Binomial tree: change of numeraire
Consider a one-step, two-state world where a stock has current price 100. After one year the stock is worth 110 with probability 0.8, and 90 with probability 0.2. One-year annually compounded interest rates are 5%.
(a) Use the fundamental theorem to find the risk-neutral probability (of the stock being worth 110) with respect to the numeraires: (i) the money market account; (ii) the ZCB with maturity 1; and (iii) the stock.
(b) Comment briefly on your answers to (a) (i) and (ii). In particular, can the riskneutral probabilities with respect to the ZCB and money market account ever differ?
(c) By assuming no-arbitrage (thus CK(m, 1)/Nm is a martingale for the appropriate numeraire and risk-neutral probability pair), price a one-year 105-strike call using the risk-neutral probabilities from (a)(i), (ii) and (iii). Verify the answers are the same.
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