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? Explain.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? Explain.

Answer 1

x = ln(P) = ln(f/g) = ln f - ln g

Then by Ito's lemma,

dx = ( αf/f dt + βf/f dz) - (γg/g dt + δg/g dz)
= (α - γ) dt + (β - δ) dz

Now I'll let you figure out how to get back to dP using Ito's lemma again.

Answer 2

JamesJ, could you please describe how to get back to dP by using Ito's lemma because I am stuck on that part. Please be detailed because I am so stuck on this little part. It would be most helpful so I would very much appreciate your effort!