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Does a Flat Book Mean the Market Maker Has No Risk Exposure?

No, a flat book primarily means the market maker has minimal directional (Delta) risk. They are still exposed to other risks, most notably volatility risk (Vega), Gamma risk, time decay risk (Theta), and execution risk (slippage, counterparty risk).

A flat book is not a risk-free position.

How Are Vega and Gamma Used Together in a Portfolio’s Risk Analysis?
Does a Delta-Neutral Portfolio Eliminate All Risk?
What Are the Limitations of Relying Solely on Delta for Hedging?
How Does a ‘Greeks’ (Delta, Gamma, Vega, Theta, Rho) Measure Option Price Sensitivity?