Black–Scholes equation

Overview

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

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

Derivation

Per the model assumptions above, the price of the underlying asset (typically a stock) follows a geometric Brownian motion. That is

d S = μ S

d t + σ S

d W

{\displaystyle dS=\mu S\,dt+\sigma S\,dW\,}

where W is a stochastic variable (Brownian motion). Note that W, and consequently its infinitesimal increment dW, represents the only source of uncertainty in the price history of the stock. Intuitively, W(t) is a process that "wiggles up and down" in such a random way that its expected change over any time interval is 0. (In addition, its variance over time T is equal to T; see Wiener process § Basic properties); a good discrete analogue for W is a simple random walk. Thus the above equation states that the infinitesimal rate of return on the stock has an expected value of μ dt and a variance of

σ

2

d t

{\displaystyle \sigma ^{2}dt}

. The payoff of an option (or any derivative contingent to stock S)

V ( S , T )

{\displaystyle V(S,T)}

at maturity is known. To find its value at an earlier time we need to know how

V

{\displaystyle V}

evolves as a function of

S

{\displaystyle S}

and

Alternative derivation

Here is an alternative derivation that can be utilized in situations where it is initially unclear what the hedging portfolio should be. (For a reference, see 6.4 of Shreve vol II). In the Black–Scholes model, assuming we have picked the risk-neutral probability measure, the underlying stock price S(t) is assumed to evolve as a geometric Brownian motion:

d S ( t )

S ( t )

= r

d t + σ d W ( t )

{\displaystyle {\frac {dS(t)}{S(t)}}=r\ dt+\sigma dW(t)}

Since this stochastic differential equation (SDE) shows the stock price evolution is Markovian, any derivative on this underlying is a function of time t and the stock price at the current time, S(t). Then an application of Itô's lemma gives an SDE for the discounted derivative process

e

− r t

V ( t , S ( t ) )

{\displaystyle e^{-rt}V(t,S(t))}

, which should be a martingale. In order for that to hold, the drift term must be zero, which implies the Black—Scholes PDE. This derivation is basically an application of the Feynman–Kac formula and can be attempted whenever the underlying asset(s) evolve according to given SDE(s).

Solving methods

Once the Black–Scholes PDE, with boundary and terminal conditions, is derived for a derivative, the PDE can be solved numerically using standard methods of numerical analysis, such as a type of finite difference method. In certain cases, it is possible to solve for an exact formula, such as in the case of a European call, which was done by Black and Scholes. The solution is conceptually simple. Since in the Black–Scholes model, the underlying stock price

S

t

{\displaystyle S_{t}}

follows a geometric Brownian motion, the distribution of

S

T

{\displaystyle S_{T}}

, conditional on its price

S

t

{\displaystyle S_{t}}

at time

t

{\displaystyle t}

, is a log-normal distribution. Then the price of the derivative is just discounted expected payoff

E [

e

− r ( T − t )

K (

See also

Bachelier model - uses arithmetic Brownian motion instead of geometric