Arbitrage-free conditions for implied volatility surface by Delta

Implied volatility surface provided by Deltas and maturities (IVS-DM) is widely used in financial fields, especially in foreign exchange options market, since it can effectively describe the characteristics of the volatilities. The main purpose of this paper is to develop arbitrage-free conditions for the IVS-DM. We propose sufficient conditions for the IVS-DM to be static arbitrage-free, which are proved to be necessary under a mild assumption. To test the arbitrage opportunities in the market data, we build a practical tool, which is more accurate than existing ones such as early warning indicator tests. For illustration, arbitrage tests are conducted on a simulated IVS-DM to show the effectiveness of the conditions. The results are consistent with the tests for the implied volatility surface provided by strikes and maturities. Furthermore, empirical examinations are implemented on EURUSD and USDJPY currency options to ensure the feasibility of the proposed conditions.