# How to show arbitrage when a European option price is greater than the no-arbitrage price?

My example is:

• Current price = 20,
• If it goes up it'll be worth 22, if it goes down it will be worth 18
• risk free rate: 12%, time = 3 months
• Strike = 21
• call option is worth 0.633

I know that if the call option value is less than 0.633 then there's an opportunity for arbitrage but what about if the option is being sold in the market at 0.7? What strategy is used in this case?

• Maybe add more context as to where you're having trouble. You did you show the arbitrage in the case that the option price is less than $0.663?$ The case where it's greater is similar only of course you sell the option rather than buying it. Sep 30, 2017 at 23:40
• If the option is overpriced you sell it and hedge that short position buying and selling the underlying. If the option is underpriced you buy it and hedge the long position. By hedging I am referring to the usual Black-Scholes delta hedging, you can find a discussion about it e.g. in Shreve Financial calculus part I. Mar 29 at 5:53

$$C = P+S-D \cdot K$$
So if instead $C > P+S-D \cdot K$, then the arbitrage profit should be $C - (P+S-D \cdot K)$.
So what do you do with the call option? Long? Short? I guess it's the opposite position of what you used when $C < P+S-D \cdot K$