Abstract
The goal of the chapter is to derive the Black-Scholes formula, one of the highlights of Mathematical finance. The proof is conducted by passing to the limit from the binomial model. The chapter begins with introducing some relevant notions: drift and volatility, continuous compounding, geometric random walk, etc. It then shows how to approximate the observed continuous-time price process with constant drift and volatility by price processes generated by suitable binomial models. The main theorem proved in the chapter establishes a general (probabilistic) version of the Black-Scholes formula for a European derivative security with a general payoff function. As a corollary to this theorem, an analytic version of the Black-Scholes formula for a European call option is obtained.
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Notes
- 1.
We use here (for x = rT) the fact that \(\left (1 + x/n\right )^{n} \rightarrow e^{x}\) as n → ∞.
- 2.
Formulas (15.7) for \(u = e^{\sigma \sqrt{\varDelta }}\) and \(d = e^{-\sigma \sqrt{\varDelta }}\) are obtained by using the approximate formula for the function e x:
$$\displaystyle{e^{x} \approx 1 + x + \dfrac{x^{2}} {2}.}$$The precise formula is \(e^{x} = 1 + x + \dfrac{x^{2}} {2} +\ldots +\dfrac{x^{m}} {m!} +\ldots\).
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Evstigneev, I.V., Hens, T., Schenk-Hoppé, K.R. (2015). From Binomial Model to Black–Scholes Formula. In: Mathematical Financial Economics. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-16571-4_15
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DOI: https://doi.org/10.1007/978-3-319-16571-4_15
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16570-7
Online ISBN: 978-3-319-16571-4
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