• Korean Journal of Mathematics
• /
• v.23 no.1
• /
• pp.81-91
• /
• 2015

This paper is about the local volatility for the price of a European quanto call option. We derive the explicit formula of the local volatility with constant foreign and domestic interest rates by adapting the methods of Dupire and Derman & Kani. Furthermore, we obtain the Dupire equation for the local volatility with stochastic interest rates.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.2
• /
• pp.507-528
• /
• 2022

We propose a path integral method to construct a time stepwise local volatility for the stock index market under Dupire's model. Our method is focused on the pricing with the Monte Carlo Method (MCM). We solve the problem of randomness of MCM by applying numerical integration. We reconstruct this task as a matrix equation. Our method provides the analytic Jacobian and Hessian required by the nonlinear optimization solver, resulting in stable and fast calculations.