Black–Scholes equation

In mathematical finance, the Black–Scholes equation, also called the Black–Scholes–Merton 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

T

{\displaystyle T}

in the future, and at maturation, it has payoff

K

(

S

T

)

{\displaystyle K(S_{T})}

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

{

∂

V

∂

t

+

1

2

σ

2

S

2

∂

2

V

∂

S

2

+

r

S

∂

V

∂

S

−

r

V

=

0

V

(

T

,

s

)

=

K

(

s

)

∀

s

{\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}}}

where

V

(

t

,

S

)

{\displaystyle V(t,S)}

is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, and

σ

{\displaystyle \sigma }

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.

Financial interpretation

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:

∂

V

∂

t

+

1

2

σ

2

S

2

∂

2

V

∂

S

2

=

r

V

−

r

S

∂

V

∂

S

{\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}=rV-rS{\frac {\partial V}{\partial S}}}

The left-hand side consists of a "time decay" term, the change in derivative value with respect to time, 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 rate of return from a long position in the derivative and a short position consisting of

∂

V

/

∂

S

{\textstyle {\partial V}/{\partial S}}

shares of the underlying asset.