Skip to content
Interview prep / Stochastic Calculus

Constant that makes W cubed minus ctW a martingale

Question

For a standard Brownian motion WW, consider Xt=Wt3−c t WtX_t=W_t^3-c\,t\,W_t with cc a constant. Which value of cc turns XX into a martingale?

Short answer

c=3c=3. Ito's formula for f(t,w)=w3−ctwf(t,w)=w^3-ctw gives a drift of ft+12fww=−cw+3wf_t+\tfrac12f_{ww}=-cw+3w. The drift vanishes for every ww exactly when c=3c=3.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Start with a free lesson: Options is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free