Black-Scholes Formula and European Option Pricing › Solving the Black-Scholes PDE

Recall from the previous chapter that the Black-Scholes PDE for a derivative $V(S,t)$ on a non-dividend-paying stock is:

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$

where:

• $V$: price of the derivative
• $t$: time
• $S$: price of the underlying asset
• $\sigma$: volatility of the underlying asset
• $r$: risk-free interest rate

This equation captures a fundamental requirement: the price of any derivative must evolve in time so no riskless arbitrage profit can be extracted. Each term in the equation plays a specific role. The first term represents how the option price changes with the passage of time alone. The second term, involving the second derivative with respect to the stock price, captures the effect of the stock's random fluctuations, which is precisely why the volatility parameter appears squared and multiplied by $S^2$. The third term accounts for the drift in the option price due to the expected growth of the stock under risk-neutral dynamics. The last term discounts the derivative value at rate $r$. Together, the PDE terms make the self-financing delta-hedged position with value $V-SV_S$ earn the risk-free rate; the unhedged derivative need not have that expected return under the physical measure.

To find the price of a European call option, we must solve this PDE subject to the boundary condition at expiration:

$$C(S,T) = \max(S - K, 0)$$

where:

• $C(S,T)$: value of the call option at expiration
• $S$: stock price at expiration
• $K$: strike price
• $T$: expiration time

This boundary condition states something intuitive: at the moment of expiration, an option is worth exactly its intrinsic value. If the stock price exceeds the strike, the call holder can buy the stock at $K$ and immediately sell it at $S$, pocketing the difference. If the stock price falls below the strike, the call expires worthless. The mathematical challenge lies in determining what the option should be worth at any time before expiration, given that it will satisfy this terminal condition.

We seek the financially admissible solution, with $0\leq C(S,t)\leq S$ and at most linear growth as $S$ increases. These restrictions exclude nonfinancial solutions on the unbounded stock-price domain.

Assume $S,K>0$, constant $\sigma>0$ and constant $r$. Introduce remaining time $\tau=T-t$, measured in years when the rate and volatility use annual units. As $\tau$ increases, we move backward from the known expiry payoff. Set $x=\ln(S/K)$, the dimensionless log-moneyness. Multiplying $S$ by a factor adds the logarithm of that factor to $x$. For example, when K=\100$, stock prices of$$50$and$$200$map to$-\ln 2$and$+\ln 2$.

Normalize and discount the option value by writing

$$V(S,t)=K e^{-r\tau}v(x,\tau).$$

Here $v$ is dimensionless; $\tau$ still has units of time. The chain rule gives

$$V_t=K e^{-r\tau}(rv-v_\tau),\qquad V_S=\frac{K e^{-r\tau}}{S}v_x,\qquad V_{SS}=\frac{K e^{-r\tau}}{S^2}(v_{xx}-v_x).$$

Substituting these derivatives into the pricing PDE and cancelling $K e^{-r\tau}$ leaves a constant-coefficient equation:

$$v_\tau=\frac{\sigma^2}{2}v_{xx}+ \left(r-\frac{\sigma^2}{2}\right)v_x.$$

Write $a=\sigma^2/2$ and $b=r-\sigma^2/2$. To remove both the drift term and the zero-order term that the next substitution creates, set

$$v=e^{\alpha x+\beta\tau}u,\qquad \alpha=-\frac{b}{2a}=\frac12-\frac{r}{\sigma^2},\qquad \beta=-\frac{b^2}{4a}=-\frac{\sigma^2\alpha^2}{2}.$$

The constant $\alpha$ is dimensionless, while $\beta$ has inverse-time units. Direct differentiation gives $u_\tau=a u_{xx}+(2a\alpha+b)u_x+(a\alpha^2+b\alpha-\beta)u$. Our choices make both extra coefficients zero, so

$$u_\tau=\frac{\sigma^2}{2}u_{xx}.$$

This is the heat equation with diffusivity $\sigma^2/2$. The terminal call payoff becomes its initial condition at $\tau=0$:

$$u(y,0)=e^{-\alpha y}(e^y-1)^+.$$

The notation $z^+=\max(z,0)$ clips negative values to zero. Thus this initial function is zero for $y\leq0$. For $\tau>0$, the whole-line heat kernel is the Gaussian density

$$G(x-y,\tau)= \frac{1}{\sigma\sqrt{2\pi\tau}} \exp\!\left(-\frac{(x-y)^2}{2\sigma^2\tau}\right).$$

It weights transformed payoff values at log-price endpoints $y$. The kernel integrates to one; $u(y,0)$ is a transformed payoff, not a probability density. Convolving the two gives

$$u(x,\tau)= \int_0^\infty \left[e^{(1-\alpha)y}-e^{-\alpha y}\right]G(x-y,\tau)\,dy.$$

To evaluate this integral, let $z=\sigma\sqrt{\tau}$ and define $I_c=\int_0^\infty e^{cy}G(x-y,\tau)\,dy$. Completing the square in its exponent gives

$$cy-\frac{(x-y)^2}{2z^2} =-\frac{(y-x-cz^2)^2}{2z^2}+cx+\frac{c^2z^2}{2}, \qquad I_c=e^{cx+c^2z^2/2}N\!\left(\frac{x+cz^2}{z}\right).$$

Here $N$ is the standard normal cumulative distribution function. Hence $u=I_{1-\alpha}-I_{-\alpha}$. Multiplying by the two scaling factors recovers $C=K e^{-r\tau}e^{\alpha x+\beta\tau}u$. Because $\beta=-\sigma^2\alpha^2/2$ and $\alpha\sigma^2=\sigma^2/2-r$, the first term simplifies to $SN(d_1)$ and the second to $Ke^{-r\tau}N(d_2)$, where

$$d_1=\frac{x+(r+\sigma^2/2)\tau}{\sigma\sqrt{\tau}}, \qquad d_2=\frac{x+(r-\sigma^2/2)\tau}{\sigma\sqrt{\tau}}.$$

The next section states the resulting prices and interprets these standardized terms.

Working backward through the transformations, reversing each change of variables in sequence, the solution for a European call option emerges as:

$$C(S,t) = S \cdot N(d_1) - K e^{-r(T-t)} N(d_2)$$

where:

• $C(S,t)$: call option price
• $S$: current stock price
• $N(\cdot)$: cumulative standard normal distribution function
• $d_1, d_2$: standardized moneyness terms
• $K$: strike price
• $r$: risk-free interest rate
• $T-t$: time to expiration

This formula expresses the option price as the difference between two terms, each involving the cumulative normal distribution function $N(\cdot)$. The function $N(x)$ gives the probability that a standard normal random variable is less than $x$, and it appears here because the underlying stock price follows a lognormal distribution under geometric Brownian motion.

The terms $d_1$ and $d_2$ are defined as:

$$d_1 = \frac{\ln(S/K) + (r + \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$$

where:

• $S$: current stock price
• $K$: strike price
• $r$: risk-free rate
• $\sigma$: volatility
• $T-t$: time to expiration

The numerator of $d_1$ contains two parts. The first, $\ln(S/K)$, measures the current moneyness of the option in logarithmic terms. A positive value indicates the option is in-the-money, while a negative value indicates it is out-of-the-money. The second part, $(r + \sigma^2/2)(T-t)$, equals the risk-neutral expected log-price drift $(r - \sigma^2/2)(T-t)$ plus the variance adjustment $\sigma^2(T-t)$. It shifts $d_2$ to $d_1$; it is not the expected log-price drift itself. The denominator, $\sigma\sqrt{T-t}$, normalizes this quantity by the standard deviation of the log-price at expiration, converting it into a standardized z-score.

The second standardized term is:

$$d_2 = d_1 - \sigma\sqrt{T-t} = \frac{\ln(S/K) + (r - \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$$

where:

• $d_1$: first standardized term
• $\sigma$: volatility
• $T-t$: time to expiration
• $S$: current stock price
• $K$: strike price
• $r$: risk-free rate

Notice that $d_2$ differs from $d_1$ only in the sign of the $\sigma^2/2$ term in the numerator. This sign change determines the probabilistic interpretation developed below. The relationship $d_2 = d_1 - \sigma\sqrt{T-t}$ shows that the two quantities differ by exactly one standard deviation of the log-return.

For a European put option, we can either solve the PDE with the put boundary condition $P(S,T) = \max(K-S, 0)$, or use put-call parity. The result is:

$$P(S,t) = K e^{-r(T-t)} N(-d_2) - S \cdot N(-d_1)$$

where:

• $P(S,t)$: put option price
• $K$: strike price
• $r$: risk-free rate
• $T-t$: time to expiration
• $S$: current stock price
• $N(\cdot)$: cumulative standard normal distribution function
• $d_1, d_2$: standardized moneyness terms

The put formula has a parallel structure to the call formula but with the terms reversed and the signs of the arguments to $N(\cdot)$ flipped. This symmetry reflects the complementary nature of calls and puts: a call benefits when the stock rises above the strike, while a put benefits when the stock falls below the strike. The relationship between the two formulas is made precise by put-call parity.

These are the celebrated Black-Scholes formulas. Let's examine what each term means.