Question

Suppose European call prices are given by

Strike 34 36 37

Call premium 5 2.1 1

(a) Identify all of the no-arbitrage violations by the above prices (Explain why).

(b) Give an arbitrage portfolio that can be used to take advantage of the above call premiums.

(c) Show that the portfolio you have written in your answer to (b) is an arbitrage portfolio.

Answer 1

a) Let C1 , C2 and C3 be equal to 5,2.1 and 1 for strike price K1=34, K2=36 and K3=37

As C2-C1= 5-2.1 = 2.9 > K2-K1 = 36-34 =2

There is a violation of arbitrage in the option prices of the pair C1 and C2

Again , since  C3-C2= 2.1-1 = 1.1 > K3-K2 = 37-36 =1

There is a violation of arbitrage in the option prices of the pair C2 and C3

Again , since  C3-C1= 5-1 = 4 > K3-K1 = 37-34 =3

There is a violation of arbitrage in the option prices of the pair C1 and C3

b) Arbitrage portfolio of C1 and C2

Long Option with Price C2 and strike K2 and Short option with price C1 and strike K1

i.e. Long 36 Call and short 34 Call to get an amount = 5-2.1 = 2.9

At maturity , if Price P<34 , both options are worthless , so there is a profit of 2.9

If 34< P<36 , the 34 Call will be exercised, you get 34. adding 2.9 earned above, buy the stock from market at P and deliver. The profit is (36.9-P), This is positive as    34< P<36 . So, there is still an arbitrage profit

If P>36 , both the Call options will be exercised, buy the stock using Call at 36 and deliver to get 34.  The profit is (2.9+34 -36) = 0.9, So, there is still an arbitrage profit

So, in all cases, there can be arbitrage profit

Similarly , it can be shown that the portfolios of

1. Long 37 Call and short 36 call and

2. Long 37 call and short 34 call

are both portfolios which can earn arbitrage profit.

c) The portfolio formed above is an arbitrage portfolio for reason explained above. The portfolio earns a profit in all possible future scenarios without investing any money, hence it is an arbitrage portfolio.