We model the feedback effect of delta hedging for the spot market volatility of the forex market (dollar-yen and dollar-euro) using an economy of two types of traders, an option market maker (OMM) and an option market taker (OMT), whose exposures reflect the total outstanding positions of all option traders in the market. A different hedge ratio of the OMM and OMT leads to a net delta hedge activity that introduces market friction and feedback effects. This friction is represented by a simple linear permanent impact model for the net delta hedge volumes that are executed in the spot market. This approach allows us to derive the dependence of the spot market volatility on the gamma exposure of the trader that hedges a larger share of her delta exposure and on the market impact of the delta hedge transactions. We reconstruct the aggregated OMM's gamma exposure by using publicly available DTCC trade repository data and find that it is negative, as expected: the OMT usually buys options with either a view on the spot price or with the desire to hedge other positions and, thus, is net long on options. As the OMM provides liquidity as a service to the market, their position is reversed compared with the OMT. Our regressions show a high goodness of fit, a highly significant parameter for the gamma exposure of the OMM and, as expected, that the volatility is increased by the OMM's short gamma exposure. Quantitatively, a negative gamma exposure of the OMM of approximately -1000 billion USD (which is around what we observe from our reconstructed OMM data) leads to an absolute increase in volatility of 0.7% in EURUSD and 0.9% in USDJPY. If we assume that the hedge ratios in the two markets are the same, the difference can be directly explained by the higher market impact of a transaction in the USDJPY spot market compared to the EURUSD spot market, as the liquidity of the EURUSD spot market is higher than that of the USDJPY spot market. Our results are in line with and empirically confirm previous theoretical work on the feedback effect of delta hedging strategies on spot market volatility.