Question

Let s follow geometric Brownian motion with expected return µ and volatility σ. Show the process followed by G(s, t) = s(a + bt), where a and b are constants, also follows geometric Brownian motion. Identify the drift and volatility terms. Please show all work

Answer 1

S(t) follows a Geometric Brownian motion (GBM)

where e is the natural exponent

G(s,t) = S(t)*(a+b*t)

G(s,t) = S(t)*a + S(t)*b*t

........this equation follows geometric brownian motion (constant multiplied by GBM results in a GBM)

S(t)*b*t = b*S(t)*t = b*S(t)*e^(lnt)

where lnt is the natural log of t

e^(lnt) = t (mathematical rule)

b*S(t)*e^(lnt) =

This equation itself follows a Geometric Brownian motion with drift =

Hence these two equations add up to create a Geometric Brownian Motion