Exercise:

Valuing a European Put Option Using a Binomial Tree

Consider a stock that is currently priced at $$100.$ Over the next two periods $($each period could represent, for example, three months$),$ the stock price can either go up by $)$ or go down by $)=(0\mathrm{.}8$ in each period. The risk$-$free interest rate is $5\%$ per annum. You are tasked with valuing a European put option on this stock with a strike price of $$100$ and a maturity that coincides with the second period.

Construct the Binomial Tree

Determine the stock prices at each node for the two periods.

Calculate the payoff of the put option at each terminal node.

Backward Induction

Use the risk$-$neutral probabilities to value the option at each node, starting from the terminal nodes and working backward to the initial node.

Calculate the risk$-$neutral probability $p,r$ is the risk$-$free rate, and $t$ is the time per period.

Option Valuation

Derive the value of the European put option at the initial node using the discounted expected payoff under the risk$-$neutral measure.

Give a Financial Interpretation.