Could you explain to me in words (no formulas) the concept of the Least Squares Monte Carlo method to price an American style option?
To compute the price of an American option or a callable instrument in general, at each potential exercise date, one is required to compare its continuation value (discounted risk-neutral expectation of what the option would pay off if it was not exercised) to the relevant exercise value/early redemption price.
By construction, lattice and finite difference methods allow a straightforward computation of the former continuation values, since they work by backward induction (starting from the terminal condition at expiry and working backwards up to inception computing risk-neutral expectations). However, these methods suffer from the curse of dimensionality (the computational burden increases rapidly with the number of underlying assets).
At the other end of the spectrum, the standard Monte Carlo method is forward in nature: one simulates realisations of the underlying price process under the risk-neutral measure, applies the payoff function and takes the discounted expectation of those paths' payouts to obtain the option price. By construction, computing continuation values at future times is then less straightforward. One could do it with nested simulations but this wouldn't be practical.
An alternative, first proposed by Longstaff and Schwartz in a celebrated paper, consists in simulating the paths and then working backwards in time to estimate the continuation values through least squares regression over a set of so-called basis functions. At each time step, the paths' payouts are updated by comparing the estimated continuation values to the exercise values and then repeating the operation. The method has some known biases when using a finite number of simulations and basis functions but is shown to converge otherwise.