If you’ve ever looked at options prices, you might have noticed something odd: the volatility you get from a Black‑Scholes formula is not the same for every strike. Deep out‑of‑the‑money puts often have a higher implied volatility than at‑the‑money calls, making a “smile” or “smirk” shape. This chapter explains why that shape appears, why it matters, and how we can model it.