaussian Ornstein-Uhlenbeck process.
The economic interpretation of this setup is that 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.
We are now interested in finding the characteristic function of the logarithmic asset price . By definition,
Here, we conditioned on the value of the time-change in the second step. Now let be the characteristic function of . Since is a Lévy process, it is given in terms of its characteristic triplet 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 be the corresponding characteristic exponent, it takes the form
see e.g. Proposition 3.2 in Cont and Tankov (2004) or Theorem 1.3.3 Applebaum (2004). Thus,
Now let be the characteristic function of the time-change , then
In the next post, we will find a general expression for .
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