Calibration of binomial model: Suppose the distribution of St is given to be lognormal:ln(StS) N(t; 2t);where = 0:04 (4% per annum), = 0:20 (20% per annum), and S = 60. Suppose youwished to approximate this distribution of the stock price three months from now usinga binomial tree with n = 100 steps. What values would you use for the parameters ofthe binomial model?2. Pricing options on binomial tree: Consider a two–period binomial example where theunderlying asset’s price movements are modeled over the next two months, each periodcorresponding to one month. The current level of the underlying is S = 100, and thesize of up– and down–moves are u = 1:05 and d = 0:95, respectively. The risk–freesimple interest rate is 0.5% per month, i.e., ¯r = 1:005.(a) Compute the risk-neutral probabilities for the above binomial example.(b) Consider a two-period European–style put option at a strike price of 100. Computeits payoffs at the end of the two periods.(c) Compute the value of the two-period European–style put option at the two nodesat time 1.(d) Compute the hedge ratios of the put option at these two nodes at time 1 andexplain the hedging strategies they imply.(e) Compute the price of the put and the hedge ratio at time 0 (today). Explain thechange in hedging strategy from time 0 to the two nodes at time 1.(f) Repeat the exercise for a two–period American–style put option.(g) Calculate using the put–call parity (do not calculate from the binomial tree) thevalue and the hedge ratios at t=0 and at t=1 nodes for a two–period Europeancall at a strike price of 100. (Hint: If you need to buy shares to replicate a put,how many shares do you need to buy to replicate a call?)1(h) Now, consider an exotic option called a “U–choose” option which gives the holderthe right to decide at a later date, in our case at time 1, whether he/she would likethe option to be a put or a call. In other words, the U–choose option is purchasedat time 0 and, at time 1, the buyer decides whether the option will be a call ora put depending on whether the price went up or down between time 0 and time1. Calculate the prices for the put and call options maturing at time 2 at the twonodes at time 1 for a strike price of 100.(i) At the up– and the down–node the holder of the option decides on the optimalchoice, that is, does he/she want the option to be a call or a put? Compare theoption values at the two nodes at time 1 to make this choice. Also, determine thehedge ratios at the two nodes.(j) Compute the price and hedge ratio of the U–Choose option at time 0.(k) Next, consider an “Asian option” which is a European–style option whose payoffis based on the average of the prices of the underlying asset in the two periods.This is a particular case of a whole class of “path-dependent” exotic options,which are options, whose payoffs depend on the whole path of prices rather thanjust the price on the expiration date. What are the payoffs and what is the priceof an Asian call option at a strike price of 100? Pay careful attention to whetherthe tree should be recombining or not.3. Executive stock options. It is March 1, 2004 and you have received a job offer froma publicly traded fund manager (stock symbol: ARB) to start in six months. Thecompany needs good people, so they are willing to leave the offer open until then.The compensation package includes 1,000 call options on ARB whose features arediscussed below. (Ignore any issues with non-transferability, etc. Assume these areordinary European options.)Denote by t = 0; 1; 2; 3 the dates March 1, 2004; Sept 1, 2004; March 1, 2005; Sept 1,2005.Assume that the company stock price today (t = 0) is S0 = 100, and that every sixmonths the stock moves up or down by 25% with equal probability. Assume the simplesemi-annual interest rate is 4%. ARB will not pay any dividends.(a) If the options are issued at-the-money on the day the offer expires (t = 1), so thatK will be S1, and expire one year later (t = 3), what is the value today of theiroffer.(b) If, instead, the strike price were fixed today, K = S0, expiring at t = 2, What isthe value of the package?2(c) Suppose ARB gives you the choice between the options described in (a) and asigning bonus of $15,000 to be paid on the date of your acceptance. Moreover,THAT choice also doesn’t have to be made until t = 1. What is the value todayof their offer?(d) Forget about (c). Suppose that the company modifies its offer in (b) by promisingto re-set the strike of the options at t = 1 to be at-the-money in case stock goesdown between now and then. What is this offer worth?3
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