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)
Thanks Examstudent - but i do know how to value the things (and yes your formulae are correct). I probably wasn't clear in my question. What I don't get is how we can "deduce" the values from the B-S formula, which is what the acted qn asked. I suppose if I first calculate the value of the cash-or-nothing (which as you say is relatively easy) then I can deduce the value of the asset-or-nothing from BS. Perhaps that's as close as I'm going to get. L
