Chapter 2: FAQs 143
Brownian motion. The model for equities, for example,
is often taken to be
dS=μSdt+σSdX+(J−1)Sdq.dqis as defined above, with intensityλ,J−1isthejump
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
∂V
∂t+^12 σ^2 S^2∂^2 V
∂S^2+rS∂V
∂S−rV+λE[V(JS,t)−V(S,t)]−λ∂V
∂SSE[J−1]= 0.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 ofJis Normally distributed. If