Question

Suppose that your stock follows the geometric Brownian motion ??=????+????. Use Ito’s Lemma to find the process followed by G = 1/S?

Answer 1

dSn=nSn−1dS+12n(n−1)Sn−2(σS)2dtdSn=nSn−1dS+12n(n−1)Sn−2(σS)2dt

L.H.S.=nSndSS+12n(n−1)Snσ2dtL.H.S.=nSndSS+12n(n−1)Snσ2dt

Dividing by SnSn,we get

dSnSn=[n(α−δ)+12n(n−1)σ2]dt+nσdZdSnSn=[n(α−δ)+12n(n−1)σ2]dt+nσdZ

Thus SnSn follows same process as SS with drift n(α−δ)+12n(n−1)σ2n(α−δ)+12n(n−1)σ2 and risk nσdZnσdZ

E[S(T)n]=S(t)ne[n(α−δ)+0.5n(n−1)σ2][T−t]