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Compute the risk-neutral probabilities

Assignment:

Pricing options on binomial tree: Consider a two-period binomial example where the underlying asset's price movements are modeled over the next two months, each period corresponding to one month. The current level of the underlying is S = 100, and the size of up- and down-moves are u = 1:05 and d = 0:95, respectively. The risk-free simple 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. Compute its payoffs at the end of the two periods.

(c) Compute the value of the two-period European-style put option at the two nodes at time 1.

(d) Compute the hedge ratios of the put option at these two nodes at time 1 and explain the hedging strategies they imply.

(e) Compute the price of the put and the hedge ratio at time 0 (today). Explain the change 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) the value and the hedge ratios at t=0 and at t=1 nodes for a two-period European call 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?)

(h) Now, consider an exotic option called a "U-choose" option which gives the holder the right to decide at a later date, in our case at time 1, whether he/she would like the option to be a put or a call. In other words, the U-choose option is purchased at time 0 and, at time 1, the buyer decides whether the option will be a call or a put depending on whether the price went up or down between time 0 and time 1. Calculate the prices for the put and call options maturing at time 2 at the two nodes 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 optimal choice, that is, does he/she want the option to be a call or a put? Compare the option values at the two nodes at time 1 to make this choice. Also, determine the hedge 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 payoff is 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 than just the price on the expiration date. What are the payoffs and what is the price of an Asian call option at a strike price of 100? Pay careful attention to whether the tree should be recombining or not.

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## Q : What is the present value of the future amounts

What is the present value of the following future amounts?