This exercise deals with obtaining martingales Suppose Xt is a geometric process with drift and diffusion parameter (a) When would the ertXt be a martingale That is, when would the following equality hold E Xt Xs,s (b) More precisely, remember from the previous derivation that or, again, E Xt Xs,s ...

a e rt X t will be a martingale whenever the following condition holds e r12 ...

This exercise deals with obtaining martingales. Suppose Xt is a geometric process with drift and diffusion

Question:

This exercise deals with obtaining martingales. Suppose Xt is a geometric process with drift Î¼ and diffusion parameter Ïƒ.
(a) When would the eˆ’rtXt be a martingale? That is, when would the following equality hold:
E[Xt|Xs,s (b) More precisely, remember from the previous derivation that

or, again,
E[Xt| Xs,s Which selection of Î¼ would make eˆ’rtXt a martingale? Would Î¼ work?
(c) How about
Î¼ = r + Ïƒ2
(d) Now try:
Î¼ = r - 1/2 Ïƒ2
Each one of these selections defines a different distribution for the eˆ’rtXt.

Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...