Inmathematical finance, theBlack–Scholes equation, also called theBlack–Scholes–Merton equation, is apartial differential equation (PDE) governing the price evolution of derivatives under theBlack–Scholes model.^[1] Broadly speaking, the term may refer to a similar PDE that can be derived for a variety ofoptions, or more generally,derivatives.

Consider a stock paying no dividends. Now construct any derivative that has a fixed maturation time${\displaystyle T}$ in the future, and at maturation, it has payoff${\displaystyle K(S_T)}$ 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

${\displaystyle \{\begin{array}{l}\frac{\mathrm{\partial }V}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }V}{\mathrm{\partial }S}-rV=0\\ V(T,s)=K(s)\mathrm{\forall }s\end{array}}$

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

The key financial insight behind the equation is that, under the model assumption of africtionless market, one can perfectlyhedge the option by buying and selling theunderlying 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 theBlack–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:

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

The left-hand side consists of a "time decay" term, the change in derivative value with respect to time, calledtheta, and a term involving the second spatial derivativegamma, 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$\mathrm{\partial }V/\mathrm{\partial }S$ shares of the underlying asset.

Black and Scholes's insight was that the portfolio represented by the right-hand side is riskless: thus the equation says that the riskless return over any infinitesimal time interval can be expressed as the sum of theta and a term incorporating gamma. For an option, theta is typically negative, reflecting the loss in value due to having less time for exercising the option (for a European call on an underlying without dividends, it is always negative). Gamma is typically positive and so the gamma term reflects the gains in holding the option. The equation states that over any infinitesimal time interval the loss from theta and the gain from the gamma term must offset each other so that the result is a return at the riskless rate.

From the viewpoint of the option issuer, e.g. an investment bank, the gamma term is the cost of hedging the option. (Since gamma is the greatest when the spot price of the underlying is near the strike price of the option, the seller's hedging costs are the greatest in that circumstance.)

Per the model assumptions above, the price of theunderlying asset (typically a stock) follows ageometric Brownian motion.^[2] That is

${\displaystyle dS=\mu Sdt+\sigma SdW}$

whereW is a stochastic variable (Brownian motion). Note thatW, and consequently its infinitesimal incrementdW, represents the only source of uncertainty in the price history of the stock. Intuitively,W(t) is aprocess that "wiggles up and down" in such a random way that its expected change over any time interval is 0. (In addition, itsvariance over timeT is equal toT; seeWiener process § Basic properties); a good discrete analogue forW is asimple random walk. Thus the above equation states that the infinitesimal rate of return on the stock has an expected value ofμ dt and a variance of${\displaystyle {\sigma }^{2}dt}$.

The payoff of an option (or any derivative contingent to stockS)${\displaystyle V(S,T)}$ at maturity is known. To find its value at an earlier time we need to know how${\displaystyle V}$ evolves as a function of${\displaystyle S}$ and${\displaystyle t}$. ByItô's lemma for two variables we have

${\displaystyle dV=\left(\mu S\frac{\mathrm{\partial }V}{\mathrm{\partial }S}+\frac{\mathrm{\partial }V}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{S}^2}\right)dt+\sigma S\frac{\mathrm{\partial }V}{\mathrm{\partial }S}dW}$

Now consider a portfolio${\displaystyle \mathrm{\Pi }}$ consisting of a short option and$\mathrm{\partial }V/\mathrm{\partial }S$ long shares at time${\displaystyle t}$. The value of these holdings is

${\displaystyle \mathrm{\Pi }=-V+\frac{\mathrm{\partial }V}{\mathrm{\partial }S}S}$

As${\displaystyle \frac{\mathrm{\partial }V}{\mathrm{\partial }S}}$ changes with time, the position in${\displaystyle S}$ is continually updated. We implicitly assume that the portfolio contains a cash account to accommodate buying and selling shares${\displaystyle S}$, making the portfolioself-financing. Therefore, we only need to consider the total profit or loss from changes in the values of the holdings:

${\displaystyle d\mathrm{\Pi }=-dV+\frac{\mathrm{\partial }V}{\mathrm{\partial }S}dS}$

Substituting${\displaystyle dS}$ and${\displaystyle dV}$ into the expression for${\displaystyle d\mathrm{\Pi }}$:

${\displaystyle d\mathrm{\Pi }=\left(-\frac{\mathrm{\partial }V}{\mathrm{\partial }t}-\frac{1}{2}{\sigma }^2{S}^2\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{S}^2}\right)dt}$

Over a time period${\displaystyle [t,t+\mathrm{\Delta }t]}$, for${\displaystyle \mathrm{\Delta }t}$ small enough, we see that

${\displaystyle \mathrm{\Delta }\mathrm{\Pi }=\left(-\frac{\mathrm{\partial }V}{\mathrm{\partial }t}-\frac{1}{2}{\sigma }^2{S}^2\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{S}^2}\right)\mathrm{\Delta }t}$

Note that the${\displaystyle dW}$ terms have vanished. Thus uncertainty has been eliminated and the portfolio is effectively riskless, i.e. adelta-hedge. The rate of return on this portfolio must be equal to the rate of return on any other riskless instrument; otherwise, there would be opportunities for arbitrage. Now assuming the risk-free rate of return is${\displaystyle r}$ we must have over the time period${\displaystyle [t,t+\mathrm{\Delta }t]}$:

${\displaystyle \mathrm{\Delta }\mathrm{\Pi }=r\mathrm{\Pi }\mathrm{\Delta }t}$

If we now substitute our formulas for${\displaystyle \mathrm{\Delta }\mathrm{\Pi }}$ and${\displaystyle \mathrm{\Pi }}$ we obtain:

${\displaystyle \left(-\frac{\mathrm{\partial }V}{\mathrm{\partial }t}-\frac{1}{2}{\sigma }^2{S}^2\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{S}^2}\right)\mathrm{\Delta }t=r\left(-V+S\frac{\mathrm{\partial }V}{\mathrm{\partial }S}\right)\mathrm{\Delta }t}$

Simplifying, we arrive at the Black–Scholes partial differential equation:

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

With the assumptions of theBlack–Scholes model, this second order partial differential equation holds for any type of option as long as its price function${\displaystyle V}$ is twice differentiable with respect to${\displaystyle S}$ and once with respect to${\displaystyle t}$.

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).^[3]

In the Black–Scholes model, assuming we have picked the risk-neutral probability measure, the underlying stock priceS(t) is assumed to evolve as a geometric Brownian motion:

${\displaystyle \frac{dS(t)}{S(t)}=rdt+\sigma dW(t)}$

Since thisstochastic differential equation (SDE) shows the stock price evolution isMarkovian, any derivative on this underlying is a function of timet 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${\displaystyle e^{-rt}V(t,S(t))}$, 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 theFeynman–Kac formula and can be attempted whenever the underlying asset(s) evolve according to given SDE(s).

Once the Black–Scholes PDE, with boundary and terminal conditions, is derived for a derivative, the PDE can be solved numerically using standard methods ofnumerical analysis, such as a type offinite difference method.^[4] 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${\displaystyle S_t}$ follows a geometric Brownian motion, the distribution of${\displaystyle S_T}$, conditional on its price${\displaystyle S_t}$ at time${\displaystyle t}$, is alog-normal distribution. Then the price of the derivative is just discounted expected payoff${\displaystyle E[e^{-r(T-t)}K(S_T)|S_t]}$, which may be computed analytically when the payoff function${\displaystyle K}$ is analytically tractable, or numerically if not.

To do this for acall option, recall the PDE above hasboundary conditions^[5]

The last condition gives the value of the option at the time that the option matures. Other conditions are possible asS goes to 0 or infinity. For example, common conditions utilized in other situations are to choose delta to vanish asS goes to 0 and gamma to vanish asS goes to infinity; these will give the same formula as the conditions above (in general, differing boundary conditions will give different solutions, so some financial insight should be utilized to pick suitable conditions for the situation at hand).

The solution of the PDE gives the value of the option at any earlier time,${\displaystyle \mathbb{E}\left[max\{S-K,0\}\right]}$. To solve the PDE we recognize that it is aCauchy–Euler equation which can be transformed into adiffusion equation by introducing the change-of-variable transformation

Then the Black–Scholes PDE becomes adiffusion equation

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\tau }=\frac{1}{2}{\sigma }^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$

The terminal condition${\displaystyle C(S,T)=max\{S-K,0\}}$ now becomes an initial condition

${\displaystyle u(x,0)=u_0(x):=K(e^{max\{x,0\}}-1)=K\left(e^x-1\right)H(x),}$

whereH(x) is theHeaviside step function. The Heaviside function corresponds to enforcement of the boundary data in theS,t coordinate system that requires whent =T,

assuming bothS,K > 0. With this assumption, it is equivalent to the max function over allx in the real numbers, with the exception ofx = 0. The equality above between themax function and the Heaviside function is in the sense of distributions because it does not hold forx = 0. Though subtle, this is important because the Heaviside function need not be finite atx = 0, or even defined for that matter. For more on the value of the Heaviside function atx = 0, see the section "Zero Argument" in the articleHeaviside step function.

Using the standardconvolution method for solving adiffusion equation given an initial value function,u(x, 0), we have

${\displaystyle u(x,\tau )=\frac{1}{\sigma \sqrt{2\pi \tau }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_0(y)\exp \left[-\frac{(x-y{)}^2}{2{\sigma }^2\tau }\right]dy,}$

which, after some manipulation, yields

${\displaystyle u(x,\tau )=K{e}^{x+\frac{1}{2}{\sigma }^2\tau }N(d_{+})-KN(d_{-}),}$

where${\displaystyle N(\cdot )}$ is thestandard normalcumulative distribution function and

These are the same solutions (up to time translation) that were obtained byFischer Black in 1976.^[6]

Reverting${\displaystyle u,x,\tau }$ to the original set of variables yields the above stated solution to the Black–Scholes equation.

The asymptotic condition can now be realized.

${\displaystyle u(x,\tau )\stackrel{x⇝\mathrm{\infty }}{\asymp }K{e}^x,}$

which gives simplyS when reverting to the original coordinates.

${\displaystyle \underset{x\to \mathrm{\infty }}{lim}N(x)=1.}$

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