Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 A. Generation of random samples.
 B. Acceleration of convergence.
 C. Longstaff-Schwartz technique.
 a. Normal Equations technique.
 D. Calculation of sensitivities.
 5 Convex Analysis.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Longstaff-Schwartz technique. his is a short summary of the technique presented in [Longstaff] .

We would like to calculate the quantity where is a stochastic process in holding all the state variables, the is some deterministic function representing the interest rate term structure, is the known payoff function depending on the path up to the moment of exercise . The functional dependence of the moment of exercise on the state variables is the subject of optimization.

Suppose the is the result of Monte-Carlo simulation of the stochastic process with being the time index, being the simulation index and being the dimension index, is the result of immediate exercise , is the discount factor between neighbor indexes .

Introduce an array and a set of functions acting .

Start with For do the following:    The answer is . We do not proceed to step k=0 because the cross sectional information collapses to a point at this step. The obtained this way value is biased high because this is a forward looking procedure. If we continue the MC simulation on the obtained strategy then we get a biased low value because the strategy is suboptimal.

The motivation for the steps above is the following. The is the value of the quantity in question at time given the information . Hence, the starting condition is obvious. The sum is used to construct the function that depends only on available information and best approximates the Y. Hence, we discount the Y from the previous step with (a), find the best approximation in (b), chose the best strategy and calculate the new Y in (c) and (d).

The step (b) may be performed using the Normal Equations technique presented in the next section.

The step (b) is unstable if the time step is small and is close to the origin. In such situation the does not contain much cross- information because all the originate from a single point and did not have time to evolve. For this reason the procedure is not effective if the exercise is immediately possible.

 a. Normal Equations technique.
 Notation. Index. Contents.