# arbitrage involving a convex/concave contract

(I doubt this knowledge has any value outside the exams.) Suppose a derivative contract is written on S(T), the terminal price of a stock. Assume a bank account with 0 interest rate either for deposit or loan. At time 0, the contract can be overpriced or under-priced, each creating a real arbitrage.

Basic realities (not assumptions) ? stock price at any time is non-negative.

— If the contract is concave, like L = log S, then a stock (+ bank account) can super-replicate the contract. (Can't subreplicate). The stock's range-of-possibilities graph is a straight tangent line touching the concave curve from above at a S(T) value equal to S(0) which is typically \$1 or \$10. The super-replication portfolio should have time-0 price higher than the contract, otherwise arbitrage by selling the contract.

How about C:=(100 ? S^2) and S(0) = \$10 and C(0) = 0? Let's try {-20S, -C, +\$200} so V(t=0) = \$0 and V(t=T) = S^2 ? 20 S +100. At Termination,

If S=10, V = 0 ←global minimum

If S=0, V= 100

If S=11, V= 1

How about C:=sqrt(S)? S(0) = \$1 and C(0) = \$1? Let's try {S, +\$1, -2C}. V(t=0) = 0. V(t=T) = S + 1 – 2 sqrt(S). At termination,

If S=0, V = 1

If S=1, V= 0 ←global minimum

If S=4, V= 1

If S=9, V= 4

— If the contract is convex, like exp(S), 2^S, S^2 or 1/S, then a stock position (+ bank account) can sub-replicate the contract. (Can't super-replicate). The replication range-of-possibilities graph is a straight tangent line touching the convex from below. This sub-rep should have a time-0 price below the contract, otherwise arbitrage by buying the contract and selling the replication.