Can anybody break down put call parity for futures, as used with the black model?
Perhaps break it down in equivalent terms of S+P = C+ X/1+r^n…
it’s the same as BSM for options, but we replace the underlying S as the Futures price, F
c = F - X with F and X PV’d at erT leads to c = Fe-rT - Xe-rT
then we add the cumulative normal risk-neutral probabilities of N(d1) to the underlying similar to BSM and N(d2) to the theoretical zero coupon bond(strike price)
c = Fe-rTN(d1) - Xe-rTN(d2) and this can be rewritten as c = e-rT [FN(d1) - XNd(2)]
with p, we have p = X- F and then again, we pv it at erT
p = Xe-rT - Fe-rT but the tricky one is that the normal probability distribution for puts for the underlying and the strike price is minus N(-d1) and N(-d2) respectively. so
p = Xe-rTN(-d2) - Fe-rTN(-d1) which can be written as p = e-rT [XN(-d2) - FN(-d1)]
Thanks Edbert