Question 1 Consider a non-dividend paying stock that is currently traded at a price of S0. Assume that the volatility of the stock is and consider the Cox, Ross and Rubinstein model for matching volatility given by u=etd=et Assume that the risk-free interest rate is r with continuous compounding. a) Explain the "no-arbitrage" and the "risk-neutral valuation" approaches for pricing European options using a one-step binomial tree. Show that the two approaches lead to the same valuation of the option. (30 marks) b) Explain the difference between European and American options and its implications for option pricing. For a one-step binomial tree, present the condition under which it is optimal to exercise American options early. Discuss both call and put options. For S0=20,r=5%,K=26,=40% and time to expiration of t=0.5 a year, determine whether early exercise is optimal. (30 marks) c) Critically discuss the validity of the statement: "The existence of volatility smiles or smirks is an indication of the limitations of option pricing models based on arbitrage. In the presence of volatility smiles arbitrageurs are bound to lose money." Assume that S0=20 and r=0% and consider a one step binomial tree with t=0.5. Assume also that six-month European call options with strike prices of 20 and $21 are traded at premiums of 2.11 and 2.38, respectively. Derive the implied volatilities of these two options. Show your calculations. (40 marks)