Abstract
We now are ready to derive the important Black–Scholes Equation [1], which is widely used to determine pricing of Calls and Puts! An outline is given next; details are developed in the next chapter.
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Notes
- 1.
Notation change! Although the literature usually uses Delta, in this book Δ represents mathematical change. So, for this hedging purpose, use the lower case δ.
- 2.
Start with Y = 1∕S, and use Itô’s Lemma to derive ΔY = −σY ΔX + [−μY + σ 2 Y ] Δt. Assuming that S has a very large value is essentially the same as assuming that Y = 0. However, the equation for ΔY is such that Y = 0 requires Y to remain zero for all time. In turn, S remains infinitely large for all time.
- 3.
The standard approach, which mimics completing the square, is to set v(x, τ) = e ax+bτ u(x, t), and select the a and b values to drop terms.
References
Black, F., and M. Scholes. 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81: 637–654.
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Saari, D.G. (2019). The Black–Scholes Equation. In: Mathematics of Finance. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-25443-8_5
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DOI: https://doi.org/10.1007/978-3-030-25443-8_5
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