Particle Method For LSV Calibration

A local-stochastic volatility (LSV) model combines a stochastic volatility process, which lets volatility itself move randomly and captures the fat tails and skew dynamics options markets show, with a local volatility "leverage function" layered on top that forces the model to exactly reproduce every observed vanilla option price. The hard part is finding that leverage function, because it depends on a conditional expectation — the expected variance given the stock price at each point in time — that has no clean closed form once stochastic volatility is added to the mix.

The particle method solves this by simulating a large population of price paths ("particles") forward through time in small steps. At each time step, the paths are grouped by their current stock price, and the leverage function needed to match the market's local volatility surface at that price and time is estimated directly from the paths that landed there — effectively computing the needed conditional expectation empirically from the simulated population rather than analytically. That estimated leverage value is then applied to move all the particles forward one more step, and the process repeats, so the leverage function is built up step-by-step, consistently with the paths the model itself is generating.

The appeal is that it calibrates an LSV model to match the entire market-quoted volatility surface essentially exactly, which matters for pricing and hedging exotic options whose value depends on the volatility smile's shape, not just its at-the-money level.