Abstract
After deriving the Black-Scholes equation for a call option from the requirement to make a portfolio risk-free, the equation is solved using a number of variable substitutions, which transforms it into a diffusion equation. Using the latter’s Green’s function is then used to value European call options. The resemblance of the solution found in this chapter to that in Chapter 4 stimulates the discussion of martingale processes. In order to better understand the mechanics of using options for hedging, a MATLAB simulation for the temporal evolution of stocks, options and bank deposits is presented.
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References
I. Steward, The Mathematical Equation That Caused the Banks to Crash, The Guardian (2012). Available online at https://www.theguardian.com/science/2012/feb/12/black-scholes-equation-credit-crunch
D. Silverman, Solution of the Black Scholes Equation Using the Green’s Function of the Diffusion Equation, unpublished note (UC Irvine, 1999)
P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives (Cambridge University Press, Cambridge, 2005)
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Ziemann, V. (2021). Black-Scholes Differential Equation. In: Physics and Finance. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-63643-2_5
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DOI: https://doi.org/10.1007/978-3-030-63643-2_5
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