Longstaff-Schwartz algorithm is a Monte Carlo method that is used for pricing American style options. The American option is an option contract that allows its holder to exercise the option either on the expiry date or at any time before it. This early exercise feature makes them more complicated to price in comparison to European options that can only be exercised on the expiry date.
The Longstaff-Schwartz algorithm addresses this issue by using a least-squares regression approach to approximate the continuation value of the option. It attempts to determine the optimal exercise strategy for the option holder at any given time before the expiry date, thus providing an option pricing estimate. The algorithm mainly consists of the following steps:
1. Generate a path of stock prices: Simulate a large number of sample paths for the underlying stock price using a stochastic process like the Geometric Brownian Motion. For a given time step ti, the stock price S(ti) can be computed as:
$$S(t_{i+1}) = S(t_i)e^{(\mu - \frac{\sigma^2}{2})(t_{i+1}-t_i) + \sigma\sqrt{t_{i+1}-t_i}Z},$$
where μ is the mean return, σ is the volatility, and Z is a standard normal random variable.
2. Calculate the cashflows: For each path, compute the cashflows resulting from exercising the option early at various time steps, using the option’s payoff function. For instance, for an American put option with a strike price K, the cashflow at time ti is given by:
cashflow(S(ti)) = max (K − S(ti), 0).
3. Estimate the continuation values: Beginning at the penultimate exercise opportunity (i.e., the second-to-last time step), regress the discounted future cashflows against the corresponding stock prices at that time step, using a polynomial basis, to obtain a continuation value function. The polynomial can be in the form:
$$\text{continuation}(S) = \sum_{k=0}^m a_k S^k,$$
where the ak’s are the polynomial coefficients.
4. Determine optimal exercise: Compare the continuation value with the immediate exercise value for each time step, and determine whether exercising the option or waiting would be more beneficial for the option holder. If the immediate exercise value is greater, set the cashflow at that time equal to the immediate exercise value and update the option value.
5. Work backward: Repeat the steps 3 and 4 for each previous time step until an optimal exercise strategy is found for every possible decision point in the time grid.
6. Calculate option value: Finally, take the average of the discounted cashflows from all paths at the first time step, and discount it back to today’s value to obtain the option price.
The Longstaff-Schwartz algorithm provides an efficient way to price American options with the flexibility of the Monte Carlo approach, but it can also be adapted for other types of options that feature early exercise or path dependency. Despite this, it is important to note that the accuracy of the algorithm depends on the number of simulation paths, the choice of regression basis functions, and the time discretization employed. Some improvements such as using control variates or conducting principal component analysis on the basis functions can help achieve better results.