Black-Scholes Equation

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A Black-Scholes Equation is a partial differential equation (PDE) that governs the price evolution of European-style options under the Black–Scholes–Merton model.



References

2024

  • (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Black–Scholes_equation Retrieved:2024-2-27.
    • In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of derivatives under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.

      Consider a stock paying no dividends. Now construct any derivative that has a fixed maturation time [math]\displaystyle{ T }[/math] in the future, and at maturation, it has payoff [math]\displaystyle{ K(S_T) }[/math] that depends on the values taken by the stock at that moment (such as European call or put options). Then the price of the derivative satisfies : [math]\displaystyle{ \begin{cases} \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 \\ V(T, s) = K(s) \quad \forall s \end{cases} }[/math] where [math]\displaystyle{ V(t, S) }[/math] is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, and [math]\displaystyle{ \sigma }[/math] is the volatility of the stock.

      The key financial insight behind the equation is that, under the model assumption of a frictionless market, one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently “eliminate risk". This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula.