Introduction to Diffusion and Jump Diffusion Processes

Asset-Prices-Simulated-using-the-Merton-Jump-Diffusion-Geometric-Brownian-Motion-Stochastic-Process
Sample Asset price paths from a jump-diffusion model.

This post is the first part in a series of posts where we will be discussing jump diffusion models. In particular, we will first introduce diffusion and jump diffusion processes (part 1/3), then we will look at how to asses if a given set of asset returns has jumps (part 2/3). The series will be concluded by looking at what options one has when it comes to calibrating/estimating jump diffusion models (part 3/3).

I have also worked on a web application in conjunction with Turing Finance and Southern Ark. We developed the web app in order to aid in understanding how various stochastic processes behave. Any feedback on how we can improve the app are most welcome.

Stochastic-Web-App-Screenshot

For a "Top of The Bell Curve" background to stochastic processes go here.

A sample space \Omega is a set of all possible elementary outcomes \omega from a particular experiment. For example, if the experiment was tossing a coin, then \Omega = \{ H,T \} . An event is a collection of elementary outcomes for which a given statement is true. For example, getting two heads in three tosses of a coin.

A \sigma-algebra F is a set that is made up of measurable events and is closed under operations of complements and unions. This then means that \{ \Omega , F \} is a measurable space. The measure that we are interested in is the probability measure P, which satisfies the usual properties of a measure, but in addition we also have that P(\Omega) =1. The triplet  \{ \Omega, F, P \} is then called a probability space.

Note that the probability measure P used in the triplet  \{ \Omega, F, P \} is of particular importance. This is because in mathematical finance we differentiate between the real world probability measure (usually represented with P) and a risk-neutral probability measure, usually represented with a Q. In this post, we will be focusing on the real world measure. The relationship between the real world and the risk neutral measure will be explored in the subsequent posts in this series.

A random variable X is a measurable function from \Omega to the real numbers. An example of a random variable is the return from a particular share/stock at a fixed point in time.

Usually in finance, we are not only interested in what is happening at a fixed point in time, but we are also interested in what is happening as we pass through time. This then brings us to the concept of a stochastic process. A stochastic process is a collection of random variables through time. That is, if X is a stochastic process, we have that X = \{X_t, t \geq 0 \}, where X_t is a random variable, and can have a different distribution to the other random variables in X. The values that the stochastic process takes on is called the state of the process. The state space can be continuous or countable.

Also note that we are only going to be looking at continuous time stochastic processes.

Associated with a stochastic process is the concept of a filtration. A filtration is a collection of \sigma-algebra's F_t. That is, if F is a filtration generated by the stochastic process, then F=\{F_t, t \geq 0 \}. Conceptually, a  filtration F_t represents all the information we have about the process up to time t.

The class of processes commonly used in stochastic models are  Markov Processes. A stochastic process is called a Markov process if the likelihood of an event happening in the future is only depended on the current state of the process, and not on the past states that the process has gone through. That is

P(X_t \in A |F_s) = P(X_t \in A | X_s),

where A is a set of real numbers and F_s is a filtration, with t \geq s. An example of a Markov process is a diffusion process. Diffusion processes are discussed in the following section.

In addition, a time homogeneous Markov Process is a Markov process which is invariant to time. That is:

 P(X_t \in A | X_s) = P(X_{t-s} \in A | X_0).

The focus of this post is given solely to  time homogeneous Markov Processes.

A diffusion process X = \{X_t, t \geq 0 \} is a continuous time Markov process that has a continuous state space and has continuous sample paths almost surely. A diffusion process is completely defined by its first two moments.

There are at least two ways of defining the evolution of a diffusion process. One  approach is to look at the transition density f(x,y)= P(X_t =x | X_s =y) of the process. This transition density f(x,y) will describe the evolution of a diffusion process via the Fokker-Planck equation. Another approach to looking at the evolution of a diffusion process is by looking at the state space of the diffusion process  X, and describing how the state of the process changes through infinitesimal changes in time. This gives rise to the stochastic differential equation (SDE) representation of a diffusion process. In this post, we are going to focus solely on the SDE representation of the diffusion process.

If X = \{X_t, t \geq 0 \} is a time homogeneous diffusion process, then its SDE representation is given as:

dX_t = \mu(X_t)dt+ \sigma(X_t)dW_t,

where \mu(X_t) is the drift coefficient ("first moment") of the process, \sigma(X_t) is the diffusion coefficient ("second moment") of the process and W_t is a Wiener process. Note that although X has continuous sample paths almost surely, it is almost surely nowhere differentiable due to the fractal nature of W_t.

Examples of diffusion processes used in finance include Geometric Brownian Motion(GBM), Ornstein Uhlenberg process and the CIR process amongst others. In this series of posts we will be focusing  on properties of GBM.

The SDE representation for GBM under the real world measure P is given as

dX_t = X_t\mu dt+ X_t \sigma dW_t,

where \mu and \sigma are constants with \sigma \geq 0.

Of particular interest to us is the solution of the above SDE.  Given the fractal nature of W_t, the above equation cannot be solved using normal calculus only. We need a special machinery called Ito calculus in order to deal properly with the W_t term. We will look into how one solves this SDE in the second post of this series.

Jump diffusion processes are  an extension of the diffusion process defined above. This extension is via adding a jump component to the diffusion process component.

The jump component that is added to the diffusion component is a jump process. The jump processes that are normally used as a jump component  are Markov processes, such as the compound Poisson process. In this series of posts, we will only be concerning ourselves with Markov jump diffusion processes.

As with diffusion processes, the evolution of a jump diffusion process can also be represented either via its transition density and the Fokker-Planck equation, or via its SDE. The SDE of  a continuous time, time homogeneous Markov jump diffusion process X = \{X_t, t \geq 0 \} is given as:

dX_t = \mu(X_t)dt+ \sigma(X_t)dW_t +dJ_t,

where dJ_t is the jump processes. The jump process that we will be focusing on is the compound Poisson process. That is, dJ_t = d(\sum_{i=0}^{N_t}(Y_i-1)), where N_t is a Poisson process with rate \lambda(X_t) and Y_i is the ith jump size.

Examples of jump diffusion processes used in the modelling of stock returns include the classic Merton jump model and Kou's jump model. The difference between the Merton model and Kou's model is in the assumption of the distribution of the jump sizes Y_is. Merton's model assumes that the Y_is are log normally distributed, while Kou's model assumes that the Y_is have a double exponential distribution.

In this post we will be focusing on Merton's jump model. The SDE for the classic Merton's model under the real world measure P is given as :

d X_t = X_t\mu dt+ X_t\sigma dW_t +X_t d(\sum_{i=0}^{N_t}(Y_i-1)),

where N_t is a Poisson process with rate \lambda and Y_i has a log normal distribution. As can be seen, the model is just an extension of GBM described in the previous section. As with GBM, we will look into how one solves this SDE in the second post of this series.

It has been observed in practice that asset returns have heavy tails. Jump diffusion processes are able to capture these heavy tails via the inclusion of the jump component in the diffusion model. Diffusion models are unable to capture extreme events, such as market crashes, as they assign small probabilities to their occurrence, that is, they produce return distributions that have short/light tails. This then means that jump diffusion models can be used to improve on the inefficiencies of the diffusion model, albeit at a cost of increased complexity.

We will be studying more  about the properties of GBM and Merton's model and their applications in finance in the coming parts of this series of posts.

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