Abstract
The value of the security changes constantly and over very short time scales: FX transactions can be measured in microseconds, or millionths of a second. The actual values of each FX trade does indeed exist, but it becomes near to impossible to describe its motion in complete detail. Financial instruments, in particular, options are modeled based on considering the underlying security to be following a stochastic process. The Black–Scholes equation, one of the cornerstones of option pricing, is studied in great detail.
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Notes
- 1.
‘Pink’ noise and ‘brown’ noise can also be defined but will not be considered.
- 2.
The term \((\partial ^2 C/\partial S \partial t)S\) can be shown to be negligible.
- 3.
Short selling entails taking a share on loan and returning it when the option matures—by buying it from the market.
- 4.
A detailed proof is given in Sect. 10.5.15.
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Baaquie, B.E. (2020). Stochastic Processes and Black–Scholes Equation. In: Mathematical Methods and Quantum Mathematics for Economics and Finance. Springer, Singapore. https://doi.org/10.1007/978-981-15-6611-0_11
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DOI: https://doi.org/10.1007/978-981-15-6611-0_11
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