The Black-Scholes Equation

An option is a contract whose payoff depends on a stock, and the insight that priced it is that the payoff can be manufactured by trading the stock and a bond in the right proportions. If a portfolio of stock and bond reproduces the option's value at every instant, then by absence of arbitrage the option must cost what the portfolio costs, and the proportions are dictated by Ito's formula. Eliminating the randomness from the hedged portfolio leaves a partial differential equation for the option value, the Black-Scholes equation, whose solution is the price. This post derives the equation and solves it [1], [2]. The stock follows the geometric Brownian motion $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$ , and a riskless bond grows at rate $r$ , $dB_t=rB_t\,dt$ .

#The hedging argument

Throughout we assume $r$ and $\sigma$ are constant, the stock pays no dividends, and the market is frictionless with continuous trading and unrestricted shorting. A European option pays a function $h(S_T)$ of the terminal price at a fixed expiry $T$ , and under the Markov dynamics of the geometric Brownian motion its no-arbitrage value before then is a function $V(t,S)$ of time and the current stock price. Apply Ito's formula to track how $V$ evolves.

Theorem1

If $V(t,S)$ is the value of an option on the stock and is twice differentiable, then absence of arbitrage forces the Black-Scholes equation

\partial_t V+rS\,\partial_S V+\tfrac12\sigma^2 S^2\,\partial_{SS}V-rV=0, \tag{1}

with the terminal condition $V(T,S)$ equal to the option's payoff. Among solutions of linear growth, $\abs{V(t,S)}\le C(1+S)$ , the price is the unique one, which is what makes the risk-neutral representation of Theorem 2 the solution rather than merely a solution.

Proof

By the Ito formula applied to $V(t,S_t)$ with $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$ and quadratic variation $dS_t^2=\sigma^2 S_t^2\,dt$ ,

dV_t=\Big(\partial_t V+\mu S_t\,\partial_S V+\tfrac12\sigma^2 S_t^2\,\partial_{SS}V\Big)dt+\sigma S_t\, \partial_S V\,dW_t. \tag{2}

Form the delta-hedged portfolio $\Pi_t=V_t-\Delta_t S_t$ holding one option short one $\Delta_t$ of stock, and choose $\Delta_t=\partial_S V$ . Read it as an admissible self-financing strategy in stock and bond, so the cost of rebalancing $\Delta_t$ is borne by the bond holding and the cross terms $S_t\,d\Delta_t+d\langle\Delta,S\rangle_t$ are absorbed there rather than entering $d\Pi_t$ . Its increment is then $d\Pi_t=dV_t-\partial_S V\,dS_t$ , and substituting Equation (2) the stochastic terms cancel,

d\Pi_t=\Big(\partial_t V+\tfrac12\sigma^2 S_t^2\,\partial_{SS}V\Big)dt, \tag{3}

a purely deterministic increment. A self-financing portfolio whose instantaneous return is deterministic must, on pain of arbitrage against the bond, earn the riskless rate, $d\Pi_t=r\Pi_t\,dt=r(V_t-\partial_S V\, S_t)\,dt$ . Equating the two expressions for $d\Pi_t$ and cancelling $dt$ ,

\partial_t V+\tfrac12\sigma^2 S^2\,\partial_{SS}V=r\big(V-S\,\partial_S V\big), \tag{4}

which rearranges to Equation (1). At expiry the option is worth its payoff, the terminal condition.

The drift $\mu$ has vanished from Equation (1), replaced everywhere by $r$ . The hedge removes not only the risk but the expected return, so the price depends only on volatility, the one feature it cannot cancel.

#Risk-neutral pricing

The disappearance of $\mu$ is the analytic face of a change of probability. Under the risk-neutral measure obtained by the Girsanov theorem, the stock drifts at $r$ rather than $\mu$ , and the solution of the Black-Scholes equation is a discounted expectation.

Theorem2

The solution of Equation (1) with terminal payoff $h(S_T)$ is $V(t,S_t)=e^{-r(T-t)}\E^{\mathbb Q} [h(S_T)\mid S_t]$ , the expectation taken under the measure $\mathbb Q$ where $dS_t=rS_t\,dt+\sigma S_t\,dW_t ^{\mathbb Q}$ .

Proof

The change of measure that turns $\mu$ into $r$ is the Girsanov transformation with the constant market price of risk $\theta=(\mu-r)/\sigma$ . Since $\theta$ is constant, $\E^{\mathbb P}[e^{\frac12\theta^2T}]=e^{\frac12\theta^2T}<\infty$ , so Novikov's condition holds, the density $Z_t=\exp(-\theta W_t-\tfrac12\theta^2t)$ is a true martingale, $d\mathbb Q=Z_T\,d\mathbb P$ defines an equivalent measure, and $W^{\mathbb Q}_t=W_t+\theta t$ is a $\mathbb Q$ -Brownian motion under which $dS_t= rS_t\,dt+\sigma S_t\,dW^{\mathbb Q}_t$ . Applying Ito and Equation (1) to the discounted value,

d\big(e^{-rt}V(t,S_t)\big)=e^{-rt}\big(\partial_tV+rS\,\partial_SV+\tfrac12\sigma^2S^2\,\partial_{SS}V-rV\big) dt+e^{-rt}\sigma S_t\,\partial_S V\,dW^{\mathbb Q}_t=e^{-rt}\sigma S_t\,\partial_S V\,dW^{\mathbb Q}_t, \tag{5}

the drift killed by the Black-Scholes equation. This is a priori only a local martingale; it is a true martingale once the integrand is square-integrable, $\E^{\mathbb Q}\!\int_0^T e^{-2rt}\sigma^2S_t^2 (\partial_SV)^2\,dt<\infty$ , which holds under the linear-growth bound on $\partial_S V$ (for the call $\partial_SV=\Phi(d_1)\in[0,1]$ ). A true martingale equals the conditional expectation of its terminal value on the filtration, $e^{-rt}V(t,S_t)=\E^{\mathbb Q}[e^{-rT}V(T,S_T)\mid\mathcal F_t]$ . Since $(S_t)$ is a time-homogeneous Markov process under $\mathbb Q$ , the expectation of a function of $S_T$ depends on $\mathcal F_t$ only through $S_t$ , so $\E^{\mathbb Q}[e^{-rT}h(S_T)\mid\mathcal F_t]=e^{-rT}\E^{\mathbb Q} [h(S_T)\mid S_t]$ , and multiplying by $e^{rt}$ gives the discounted expectation.

#The Black-Scholes formula

For a call the payoff is $h(S_T)=(S_T-K)^+$ , and the expectation is a lognormal integral with a closed form.

Theorem3

The value of a European call with strike $K$ and expiry $T$ is, writing $\tau=T-t$ ,

V=S\,\Phi(d_1)-Ke^{-r\tau}\Phi(d_2),\qquad d_{1,2}=\frac{\ln(S/K)+(r\pm\tfrac12\sigma^2)\tau}{\sigma\sqrt \tau}, \tag{6}

where $\Phi$ is the standard normal distribution function.

Proof

Under $\mathbb Q$ the solution of the stock equation is $S_T=S\exp\big((r- \tfrac12\sigma^2)\tau+\sigma\sqrt\tau\,Z\big)$ with $Z$ a standard normal, so $S_T>K$ exactly when $Z>-d_2$ . By Theorem 2,

V=e^{-r\tau}\E^{\mathbb Q}\big[(S_T-K)^+\big]=e^{-r\tau}\int_{-d_2}^\infty\Big(S\,e^{(r-\frac12\sigma^2)\tau +\sigma\sqrt\tau z}-K\Big)\varphi(z)\,dz, \tag{7}

where $\varphi$ is the standard normal density. Since $\E^{\mathbb Q}[S_T]=Se^{r\tau}<\infty$ (the lognormal first moment), $S_T\mathbf 1_{\{Z>-d_2\}}$ is integrable and the integral splits into two finite pieces. The $K$ term integrates to $Ke^{-r\tau}\int_{-d_2}^\infty\varphi=Ke^{-r\tau}\Phi(d_2)$ by $\int_{-a}^\infty \varphi=\Phi(a)$ . For the $S$ term, completing the square turns $e^{\sigma\sqrt\tau z}\varphi(z)=e^{\sigma^2\tau/2}\varphi(z-\sigma\sqrt\tau)$ , so

e^{-r\tau}S\,e^{(r-\frac12\sigma^2)\tau}\int_{-d_2}^\infty\varphi(z-\sigma\sqrt\tau)\,dz=S\int_{-d_2-\sigma \sqrt\tau}^\infty\varphi(u)\,du=S\,\Phi(d_2+\sigma\sqrt\tau)=S\,\Phi(d_1), \tag{8}

the exponential prefactors collapsing to $1$ and $d_2+\sigma\sqrt\tau=d_1$ . Subtracting gives Equation (6).

The formula prices the call from five inputs, the stock price, the strike, the time, the rate, and the volatility, of which only volatility is unobserved, so option markets quote volatility. The hedge ratio is $\partial_S V=\Phi(d_1)$ , the delta, the amount of stock the replicating portfolio holds, and following it continuously is the trading strategy that manufactures the option. The Black-Scholes equation is the meeting point of the whole curriculum, the Ito formula supplying the dynamics, the change of measure supplying the risk-neutral expectation, the Gaussian supplying the lognormal integral, and the absence of arbitrage supplying the principle, the abstract machinery of stochastic analysis resolving into a formula a trader can evaluate.

[1]

F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, vol. 81, no. 3, pp. 637–654, 1973.

[2]

S. E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models. Springer, 2004.

Explore connections

see in the atlas

cite

@misc{black-scholes-pde,
  author = {Zac Kienzle},
  title  = {The Black-Scholes Equation},
  year   = {2026},
  month  = {06},
  url    = {https://zackienzle.com/blog/black-scholes-pde}
}

#The hedging argument

Theorem1

If $V(t,S)$ is the value of an option on the stock and is twice differentiable, then absence of arbitrage forces the Black-Scholes equation

\partial_t V+rS\,\partial_S V+\tfrac12\sigma^2 S^2\,\partial_{SS}V-rV=0, \tag{1}

Proof

By the Ito formula applied to $V(t,S_t)$ with $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$ and quadratic variation $dS_t^2=\sigma^2 S_t^2\,dt$ ,

dV_t=\Big(\partial_t V+\mu S_t\,\partial_S V+\tfrac12\sigma^2 S_t^2\,\partial_{SS}V\Big)dt+\sigma S_t\, \partial_S V\,dW_t. \tag{2}

d\Pi_t=\Big(\partial_t V+\tfrac12\sigma^2 S_t^2\,\partial_{SS}V\Big)dt, \tag{3}

\partial_t V+\tfrac12\sigma^2 S^2\,\partial_{SS}V=r\big(V-S\,\partial_S V\big), \tag{4}

which rearranges to Equation (1). At expiry the option is worth its payoff, the terminal condition.

#Risk-neutral pricing

Theorem2

Proof

d\big(e^{-rt}V(t,S_t)\big)=e^{-rt}\big(\partial_tV+rS\,\partial_SV+\tfrac12\sigma^2S^2\,\partial_{SS}V-rV\big) dt+e^{-rt}\sigma S_t\,\partial_S V\,dW^{\mathbb Q}_t=e^{-rt}\sigma S_t\,\partial_S V\,dW^{\mathbb Q}_t, \tag{5}

#The Black-Scholes formula

For a call the payoff is $h(S_T)=(S_T-K)^+$ , and the expectation is a lognormal integral with a closed form.

Theorem3

The value of a European call with strike $K$ and expiry $T$ is, writing $\tau=T-t$ ,

V=S\,\Phi(d_1)-Ke^{-r\tau}\Phi(d_2),\qquad d_{1,2}=\frac{\ln(S/K)+(r\pm\tfrac12\sigma^2)\tau}{\sigma\sqrt \tau}, \tag{6}

where $\Phi$ is the standard normal distribution function.

Proof

Under $\mathbb Q$ the solution of the stock equation is $S_T=S\exp\big((r- \tfrac12\sigma^2)\tau+\sigma\sqrt\tau\,Z\big)$ with $Z$ a standard normal, so $S_T>K$ exactly when $Z>-d_2$ . By Theorem 2,

V=e^{-r\tau}\E^{\mathbb Q}\big[(S_T-K)^+\big]=e^{-r\tau}\int_{-d_2}^\infty\Big(S\,e^{(r-\frac12\sigma^2)\tau +\sigma\sqrt\tau z}-K\Big)\varphi(z)\,dz, \tag{7}

e^{-r\tau}S\,e^{(r-\frac12\sigma^2)\tau}\int_{-d_2}^\infty\varphi(z-\sigma\sqrt\tau)\,dz=S\int_{-d_2-\sigma \sqrt\tau}^\infty\varphi(u)\,du=S\,\Phi(d_2+\sigma\sqrt\tau)=S\,\Phi(d_1), \tag{8}

the exponential prefactors collapsing to $1$ and $d_2+\sigma\sqrt\tau=d_1$ . Subtracting gives Equation (6).

[1]

F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, vol. 81, no. 3, pp. 637–654, 1973.

[2]

S. E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models. Springer, 2004.

Explore connections

see in the atlas

cite

@misc{black-scholes-pde,
  author = {Zac Kienzle},
  title  = {The Black-Scholes Equation},
  year   = {2026},
  month  = {06},
  url    = {https://zackienzle.com/blog/black-scholes-pde}
}