You Have Three European 3-Month Xyz Calls with Exercise Prices 100,120 and 130. the Calls Are at $8, 5 and 3, Respectively. Do We See Any Arbitrage Opportunity?

1. You have three European 3-month XYZ calls with exercise prices 100,120 and 130. The calls are at $8, 5 and 3, respectively. Do we see any arbitrage opportunity? Justify your answer.

Let 100C1@$8, 120C2@$5, and 130C3@$3.

Since these three call options are on the same underlying asset, with the same expiration date and different exercise prices (100<120<130), no arbitrage opportunity can be made if:

λC1+ (1-λ) C3-C2>0     (λ= (K3-K2) / (K3-K1))

 In our case, λ=(130-120)/(130-100)=1/3. 

λC1+ (1-λ) C3-C2=1/3*8+2/3*3-5=(-1/3)<0. Obviously, the arbitrage bound is being violated. We can make instant profit $1 by buying this portfolio (C1-3C2+2C3), that is, buying one C1, selling three C2, and buying two C3.

   To justify my answer, I would like to plot the payoffs of the entire strategy on the graph below:

Thus, we will never lose money in the future regardless of the price of the stock.

1. You have XYZ trading at $42. European 6-month 40 calls and puts are traded at:

Assuming the risk-free rate is 0%, do you see any arbitrage opportunity? Justify your answer.

According to the Put-Call Parity, no arbitrage opportunity occurs if: C+PV(X)-S-P=0.

In our case, we can buy the call @$5.5 and sell the put @$2.75. The cost of buying the portfolio is C+PV(X)-S-P=5.5+40/(1+0%)0.5-42-2.75=0.75>0, which is not negative. Therefore, there’s no arbitrage opportunity if we buy the portfolio.

On the other hand, we can sell the call@$5 and buy the put@$3.25. The cost of selling the portfolio is –[C+PV(X)-S-P]= –[5+40/(1+0%)0.5-42-3.25]=0.25>0, which is not negative. Therefore, there’s no arbitrage opportunity if we sell the portfolio.

To conclude, we cannot find any arbitrage opportunity neither from buying nor from selling the portfolio.

1. Create a portfolio with the following payoffs:

   We can achieve the payoffs by creating two spreads. Let us assume two calls and two puts. The exercise price of C1 and P2 is 30, and that of C2 and P2 is 60. Consider a portfolio: mC1+nC2+aP1+bP2. The payoffs table of the portfolio should be:

In order to make a(30-St)+b(60-St)=30, a should equal to -1, b should equal to 1;

In order to make m(St-30)+b(60-St)=St, m should equal to 2.

In order to make m(St-30)+n(St-60)=60, n should equal to -2

   As a result, our portfolio should be 2C1-2C2+P2-P1.(X1=30, X2=60) That is, buying two call options with strike price 30, selling two call options with strike price 60, buying one put option with strike price60, and selling one put option with strike price 30.

1. Show that options are riskier than the underlying asset.

η is a measurement of an option’s risk relative to the risk of an underlying asset.

The call’s η is:  ηC=(δC/C)/( δS/S)= (δC/δS)(S/C)

We can substitute the call option formula to SN(d1)-Xe-rtN(d2).