Abstract
For the Black-Scholes model, as introduced in the last chapter, we can now derive the no-arbitrage price of a European-style option – the so-called Black-Scholes formula. In Section 7.1, we will discuss a direct approach to obtaining the Black-Scholes formula as the solution of a partial differential equation.
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- 1.
The notation \({ d \atop \rightarrow }\) indicates that the distribution of Z(n) converges to the distribution of Z. Under the assumptions of this version of the Central Limit Theorem, the so-called Lindeberg condition , \(\lim _{n\rightarrow \infty } \frac{1} {{n\sigma }^{2}(n)}\sum _{k=1}^{n}\mathbb{E}\left [Y _{k}{(n)}^{2}\mathbf{1}_{\{\vert Y _{k}(n)\vert >\epsilon \}}\right ]\;\text{for all }\epsilon > 0,\) is fulfilled, under which the result holds. For details consult Feller [31].
- 2.
f(n) = O(g(n)) means that there exist M, n 0 > 0, such that f(n) ≤ M ⋅g(n) for all n ≥ n 0. The notation f(n) = o(g(n)), on the other hand, is used if f(n) ∕ g(n) → 0 for n → ∞.
- 3.
Formally, pulling the limit inside the expectation (i.e. the integral) as in (7.6) requires further justification. It can either be proven that \({(S_{0}\,{e}^{Z(n)} - {e}^{-rT}K)}^{+}\) is uniformly integrable, or one can first derive the formula for a put option (in which case the interchange is justified by dominated convergence) and subsequently apply the put-call parity.
- 4.
Heuristically, this can be seen by taking the limit in the CRR model. For a formal proof using Girsanov’s Theorem consult the references at the end of the chapter.
- 5.
This assumption is better suited for indices.
- 6.
This portfolio is commonly referred to as \(\Delta \) -neutral portfolio.
- 7.
In practice, lowering \(\Gamma \) could, e.g., be achieved by adding options of different strikes.
- 8.
\(\Phi (x) = \frac{1} {\sqrt{2\pi }}{e}^{-{x}^{2}/2 }\) denotes the probability density function of the standard normal distribution.
References
J. Andreasen, B. Jensen, and R. Poulsen. Eight valuation methods in financial mathematics: The Black-Scholes formula as an example. Mathematical Scientist, 23(1):18–40, 1998.
M. Baxter and A. Rennie. Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press, Cambridge, 1996.
D. Duffie. Dynamic Asset Price Theory. 3rd edition. Princeton University Press, Princeton, NJ, 2001.
R. J. Elliott and E. P. Kopp. Mathematics of Financial Markets. 2nd edition. Springer, New York, 2005.
W. Feller. An Introduction to Probability Theory and Its Applications, volume 2. Wiley, New York, 1970.
M. B. Garman and S. W. Kohlhagen. Foreign currency option values. Journal of International Money and Finance, 2(3):231–237, 1983.
P. Wilmott. Introduces Quantitative Finance. Wiley, Chichester, 2007.
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Albrecher, H., Binder, A., Lautscham, V., Mayer, P. (2013). The Black-Scholes Formula. In: Introduction to Quantitative Methods for Financial Markets. Compact Textbooks in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0519-3_7
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