Abstract
Numerical techniques prove particularly useful when explicit solution formulas in a certain model cannot be derived even for simple derivatives (e.g. in the Black-Karasinski model) or when the to-be-priced financial instrument has a complex structure so that analytical methods fail (e.g. if multiple cancelation rights exist).
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Notes
- 1.
Such a digital condition could be ‘the option value remains 0, once the underlying stock price exceeds a certain barrier’.
- 2.
A detailed analysis of this aspect can be found in Zulehner [76].
- 3.
Such non-structured grids could, for example, include triangular or tetrahedral grids, or grids that become finer in certain parts of the computational region.
- 4.
It will be practical if W only takes non-zero values on a small interval (see Figure 10.7). In this case the first term on the right-hand side of the above equation disappears, while the second (integral) term has a correspondingly small integration domain.
- 5.
In the Black-Scholes differential equation the term with the second derivative with respect to S is the diffusion term, which has smoothing properties. The term with the first derivative with respect to S is the so-called convection term. Heat transmission, for example, can occur by heat conduction (diffusion) or by the flow of fluids such as liquids or gases (convection). A central heating system in a house would be an example of convection. It is often challenging to treat convection numerically. The so-called upwind techniques or streamline diffusion techniques increase the stability of problems that show dominant convection. In the case of mean-reverting interest rate models, convection can be significant.
- 6.
This is the case for most stock price models discussed here, in particular for the Heston and the Merton model.
- 7.
This can be justified, as the integrand is absolutely integrable under the given assumption for α.
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Albrecher, H., Binder, A., Lautscham, V., Mayer, P. (2013). Numerical 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_10
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