Please help me answer the following problems step by step

Question 2.18 Near market closing time on a given day, you lose access to stock prices, but some European call and put prices for a stock are available as follows: Strike Price Call Price Put Price $45 $12 $4 $55 $7 $60 $4 $12 All 6 options have the same expiration date. The risk-free interest rate is zero. After reviewing the information above, Jill tells Sabrina and Kelly that one could use the following zero-cost portfolio to obtain arbitrage profit: Short one put option with strike price 45; long 3 put options with strike price 55; lend $1; and short some number of put options with strike price 60. Sabrina claims that the following zero-cost portfolio can produce arbitrage profit: Long one call option with strike price 45; short 3 call options with strike price 55; lend $1; and long some number of call options with strike price 60. Kelly claims that the following zero-cost portfolio can produce arbitrage profit: Long 2 calls and short 2 puts with strike price 60; long 1 call and short 1 put with strike price 45; lend $2; and short some calls and long the same number of puts with strike price 55. Which of the following statements is correct? A Only Jill is correct. B Only Sabrina is correct. C Only Kelly is correct. D Only Sabrina and Kelly are correct. E None of them is correct. ActuarialBrew.com 2015 Page 2.03 Exam MFE/3F Questions Chapter 2- Comparing Options Question 2.19 You are given: (i) C(K,T) denotes the current price of a K-strike T-year European call option on a nondividend-paying stock. (ii) P(K,T) denotes the current price of a K-strike T-year European put option on the same stock. (iii) S denotes the current price of the stock. (iv) The continuously compounded risk-free interest rate is r, and r > 0. Which of the following is (are) correct? I O S P(85, T) - P(80, T) 5 5e FT II 75e-FT s P(75, T) - C(80,T) + SS 80e-TT III 80e- s P(75,T) - C(80,T) + Ss 85 A I only B II only C III only D I and II only E I and III onlyProblem 1 (5 points) : Consider the following discrete-time Markov chains. 0.4 0.4 0.6 1 0.2 5 0.2 0.4 0.4 0.2 0.4 1 0.5 0.5 0.2 3 1 2 0.6 0.3 1 0.2 0.4 0.1 (a) (b) Figure 1: DTMCs for Problem 1 For each of them answer the following questions: 1. Is the chain irreducible? 2. How many state classes does it have, and what are the states in each class (transient, positive recurrent or null-recurrent)? 3. Is the chain time-reversible? 4. If we start at state 1, find the limiting probability of lim, Ply the probability that we are at state i after a large number of steps) (if the chain is ergodic, this is the stationary probability mi-)Problem 2 (5 points): Jobs arrive into a system consisting of two servers, according to a Poisson process with rate A jobs/sec. Job sizes are exponentially distributed. A job arriving is sent to server 1 with probability p, which can serve jobs with rate m jobs/sec. Otherwise, it is sent to server 2, which can serve jobs with rate po jobs/sec. Each server has its own queue, and can fit up to K total jobs (i.e. 1 in service and K - 1 waiting). Jobs arriving to a full server get rejected. Answer the following questions: 1. Is the condition A