Mathematical Finance 2
The current price of a stock is $100. The risk-free interest rate is 4% per annum. Consider a one-year European call option on this stock with strike price that is also $100.
(a) Divide the one-year period into four three-month intervals, and use the recombinant tree with u = 1.25 and d =0.82. Calculate the risk-neutral probabilities (that is,calculate q). What is the option value?
(b) (MODIFIED) Suppose the market price of the option is 16. Assuming the market is truly described by the tree of part (a), there must be an arbitrage. Explain in detail (specifying all trades) how you can take advantage of the \incorrect" market price to earn a risk-free profit, supposing that the stock price follows the trajectory
100; 125; 102.5; 84.05; 105.06 (recall that it doesn't actually matter what trajectory the stock price takes, there is an arbitrage opportunity no matter which trajectory it takes, but specifying one trajectory allows you to give concrete answers