Black–Scholes equation
that depends on the values taken by the stock at that moment (such as European call or put options).
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.
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 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.
Black and Scholes' 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).
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.
Per the model assumptions above, the price of the underlying asset (typically a stock) follows a geometric 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
By Itô's lemma for two variables we have Replacing the differentials with deltas in the equations for dS and dV gives: Now consider a portfolio
, the total profit or loss from changes in the values of the holdings is: Substituting
Thus uncertainty has been eliminated and the portfolio is effectively riskless, i.e. a delta-hedge.
we obtain: Simplifying, we arrive at the Black–Scholes partial differential equation: With the assumptions of the Black–Scholes model, this second order partial differential equation holds for any type of option as long as its price function
Here is an alternative derivation that can be utilized in situations where it is initially unclear what the hedging portfolio should be.
[3] 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: 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
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).
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.
The solution of the PDE gives the value of the option at any earlier time,
The Heaviside function corresponds to enforcement of the boundary data in the S, t coordinate system that requires when t = T, assuming both S, K > 0.
With this assumption, it is equivalent to the max function over all x in the real numbers, with the exception of x = 0.
Though subtle, this is important because the Heaviside function need not be finite at x = 0, or even defined for that matter.
Using the standard convolution method for solving a diffusion equation given an initial value function, u(x, 0), we have which, after some manipulation, yields where
is the standard normal cumulative distribution function and These are the same solutions (up to time translation) that were obtained by Fischer Black in 1976.
to the original set of variables yields the above stated solution to the Black–Scholes equation.
