An underlying asset is priced today at $100. A European call option on the asset has strike price S105. Consider a three-step binomial tree in which at each step, the price either increases 4% or decreases 4%. Assume that the continuously compounded risk-free rate is 4% per annum and the time between steps is δt = 0.25 year, so that one dollar increases in value to exp(0.04 × 0.25) dollars after one time step.
a What are the risk-neutral probabilities? Are they the same at all nodes? Explain your reasoning.
b What is the initial price of the option? Obtain the price by work- ing backwards through the nodes and also by using the binomial distribution without calculating the prices at the intermediate nodes
c How much stock and risk-free asset would the replicating portfolio hold initially? Explain your reasoning.
d How would the replicating portfolio be adjusted if the price of the stock went up on the first step? Confirm that these adjustments to the replicating portfolio are self-financing.