142 Frequently Asked Questions In Quantitative Finance
What is a Jump-Diffusion Model
and How Does It Affect Option Values?
Short Answer
Jump-diffusion models combine the continuous Brown-
ian 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 asdqwheredqis the
jump in a random variableqduring timettot+dt.dq
is 0 with probability 1−λdtand 1 with probabilityλdt.
Note how the probability of a jump scales with the time
period over which the jump may happen,dt. The scale
factorλis known as theintensityof the process, the
largerλthe 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
