Question

Suppose f and g are prices of traded securities that depend on the same source of uncertainty. Their evolution is governed by the following geometric Brownian motions:
df = αfdt + βfdz
dg = γgdt + δgdz
Let P be the relative price of g in terms of f:
P == f/g

(a) (10 points) Find the expression for dP [Hint: Proceed in the following steps.
(1) Define x == In(P) and use the Lognormal Property (or Ito's Lemma if you prefer) to find dx. (2) Note that P = e^x: and use Ito's lemma to find dy = dP.

(b) (5 points) What restriction on the parameters α, β, γ, and δ would make P a martingale? Suppose f and g are prices of traded securities that depend on the same source of uncertainty. Their evolution is governed by the following geometric Brownian motions:
df = αfdt + βfdz
dg = γgdt + δgdz
Let P be the relative price of g in terms of f:
P == f/g

(a) (10 points) Find the expression for dP [Hint: Proceed in the following steps.
(1) Define x == In(P) and use the Lognormal Property (or Ito's Lemma if you prefer) to find dx. (2) Note that P = e^x: and use Ito's lemma to find dy = dP.

(b) (5 points) What restriction on the parameters α, β, γ, and δ would make P a martingale? @Financ