# Riskless Pricing of an European Option; How to calculate?

I don't know if this question could be asked/answered here but since there is a "finance" tag...I'll give it a shot.

I am asked to show that the value of an option below obtained by 1. Risk-neutral 2. Risk-less portfolio argument are the same. I managed the value of the option calculated by the risk-neutral version.

Current price of a stock is $\$40$. It is known that it either increases of decreases by 12.5% every 3 months over the next 6 month period. The risk-free interest is$8%$by continuous compounding. Verify that the values of a 6-month European put option on the stock with strike price$\$40$ obtained by the riskless portfolio and the risk-neutral arguments are the same.

Simply, my notes mention a super tiny bit of the pricing via risk-less portfolio method. And further, the example it goes through is of a binomial 1-period model, so it doesn't really help me much as I have to consider a 2-period model in this question.

If my calculations are correct, I have $\$1.87522...$as my price for the option in risk-neutral arguments. So my risk-less argument should give me the same value for the option but with the information I am given, and the scarce information found on the internet, I don't know what to do.... Can someone please tell me how this works for that part please? Thanks... • Which book are you using? To do binomial on 2 steps, just draw the tree for these two steps, and do this backwards. You know payoffs at the very end, for all (how many) possible different final stock prices. – Ilya Nov 13 '15 at 21:22 • Hi, the notes are prepared by my prof, but he said that it is based on Tomas Bjork's "Arbitrage theory in continuous time." I did that method(tree backwards) for the risk-neutral bit. I find the option values at N=2(N is the period) for each nodes and then find the martingale probabilities denoted$q_u$and$q_d$in my notes, use them to find the price at N=1, and then N=0. My question for risk"LESS" version is, how exactly does this work? I used$q_u$,$q_d$for the risk-neutral argument but what does this become in the risk-less argument? How exactly do I proceed? – Melba1993 Nov 13 '15 at 21:27 • I thought, risk-neutral method is: get martingale measure, compute the expectation. Risk-less method is: compute which amount of Stock (Delta) you need to hold in each node to hedge, assume that you do this transaction, and see how much of initial capital is needed. – Ilya Nov 13 '15 at 21:30 • For the$\Delta$method, the example in my notes with N=1 assumes that the values at the nodes$S_1(u),S_1(d)$gives$S_1(u)\Delta-\Phi(S_1(u))=S_1(d)\Delta-\Phi(S_1(d))$and then solves for$\Delta$. How do I do this for N=2? I have$3$nodes now, yes?$S_2(uu),S_2(ud),S_2(dd)$. I tried doing it separately, for$S_2(uu),S_2(ud)$and$S_2(ud),S_2(dd)$and got different$\Delta$s...and what would these 2$\Delta\$s do to calculate the final option price at N=0...? I'm sorry I am very very lost.... – Melba1993 Nov 13 '15 at 21:35 