Monte Carlo Simulations
The algorithm estimates the continuation value of an option at various time steps using simulated paths. Let \( C(t) \) denote the continuation value at time \( t \), and let \( P(t) \) represent the immediate exercise payoff. The decision rule for early exercise is given by:
Monte Carlo simulations generate multiple paths of the underlying asset price, \( S_t \), which evolves according to a stochastic process:
Here, \( \mu \) is the drift rate, \( \sigma \) is the volatility, and \( W_t \) is a Wiener process representing the random component.
To estimate the continuation value \( C(t) \), the algorithm applies regression techniques. Using simulated asset price paths, it fits a regression model to approximate the expected value of the payoff:
This approach enables the calculation of the continuation value based on the observed states \( S_t \) at each time step.
The Longstaff-Schwartz algorithm offers several advantages:
- It handles the complexity of early exercise decisions in American options efficiently.
- It is flexible and can be applied to a wide range of options and derivatives.
- It eliminates the need for a closed-form solution, which is often unavailable for American options.
Despite its strengths, the algorithm has limitations and is often misunderstood:
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Misconception 1: The algorithm inherently uses neural networks.
Correction: It employs regression-based techniques, though machine learning methods may enhance advanced implementations. -
Misconception 2: The early exercise boundary is modeled as a Brownian motion.
Correction: The boundary is inferred from the simulated paths and continuation value estimates. -
Misconception 3: The algorithm relies on quantum computing.
Correction: It uses classical computational methods, with no dependence on quantum principles.
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