While specific titles vary by author (with notable contributions in this field often coming from publishers like Springer or specialized academic presses), a volume dedicated to volatility and stochastic analysis generally covers three pillars:
Local Volatility Models: Developed by Bruno Dupire, these models make volatility a function of the current asset price and time, allowing for a perfect fit to the current market prices of vanilla options.Stochastic Volatility Models: Models like Heston or SABR treat volatility itself as a random process. This acknowledges that volatility is not just a function of price but an independent source of uncertainty that can "jump" or trend. Foundations of Stochastic Analysis
You need a solid grasp of calculus (partial derivatives), probability theory (martingales), and basic stochastic calculus (Ito’s lemma). If you’ve read Hull’s Options, Futures and Other Derivatives , you are ready for this book.