1. 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. ) Comment briefly on your answers to (a) (i) and (ii). In particular, can the risk- neutral probabilities with respect to the ZCB and money market account ever differ? b EXERCISES 89 (c) By assuming no-arbitrage (thus C (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. 1. 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. ) Comment briefly on your answers to (a) (i) and (ii). In particular, can the risk- neutral probabilities with respect to the ZCB and money market account ever differ? b EXERCISES 89 (c) By assuming no-arbitrage (thus C (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