Lévy process

A Lévy process \(x_t\) is a stochastic process which satisfies the following properties

• \(x_0 = 0\)

• independent increments: \(x_t - x_s\) is independent of \(x_u; u \le s\ \forall\ s < t\)

• stationary increments: \(x_{s+t} - x_s\) has the same distribution as \(x_t - x_0\) for any \(s,t > 0\)

This means that the shocks to the process are independent, while the stationarity assumption specifies that the distribution of \(x_{t+s} - x_s\) may change with \(s\) but does not depend upon \(t\).

Remark: The properties of stationary and independent increments implies that a Lévy process is a Markov process. Thanks to almost sure right continuity of paths, one may show in addition that Lévy processes are also Strong Markov processes. See (Markov property).

Characteristic function

The independence and stationarity of the increments of the Lévy process imply that the characteristic function of \(x_t\) has the form

(1)\[\begin{equation} \Phi_{x_t, u} = {\mathbb E}\left[e^{i u x_t}\right] = e^{-\phi_{x_t, u}} = e^{-t \phi_{x_1,u}} \end{equation}\]

where the characteristic exponent \(\phi_{x_1,u}\) is given by the Lévy–Khintchine formula.

There are several Lévy processes in the literature, including, importantly, the Poisson process, the compound Poisson process, and the Brownian motion.

Time Changed Lévy Processes

We follow the paper by Carr and Wu [CW02] to defined a continuous time changed Lévy process \(y_t\) as

(2)\[\begin{align} y_t &= x_{\tau_t}\\ \tau_t &= \int_0^t \lambda_s ds \end{align}\]

where \(x_s\) is a Lévy process and \(\lambda_s\) is a positive and integrable process which we refer to stochastic intensity process. While \(\tau_t\) is always continuous, \(\lambda\) can exhibit jumps. Since the time-changed process is a stochastic process evaluated at a stochastic time, its characteristic function involves expectations over two sources of randomness:

(3)\[\begin{equation} \Phi_{y_t, u} = {\mathbb E}\left[e^{i u x_{\tau_t}}\right] = {\mathbb E}\left[{\mathbb E}\left[\left.e^{i u x_s}\right|\tau_t=s\right]\right] \end{equation}\]

where the inside expectation is taken on \(x_{\tau_t}\) conditional on a fixed value of \(\tau_t = s\) and the outside expectation is on all possible values of \(\tau_t\). If the random time \(\tau_t\) is independent of \(x_t\), the randomness due to the Lévy process can be integrated out using the characteristic function of \(x_t\):

(4)\[\begin{equation} \Phi_{y_t, u} = {\mathbb E}\left[e^{-\tau_t \phi_{x,u}}\right] = {\mathbb L}_{\tau_t}\left(\phi_{x_1,u}\right) \end{equation}\]

Remark: Under independence, the characteristic function of a time-changed Lévy process \(y_t\) is the Laplace transform of the cumulative intensity \(\tau_t\) evaluated at the characteristic exponent of \(x\).

Therefore the characteristic function of \(y_t\) can be expressed in closed form if

• the characteristic exponent of the Lévy process \(x_t\) is available in closed from

• the Laplace transform of \(\tau_t\), the integrated intensity process, is known in closed from

Leverage Effect

To obtain the Laplace transform of \(\tau_t\) in closed form, consider its specification in terms of the intensity process \(\lambda_t\):

(5)\[\begin{equation} {\mathbb L}_{\tau_t}\left(u\right) = {\mathbb E}\left[e^{- u \int_0^t \lambda_s ds}\right] \end{equation}\]

This equation is very common in the bond pricing literature if we regard \(u\lambda_t\) as the instantaneous interest rate. In the general case, the intensity process is correlated with the Lévy process of increments, this is well known in the literature as the leverage effect.

Carr and Wu [CW02] solve this problem by changing the measure from an economy with leverage effect to one without it.

(6)\[\begin{align} \Phi_{y_t, u} &= {\mathbb E}\left[e^{i u y_t}\right] \\ &= {\mathbb E}\left[e^{i u y_t + \tau_t \phi_{x_1, u} - \tau_t \phi_{x_1, u}}\right] \\ &= {\mathbb E}\left[M_{t, u} e^{-\tau_t \phi_{x_1,u}}\right] \\ &= {\mathbb E}^u\left[e^{-\tau_t \phi_{x_1,u}}\right] \\ &= {\mathbb L}_{\tau_t}^u\left(\phi_{x_1,u}\right) \end{align}\]

where \(E[\cdot]\) and \(E^u[\cdot]\) denote the expectation under probability measure \(P\) and \(Q^u\), respectively. The two measures are linked via the complex-valued Radon–Nikodym derivative

(7)\[\begin{equation} M_{t, u} = \frac{d Q^u}{d P} = \exp{\left(i u y_t + \tau_t \phi_{x_1, u}\right)} = \exp{\left(i u y_t + \phi_{x_1, u}\int_0^t \lambda_s ds\right)} \end{equation}\]

Affine definition

In order to obtain analytically tractable models we need to impose some restriction on the stochastic intensity process. An affine intensity process takes the general form

(8)\[\begin{equation} v_t = r_0 + r z_t \end{equation}\]

where \(r_0\) and \(r_1\) are contants and \({\bf z}_t\) is a Markov process called the state process. When the intensity process is affine, the Laplace transform takes the following form.

(9)\[\begin{equation} {\mathbb L}_{\tau_t}\left(z\right) = {\mathbb E}\left[e^{- z \tau_t}\right] = e^{-a_{u, t} - b_{u, t} z_0} \end{equation}\]

where coefficients \(a\) and \(b\) satisfy Riccati ODEs, which can be solved numerically and, in some cases, analytically.