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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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 MATH where $X_{t}$ is a stochastic process in $R^{N}$ holding all the state variables, the $r\left( x\right) $ is some deterministic function representing the interest rate term structure, $h$ is the known payoff function depending on the path MATH up to the moment of exercise $\tau$ . The functional dependence of the moment of exercise $\tau$ on the state variables MATH is the subject of optimization.

Suppose the MATH is the result of Monte-Carlo simulation of the stochastic process $X_{t}$ with $k$ being the time index, $\omega$ being the simulation index and $n$ being the dimension index, MATH is the result of immediate exercise MATH , MATH is the discount factor between neighbor indexes MATH .

Introduce an array MATH and a set of functions MATH acting MATH .

Start with MATH

For $k=K-1,K-2,...,1$ do the following: MATH MATH MATH MATH The answer is MATH . We do not proceed to step k=0 because the cross sectional information MATH 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 MATH then we get a biased low value because the strategy is suboptimal.

The motivation for the steps above is the following. The $Y_{k,\omega}$ is the value of the quantity in question at time $k$ given the information MATH . Hence, the starting condition is obvious. The sum MATH 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 $k$ is close to the origin. In such situation the MATH does not contain much cross- $\omega$ information because all the $X_{k,\omega}$ originate from a single point $X_{0}$ 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.


















Copyright 2007