Abstract
Having used arbitrage considerations to derive various properties of derivatives, in particular of option prices (upper and lower bounds, parities, etc.), we now demonstrate how such arbitrage arguments, with the help of results from stochastic analysis, namely Ito’s formula 2.22, can be used to derive the famous Black-Scholes equation.
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Notes
- 1.
Path independence is necessary for the derivative’s value V to be only a function of t and the current underlying price S. Path-dependent prices V are functions of further variables which themselves are dependent on the history of the underlying. The total differential of V is then no longer given by Eq. 2.22; additional terms must be included in the expression.
- 2.
The fact that the “initial condition” in our case belongs really to the end of the time period under consideration is just a question of vocabulary and should not confuse the reader. Alternatively one can introduce a new time variable \(\widetilde {t}:=T-t\) (sometimes called term to maturity). Then the initial condition belongs to the lowest value of \(\widetilde {t}\), namely \(\widetilde {t}=0\).
- 3.
In contrast, for European options, the Black-Scholes differential equation is always solved on the set S = 0 to S = ∞. The boundary is thus always fixed and known for European options.
- 4.
The Black-Scholes equation simply means that the yield from a delta-hedged portfolio must equal the yield from a risk-free investment.
- 5.
In applications, this function f(S, T) is often the payoff profile of a derivative.
- 6.
We have also used the following general property of the logarithm:
$$\displaystyle \begin{aligned} \frac{\partial}{\partial t}\ln\left(B\right) =\frac{1}{B} \frac{\partial B}{\partial t}\;. \end{aligned}$$ - 7.
In the second equation, Eq. 2.26 was used in the conversion to the drift \(\mu =\widetilde {\mu }-\sigma ^{2}/2\) of the stochastic process for the returns \(\ln (S)\).
- 8.
Note that by definition the volatility has the dimension \(1/\sqrt {\text{time}}\), i.e. σ2 has the dimension 1∕time, thus making the variable z dimensionless.
References
M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1972)
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Deutsch, HP., Beinker, M.W. (2019). The Black-Scholes Differential Equation. In: Derivatives and Internal Models. Finance and Capital Markets Series. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-22899-6_7
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DOI: https://doi.org/10.1007/978-3-030-22899-6_7
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