Question

A stock is currently trading at a price of 22. You observe the following prices for
European put options on the stock (the strikes are in parentheses): C(20) =3.35
and.C(22) = 1.95. Given this information, you can conclude that the minimum
price of the 24-strike call consistent with no-arbitrage is

Answer 1

Sol:

Let the price of an option at strike K be given by V(k). To say that the price is convex in the strike for all K>0 and S>0. Let’s assume that the opposite is true, i.e. that there exist tradable option contracts expiring on the same date such that

                          V(K – s) + V(K + s) < 2V(K)

Therefore we have to buy K+s and one at K-s, and purchase by selling two o the options at K

(Which we can do, because the two options struck at K are at least as expensive as the other two combined)

At expiry the price of the stock is S, and my total payout is

                        P = (S – (K-s)) + (S – (K+s) – 2(S-k)

From the given problem C (20 - s) +C (24 - s) < C (22 - s)

                           = (20 - 3.35) + (24 – s) – 2(22 - 1.95)

                           = 16.65 + 24 – s – 40.10

Therefore ‘s’    = 0.55

So we have the possibility of making aa proit, but no possibility of making a loss – which is an arbitrate. Since no arbitrage exist, the option price must be convex in the strike price.

We can conclude the minimum price of the 24 – strike call with no – arbitrage is at 0.55.

Thank you.