Question

Consider a two-step binomial tree for a European put option on a non-dividend paying stock “XY”. The current price of stock “XY” is $60. Over each of the next two 6-month periods the stock price is expected to go up by 10% or down by 10%. The risk-free rate of interest is 8% per annum with continuous compounding. The European put option will expire in 1 year and has an exercise price of $55.

a) Calculate the probabilities that the stock price goes up and down in the risk neutral world. [4 marks]

b) Calculate the stock price at each node of the binomial tree. [7 marks]

c) Use the binomial option pricing formula to calculate the value of the put option at each node of the tree. [12 marks]

d) Can you explain why, in your calculations of the option price, you are allowed to use the risk-free rate of interest? [7 marks]

Answer 1

d) This is because the hedging strategy will comprise the same amount of the underlying asset but a smaller cash loan, because, if the risk-free rate increases then all other things being equal the value of the call option will rise since the latter will accrue interest faster. i.e., buyer will have to pay more amount of money up front in order that the seller can finance the hedging strategy.

The situation is more complicated if the price of the underlying asset changes, as it probably will. But remember that the model we're looking at is simplified in several respects and it's not unreasonable to consider the effect of a change in the risk-free rate in isolation.