In Q&A 4.11 iii): the answer given seems more of an intuition than a deduction to me. Anyone got a way to mathematically deduce the formulae for cash-or-nothing /asset-or-nothing prices from B-S formula?
The only way I can seem to get there is to go back to replicating portfolios.
Hi
i dont have these notes .. have you tired Hull???
1) I presume asset or nothing option has formula SN(d1)
2) I presume cash or nothing is Kexp(-rt)N(d2)
intutitively these make sense.
for the cash or nothing option, it is quite simple
payoff * pv * probability of exercise (under Q)
K * EXP (-RT) * N(d2)
from what i remember this was a Hull problem - proving risk neutral probability of exercise of N(d2)
The formal way to do it (sorry i dont know how to type math symbols) is to follow the proof of black scholes formula using integrals - ( d2 is the lower limit, standard normal etc), i remember this was detailed in the ST6 core reading, if not see this
http://www.gronnemosegaard.dk/Mortens hjemmeside/Papers/Black scholes.pdf or i am sure baxter & rennie detail it.
for these options, you are not integrating MAX (S-K,0).
for asset or nothing, your integrating S when S>K, so get SN(d1)
for cash or nothing, your integrating K when S>K. so get Kexp(-rt)N(d2)
