120 Frequently Asked Questions In Quantitative Finance
This simply means the probability that the random
variableylies betweenaandbat timet′in the future,
given that it started out with valueyat timet. You can
think ofyandtas being current or starting values with
y′andt′being future values.
The transition probability density functionp(y,t;y′,t′)
satisfies two equations, one involving derivatives with
respect to the future state and time (y′andt′)and
called the forward equation, and the other involving
derivatives with respect to the current state and time
(yandt) and called the backward equation. These two
equations are parabolic partial differential equations not
dissimilar to the Black–Scholes equation.
The forward equation Also known as theFokker–Planckor
forward Kolmogorov equationthis is
∂p
∂t′
=^12
∂^2
∂y′^2
(B(y′,t′)^2 p)−
∂
∂y′
(A(y′,t′)p).
This forward parabolic partial differential equation
requires initial conditions at timetandtobesolved
fort′>t.
Example: An important example is that of the distri-
bution of equity prices in the future. If we have the
random walk
dS=μSdt+σSdX
then the forward equation becomes
∂p
∂t′
=^12
∂^2
∂S′^2
(σ^2 S
′ 2 p
)−
∂
∂S′
(μS′p).
A special solution of this representing a variable that
begins with certainty with valueSat timetis
p(S,t;S′,t′)
=
1
σS′
√
2 π(t′−t)
e
−
(
ln(S/S′)+(μ−^12 σ^2 )(t′−t)
) 2
/ 2 σ^2 (t′−t)
.
This is plotted as a function of bothS′andt′below.
