What is a Poisson Process and What are its Uses in Finance?

Short answer

The Poisson process is a model for a discontinuous random variable. Time is continuous, but the variable is discrete. The variable can represent a 'jump' in a quantity or the occurrence of an 'event.'

Example

The Poisson process is used to model radioactive decay. In finance it can be used to model default or bankruptcy, or to model jumps in stock prices.

Long answer

The most important stochastic process in quantitative finance is Brownian Motion (the Wiener process) used to model continuous asset paths. The next most useful stochastic process is the Poisson process. It is used to model discontinuous jumps in an asset price or to model events such as bankruptcy.

A Poisson process dq is defined by the limit as dt goes to zero of

There is therefore a probability X dt of a jump in q in the time step dt. The parameter X is called the intensity of the Poisson process, it is a parameter, possibly time dependent, or dependent on other variables in a model, that represents the likelihood of the jump.

When a model has both a Wiener process dX term and a Poisson process dq term it is called a jump-diffusion model.

If the Poisson process is used to model events, such as the arrival of buses at a bus-stop, then we can answer questions about the number of buses arriving with a certain time using the following result:

This is the probability of exactly n buses having arrived (or there having been n asset price jumps) in a time t.

References and Further Reading

Ross, SM 1995 Stochastic Processes. John Wiley & Sons Ltd

What is a Jump-Diffusion Model and How does it Affect Option Values?

Short answer

Jump-diffusion models combine the continuous Brownian motion seen in Black-Scholes models (the diffusion) with prices that are allowed to jump discontinuously. The timing of the jump is usually random, and this is represented by a Poisson process. The size of the jump can also be random. As you increase the frequency of the jumps (all other parameters remaining the same), the values of calls and puts increase. The prices of binaries, and other options, can go either up or down.

Example

A stock follows a lognormal random walk. Every month you roll a dice. If you roll a one then the stock price jumps discontinuously. The size of this jump is decided by a random number you draw from a hat. (This is not a great example because the Poisson process is a continuous process, not a monthly event.)

Long answer

A Poisson process can be written as dq where dq is the jump in a random variable q during time t to t + dt. dq is 0 with probability 1 - X dt and 1 with probability X dt. Note how the probability of a jump scales with the time period over which the jump may happen, dt. The scale factor X is known as the intensity of the process, the larger X the more frequent the jumps.

This process can be used to model a discontinuous financial random variable, such as an equity price, volatility or an interest rate. Although there have been research papers on pure jump processes as financial models it is more usual to combine jumps with classical Brownian motion. The model

for equities, for example, is often taken to be

dq is as defined above, with intensity X, J — 1 is the jump size, usually taken to be random as well. Jump-diffusion models can do a good job of representing the real-life phenomenon of discontinuity in variables, and capturing the fat tails seen in returns data.

The model for the underlying asset results in a model for option prices. This model will be an integro-differential equation, typically, with the integral term representing the probability of the stock jumping a finite distance discontinuously. Unfortunately, markets with jumps of this nature are incomplete, meaning that options cannot be hedged to eliminate risk. In order to derive option-pricing equations one must therefore make some assumptions about risk preferences or introduce more securities with which to hedge.

Robert Merton was the first to propose jump-diffusion models. He derived the following equation for equity option values

E[] is the expectation taken over the jump size. In probability terms this equation represents the expected value of the discounted payoff. The expectation being over the risk-neutral measure for the diffusion but the real measure for the jumps.

There is a simple solution of this equation in the special case that the logarithm of J is Normally distributed. If the logarithm of J is Normally distributed with standard deviation a' and if we write

then the price of a European non-path-dependent option can be written as

In the above

and Vbs is the Black-Scholes formula for the option value in the absence of jumps. This formula can be interpreted as the sum of individual Black-Scholes values each of which assumes that there have been n jumps, and they are weighted according to the probability that there will have been n jumps before expiry.

Jump-diffusion models can do a good job of capturing steepness in volatility skews and smiles for short-dated option, something that other models, such as stochastic volatility, have difficulties in doing.

References and Further Reading

Cox, J & Ross, S 1976 Valuation of options for alternative stochastic processes. Journal of Financial Economics 3 145-166