Geometric return (Hull chapter 14)

[ancl9]

2 Formulas are being used to calculate geometric return in Hull chapter 14:

1. geometric return - mu- (sigma^s/2)
2. geometric return = (Price at T/price at t)^(1/(T-t))

For the example on study note page 60, those 2 formulas do not result in the same result.
Can someone clarify when to use which formula?

Thank you,

 

[ami44]

I can't find your formulas in my (old) edition of Hull, and I can't see the study note, but let me try to answer anyway:

Formula 1 is the mean geometric return of a geometric Brownian motion (I assume it is μ - σ^2/2). To be precise, it is the mean of the logarithm of a geometric Brownian motion defined as Xt = exp((μ - σ^2/2) * t + σ * Wt) with Wt a Wiener Process.

Formula 2 is the geometric return if you have two historical prices at the time T and t. A further difference is, that Formula 1 uses continuous compounding and formula 2 annual compounding and formula 1 has no time included i.e. the time interval is assumed to be 1

hope that helped