New formula predicts future market trends with stochastic interest rates.
The article extends the Dupire formula to handle cases with stochastic interest rates and local volatility. It provides formulas for different scenarios, like when short rates are stochastic or when there are multiple stochastic local volatility terms. The formulas can be used to calculate and calibrate models with stochastic local volatility, even though they are implicit. The research helps in pricing financial instruments accurately in various market conditions.