The price of the stock S follows the one-step binomial model with S(0) = $100 and S(1)

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The price of the stock S follows the one-step binomial model with S(0) = $100 and S(1) = { Su = $105, if S goes up; Sd= $98, if S goes down }. Moreover, we have P({S(1) =Su}) =1/3. The interest rate is r= 2% and B(0,1) = 100. We are interested in pricing a European call and a European put with strike price X = 102 and exercise time isT=

(a )Verify that there are no arbitrage opportunities in the market {S, B}.

(b) Determine the replicating portfolios VC and VP, where VC, respVP, is the replicating portfolio of the European call, resp.put.

(c) Determine the premia,CE and PE, where CE, resp.PE, is the premium of the European call, resp.put. Now that you have priced the two options, you are called to determine the expected return and risk of portfolios in the market {S, B, CE},i.e., we can buy/shortsell shares S, buy/issue bonds B, and buy/write European call options written on Swith strike price X and exercise time T= 1; for simplicity we omit the European put option.

(d) Assume that your initial capital is $500. Determine the expected return and risk of the portfolios W1, W2andW3, where W1is the portfolio consisting of shares only; W2 is the portfolio consisting of European call options only.W3 is the portfolio where you split the initial capital between stock and option.

(e) For the portfolio W4 youissue the portion of the bond B whose worth is equal to CE(in other words, you borrow CE dollars) and you buy a European call option.Determine its expected value at time1.