The current price of Hatha Corp. (Symbol: HACK) stock is $10. The current date is t=0, and future times are t=1, t=2, and t=3.  In each of the next three periods, HACK stock can increase by 20% or fall by 10%. HACK stock pays no dividends, and the risk-free interest rate is 4%. Perfect capital markets apply in the setting of this question.

 

(a)    Draw the binomial tree that represents the evolution of HACK stock price. Using the binomial price model, determine the risk-neutral probability of each move at each of the nodes in the tree.

 

(b)    Using the binomial pricing model, determine the arbitrage-free value of a European call option with strike price $12 and expiration date T=3 at each node of the tree you drew in part (a). Also determine the replicating portfolio for this call option at each node of the tree.

 

(c)     Using the binomial pricing model, determine the arbitrage-free value of a European put option with strike price $12 and expiration date T=3 at each node of the tree you drew in part (a). Also determine the replicating portfolio for this put option at each node of the tree.

 

(d)    Verify that put-call parity holds at the initial node of the tree for the options you priced in items (b) and (c).

A bullish call spread on HACK stock consists of a long position on a call option on HACK with strike price $8 and a short position on a call option on HACK with strike price $12. Both options are European, and both expire at T=3.

 (e)    Using the binomial pricing model, determine the arbitrage-free value of the bullish call spread at each node of the tree you drew in part (a).