Sunday, August 26, 2018

How to derive Black's formula for the valuation of an option on a future?


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)KerTN(d2)

.


Where d1=ln(S0K)+(r+12σ2)TσT and d2=d1σT


I actually don't know how's possible to get the famous black formula on a forward contract:


C=erT(FN(d1)KN(d2))

.


where now d1=ln(FK)+12σ2TσT and d2=d1σT


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)+erT[vn(d)KN(d)]

.


where d=S0erTKv and v=erTσ1e2rT2r 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=erT[(FK)N(d)σTn(d)]


where now d=FKσT


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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