Consider the average asset A defined in (9.1). (a) Find dAY (t) in the geometric Poisson process...

Consider the
average asset A defined in (9.1). (a) Find dAY (t) in the geometric Poisson
process model. (b) Let u Y (t, XY (t), AY (t)) be the price of the Asian option
with payoff f Y (XY (T ), AY (T )). Using the Ito’s formula for jumps (with 2
spatial variables), find the integro-differential for u Y (t, x, y). Show that
u Y (t, x, y) = 1, u Y (t, x, y) = x, and u Y (t, x, y) = y are solutions of
this . Also check that when the contract does not depend on A, the price
function u Y (t, x, y) does not depend on the variable x, and the
integro-differential simplifies to (10.44). (c) Find dAX (t) and the
integro-differential for u X(t, x, y). Show that u X(t, x, y) = 1, u X(t, x,
y) = x, and u X(t, x, y) = y are solutions of this . Note that when the payoff
does not depend on the asset Y , the integro-differential does not depend on
the variable x, and the simplifies. The price processes YA and XA do not
preserve the stationarity property, and thus the for u A cannot be written in
terms of a compensator ν A as the compensator does not exist.