Problem 3: Binomial Trees and Properties of Options

In this question, you need to price options with binomial trees. You will consider puts and calls on a share with spot price of $30. Strike price is $34. Furthermore, assume that over each of the next two four-month periods, the share price is expected to go up by 11% or down by 10%. The risk-free interest rate is 6% per annum with continuous compounding.

a. Use a two-step binomial tree to calculate the value of an eight-month European call option using the no-arbitrage approach. [3 marks]

b. Use a two-step binomial tree to calculate the value of an eight-month European put option using the no-arbitrage approach. [3 marks]

c. Show whether the put-call-parity holds for the European call and the European put prices you calculated in a. and b. [1 mark]

d. Use a two step-binomial tree to calculate the value of an eight-month European call option using risk-neutral valuation. [1 mark]

e. Use a two step-binomial tree to calculate the value of an eight-month European put option using risk-neutral valuation. [1 mark]

f. Verify whether the no-arbitrage approach and the risk-neutral valuation lead to the same results. [1 mark]

g. Use a two-step binomial tree to calculate the value of an eight-month American put option. [1 mark]

h. Without calculations: What is the value of an eight-month American call option with a strike price of $34? Why? [2 marks]

i. Without calculations: Consider an at-the-money European put on the same share and a timeto-maturity of 8 months. Will the price of the at-the-money put be higher or lower compared to the put price you calculated in e.? Why? [2 marks]

j. Without calculations: What would happen to the option prices you calculated in d. and e. if the interest rate drops to 4%? Why? [2 marks]

Note: When you use no-arbitrage arguments, you need to show in detail how to set up the riskless portfolios at the different nodes of the binomial tree.