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The market price of a security can be modelled by assuming that it will either increase by 12% or decrease by 15% each month, independently of the price movement in other months. No dividends are payable during the next two months. The continuously compounded monthly risk- free rate of interest is 1%. The current market price of the security is 127. (i) Use the binomial model to calculate the value of a two-month European put option on the security with a strike price of 125. [3] (ii) Calculate the value of a two-month American put option on the same security with the same strike price. [3] (iii) Calculate the value of a two-month American call option on the same security with the same strike price. [2] [Total 8] 2 A company share price is to be modelled using a 5-step recombining binomial tree, with each step in the tree representing one day. Each day, it is assumed that the share price: increases by 2%, or decreases by 1%. Assume that the risk-free force of interest is 6 =5.5% po and that there are 365 days in a year. No dividends are to be paid over the next five days. (i) Calculate the risk-neutral probability of an up-step on any given day. [2] (W) Calculate the fair price of a 5-day at-the-money call option on f10,000 worth of shares in this company. [5] A special option is available where the payoff after 5 days is: max $5 - K, O where $5 is the arithmetic average share price recorded at the end of each of the 5 days and K is the strike price. (ii) Calculate the fair price of the special option (strike price K =1.065, ) on $10,000 worth of shares in this company. [4] [iv) Explain whether an at-the-money special option is likely to have a higher value of vega than a standard call option. [3] [Total 14]3 In a one-step binomial tree model it is assumed that the initial share price of 260 will either increase to 285 or decrease to 250 at the end of one year. Assume that the annual risk-free force of interest is 0.05 and that no dividends are payable. (1) Calculate the price of a one-year European call option with a strike price of 275, using each of the following: (a) a replicating portfolio method (b ) risk-neutral valuation (c) a risk-free portfolio method. (ii) Repeat your calculations in (i) for a one-year European put option with a strike price of 275. (mii) Verify numerically that the put-call parity relationship holds in this case. 4 The market price of a non-dividend-paying security with current market price $ is being modelled using a one-step binomial tree in which the proportionate changes in the security price following an up- and a down-movement are denoted by u and d . The risk-free force of interest over the period is r. Show that if an option on this security has a payoff of Z, following an up-movement and a payoff of zy following a down-movement, then the option can be replicated exactly using a portfolio consisting of A securities, where AS=-", and an amount of cash, w , which you should u-d specify. 5 The increase in the price of a share over the next year is believed to have a mean of 10% and a standard deviation of 10%. (i) Determine the values of u and d for a one-step binomial tree model that are consistent with the mean and standard deviation of the return on the underlying share, assuming that the share price is twice as likely to go up than to go down. (ii) Hence calculate the value of each of the following options, given that the current share price is 250, the risk-free force of interest is 7%% per annum and dividends can be ignored: (a) a one-year European call option with a strike price of 275 (b) a one-year European put option with a strike price of 300