Abstract
Many problems that arise in financial mathematics are structurally complex, so that one often cannot obtain explicit results (such as explicit pricing formulas for derivatives) or successfully apply numerical methods as outlined in Chapter 10. In such cases stochastic simulation can offer an efficient and powerful alternative for obtaining numerical estimates for specific quantities.
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Notes
- 1.
In the sequel, possibly multi-variate random variables and vectors will denoted by bold letters or symbols.
- 2.
In the case of a European call option with strike K and maturity T we would set Z = S T . The function g(Z) would be the discounted pay-off \({e}^{-rT}{(Z - K)}^{+}\) of the option and F Z the (one-dimensional, i.e. s = 1) distribution function of the price S T of the underlying asset under the risk-neutral measure.
- 3.
Let A be some event (e.g. that a random variable Z produces a negative realization, or that a sample path crosses a given barrier). We can then define the indicator function
$$\displaystyle{\mathbf{1}_{A}(\mathbf{Z}) = \left \{\begin{array}{rl} 1,&\text{if }\mathbf{Z} \in A,\\ 0, &\text{if } \mathbf{Z} \notin A. \end{array} \right.}$$As \(\mathbb{P}(A) = \mathbb{E}[\mathbf{1}_{A}(\mathbf{Z})]\), a method to evaluate (11.1) will also allow to compute probabilities of events. In finance and insurance, such events of interest include the default of a bond (i.e. promised payments cannot be made in full), the bankruptcy of a company or the knock-in/knock-out event of a barrier option, to name a few.
- 4.
The Monte Carlo method was developed in the Manhattan project in the 1940s. The name is related to the randomness involved in the method, and finds its origin in the name of Monaco’s administrative area Monte Carlo with its casino.
- 5.
In practice, one will produce samples through a deterministic algorithm which imitates the uniform distribution well. This imitation is referred to as ‘pseudo-random-number’ algorithm and its quality can be assessed by statistical tests. The Mathematica command RandomReal[1,n] produces a set of n sample points from a U(0, 1) distribution. For mathematical background on how to generate U(0, 1) pseudo-random numbers see Korn et al. [48].
- 6.
This is the reason why Monte Carlo methods are typically preferred over numerical integration methods in dimensions s ≥ 5.
- 7.
f(Z) ∕ f I (Z) is the so-called Radon-Nikodým derivative or likelihood ratio.
- 8.
Concretely, it is the total variation in the sense of Hardy and Krause.
- 9.
For T = 1 and a time step \(\Delta t = 0.01\), one will have to sample M = 100 normally distributed random variables to generate one sample path. Thus, the simulation of the price of an option with N = 1, 000 sample paths will require 1,000 points of a 100-dimensional point sequence.
- 10.
As it corresponds to the Euler approximation for the numerical solution of ordinary differential equations.
References
S. Asmussen and P. W. Glynn. Stochastic Simulation: Algorithms and Analysis. Springer, New York, 2007.
M. Drmota and R. Tichy. Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, 1651. Springer-Verlag, Berlin, 1997.
P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer, New York, 2004.
R. Korn, E. Korn, and G. Kroisandt. Monte Carlo Methods and Models in Finance and Insurance. Chapman & Hall/CRC, Boca Raton, FL, 2010.
A. J. McNeil, R. Frey, and P. Embrechts. Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton, NJ, 2005.
H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
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Albrecher, H., Binder, A., Lautscham, V., Mayer, P. (2013). Simulation Methods. In: Introduction to Quantitative Methods for Financial Markets. Compact Textbooks in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0519-3_11
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