Why Equality Can Still Create Arbitrage in a Call Bound

Summary

The document asks why a European call’s value must be strictly greater than the discounted-strike intrinsic bound in an arbitrage argument, and what trade would exploit equality. The accepted answer explains arbitrage in terms of a portfolio that cannot lose and has a positive probability of gaining; guaranteed profit is not required.

At equality, the suggested position is to buy the call and short the corresponding intrinsic-value position, creating a zero-cost portfolio under the stated bound. If the underlying has a chance of crossing the strike, this position can produce a positive payoff in some outcomes while avoiding a loss, which is enough for arbitrage under the assumptions. The explanation is brief and relies on that chance of crossing; it does not discuss market frictions, constraints on shorting, or other conditions that could affect whether the trade is practically available.

Key ideas

• An arbitrage can qualify when losses are impossible and gains have positive probability.
• Equality at the stated call-price bound permits a zero-cost long-call and short-intrinsic position.
• The argument assumes the underlying has a chance to cross the strike.
• Trading frictions and portfolio constraints are not addressed.

Tags

# Why is this inequality strict for arbitrage argument for European call?


# Why is this inequality strict for arbitrage argument for European call?












in the notes about arbitrage arguments I am reading, I notice the statement

> We can also see that $$C^E_t>(S_t-K\mathrm{e}^{-r(T-t)})^+$$ Notice that the inequality holds STRICTLY!

I don't particularly understand why the inequality must be strict. What arbitrage can occur when equality occurs? What exactly should I be containing in my portfolio to replicate this?

## Answer by dm63 (score 5, accepted)

https://quant.stackexchange.com/a/71479

It is because to show the existence of arbitrage, it suffices to show that there is no chance of losing money,and a positive chance of making money. Arbitrage does not imply you are certain to make money. Thus, equality in your equation implies that we can create for zero cost a portfolio long the option/ short the intrinsic, which will have a positive chance of making money as long as the underlying stock has a chance of crossing the strike.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.