Black–Scholes equation

In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put 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.

For a European call or put on an underlying stock paying no dividends, the equation is:

where V is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, and ${\displaystyle \sigma }$ is the volatility of the stock.

The key financial insight behind the equation is that 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.

The equation has a concrete interpretation that is often used by practitioners and is the basis for the common derivation given in the next subsection. The equation can be rewritten in the form:

The left hand side consists of a "time decay" term, the change in derivative value due to time increasing called theta, and a term involving the second spatial derivative gamma, the convexity of the derivative value with respect to the underlying value. The right hand side is the riskless return from a long position in the derivative and a short position consisting of ${\displaystyle {\frac {\partial V}{\partial S}}}$ shares of the underlying.

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