I've got a question about 1976 Black Model and Bachelier model.
I know that a geometric brownian motion in the P measure dSt=μStdt+σStdWPt for a stock price St leads (after a change of measure) to the Black-Scholes formula for a Call:
C=S0N(d1)−Ke−rTN(d2)
Where d1=ln(S0K)+(r+12σ2)Tσ√T and d2=d1−σ√T
I actually don't know how's possible to get the famous black formula on a forward contract:
C=e−rT(FN(d1)−KN(d2))
where now d1=ln(FK)+12σ2Tσ√T and d2=d1−σ√T
Should I simply insert F(0,T)=S0erT in the first BS formula to get the second one?
I'm asking this because I've tried to derive the BS formula using an arithmetic brownian motion like dSt=μdt+σdWPt, and I get:
C=S0N(d)+e−rT[vn(d)−KN(d)]
where d=S0erT−Kv and v=erTσ√1−e−2rT2r and remembering that N(d) and n(d) are the CDF and PDF.
but the previous substitution F(0,T)=S0erT doesn't seems to lead to the known result C=e−rT[(F−K)N(d)−σ√Tn(d)]
where now d=F−Kσ√T
I think I could reach the equations on forward both in the geometric brownian motion and arithmetic brownian motion using the equations
dF=FσdWQt and dF=σdWQt but I don't know how justify the use of them.
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