Python for Finance: Analyze Big Financial Data

Further Reading

References for the topics of this chapter in book form are:

Glasserman, Paul (2004): Monte Carlo Methods in Financial Engineering. Springer,

New York.

Hilpisch, Yves (2015): Derivatives Analytics with Python. Wiley Finance, Chichester,

England. http://derivatives-analytics-with-python.com.

Original papers cited in this chapter:

Cox, John, Stephen Ross, and Mark Rubinstein (1979): “Option Pricing: A

Simplified Approach.” Journal of Financial Economics, Vol. 7, No. 3, pp. 229–263.

Kohler, Michael (2010): “A Review on Regression-Based Monte Carlo Methods for

Pricing American Options.” In Luc Devroye et al. (eds.): Recent Developments in

Applied Probability and Statistics. Physica-Verlag, Heidelberg, pp. 37–58.

Longstaff, Francis and Eduardo Schwartz (2001): “Valuing American Options by

Simulation: A Simple Least Squares Approach.” Review of Financial Studies, Vol.

14, No. 1, pp. 113–147.

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For details on how to estimate Greeks numerically by Monte Carlo simulation, refer to Chapter 7 of Glasserman

(2004). We only use forward-difference schemes here since this leads to only one additional simulation and revaluation

of the option. For example, a central-difference approximation would lead to two option revaluations and therefore a

higher computational burden.

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American exercise refers to a situation where exercise is possible at every instant of time over a fixed time interval

(at least during trading hours). Bermudan exercise generally refers to a situation where there are multiple, discrete

exercise dates. In numerical applications, American exercise is approximated by Bermudan exercise, and maybe letting

the number of exercise dates go to infinity in the limit.

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That is why their algorithm is generally abbreviated as LSM, for Least-Squares Monte Carlo.

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Kohler (2010) provides a concise overview of the theory of American option valuation in general and the use of

regression-based methods in particular.

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See also Chapter 6 of Hilpisch (2015).

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The main reason is that the “optimal exercise policy” based on the regression estimates of the continuation value is

only “suboptimal.”

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Cf. Chapter 6 in Hilpisch (2015) for a dual algorithm leading to an upper bound and a Python implementation

thereof.