5) [20 points] The current price of a stock is 165.38 dollars per share (S(t) = 165.38). The interest rate is 3% per year (with continuous com- pounding). The following table shows the prices of American put options for different strike prices. These options expire three months from today. Option Strike Price K Price PA 155 6.50 160 8.52 165 12.66 170 14.90 175 15.75 Find an arbitrage opportunity. To receive full credit, explain what you would do at time t = 0 and why the strategy that you propose is indeed an arbitrage oportunity. ) (15 pts) The price of a stock is S(0) = $40. You are to price a call and put option, using a one-step binomial model. The stock is assumed to either increase to $50 or decrease to $25. Both options expire in three months, have a strike price equal to the forward price of the stock, and the risk-free rate is r = 5%. Illustrate the model, determine the risk-neutral probabilities, evaluate the price of the call and put, and verify that put-call parity is met. 5. Use your words or examples to explain why the value of a call option increases as the market interest rate increases, and why the value of a put option increases as the market interest rate decreases. (8 points) 6.6. European options can only be exercised on the expiration date, while American options can be exercised on any date before the expiration. Intuitively, American options should be more popular among investors. However, we rarely see American options being traded on today's market. Use your words to explain why most of the options on the market are European options. (8 points) 7.7. Black-Scholes model shares common intuitions with risk-neutral option pricing model (also known as the binomial option pricing model). One of the biggest underlying assumptions of risk-neutral (binomial) model is that we live in a risk-neutral world. In a risk-neutral world, all investors only demand risk-free return on all assets. Although the risk-neutral assumption is counterfactual, it is brilliant and desirable because the prices of an option estimated by risk-neutral approach are exactly the same with or without the risk-neutral assumption. Use your words to explain why that is the case, and how risk-neutral assumption greatly simplifies the calculations of risk-neutral option pricing approach. (8 points)