102 Frequently Asked Questions In Quantitative Finance
We can understand this (if not entirely legitimately
derive it) via Taylor series by using the rules of thumb
dXi^2 =dt and dXidXj=ρijdt.
Another extension that is often useful in finance is to
incorporate jumps in the independent variable. These
are usually modelled by a Poisson process. This isdq
suchdq=1 with probabilityλdtand is 0 with probabil-
ity 1−λdt. Returning to the single independent variable
case for simplicity, supposeysatisfies
dy=a(y,t)dt+b(y,t)dX+J(y,t)dq
wheredqis a Poisson process andJis the size of the
jump or discontinuity iny(whendq=1) then
df=
(
∂f
∂t
+a(y,t)
∂f
∂y
+^12 b(y,t)^2
∂^2 f
∂y^2
)
dt+b(y,t)
∂f
∂y
dX
+(f(y+J(y,t))−f(y,t))dq.
AndthisisItˆo in the presence of jumps.
References and Further Reading
Joshi, M 2003The Concepts and Practice of Mathematical
Finance.CUP
Neftci, S 1996An Introduction to the Mathematics of Financial
Derivatives. Academic Press
Wilmott, P 2001Paul Wilmott Introduces Quantitative Finance.
John Wiley & Sons
