Black-Scholes equation

The price $V(t,s)$ of a sufficiently regular European claim satisfies

Differentiating the displayed transformation shows that it preserves the Black-Scholes equation. The first derivative coefficient fixes the exponent $1-2r/{\sigma }^2$, which also makes the zero-order coefficient correct. Subtracting this reflected image of a European call option from the call itself enforces a zero boundary at $S=B$. For strike price $K\ge B$, the reflected maturity contingent claim payoff vanishes on $S>B$, giving the no-rebate down-and-out option formula. Prices using this reflection are conditional on the barrier not having been hit already.

If holding one unit of a commodity costs a constant cash amount $\delta dt$ for storage, its gain is $dS-\delta dt$. A delta hedge long a derivative and short $V_S$ commodity units has riskless drift $V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}+\delta V_S$ and value $V-S{V}_S$. Equating the drift to the bank return gives the displayed Black-Scholes equation. This is an additive cash carry term, unlike a proportional dividend or storage yield.

The corresponding pricing diffusion is $d\hat{S}=(r\hat{S}+\delta )dt+\sigma \hat{S}dW$. A fixed-strike forward has value $S+\delta (1-e^{-r\tau })/r-K{e}^{-r\tau }$, and put-call parity gives that same value for a matching call minus put. Replication buys the commodity and a bank reserve equal to the present value of its future storage payments. For $r=0$, the reserve is $\delta \tau$.

For the dividend-free Black-Scholes equation, the displayed transform preserves solutions when $B>0$ and $\sigma >0$. Put $x=\log (S/B)$ and $v(x,t)=V(B{e}^x,t)$. The equation becomesSubstitution shows that $\hat{v}(x,t)=e^{qx}v(-x,t)$ solves the same equation when $q=-2b/{\sigma }^2=1-2r/{\sigma }^2$: its first-derivative coefficient becomes $-{\sigma }^{2}q-b=b$, and the additional zeroth-order term ${\sigma }^2{q}^2/2+bq$ vanishes. Subtracting the reflected solution enforces a zero boundary value at $S=B$. Its use in a killed barrier option is restricted to the alive side of the barrier; the algebraic extension to the other side is generally not a valid price.

In the Black-Scholes equation with continuous stock dividends, separation of variables gives $g^{\prime }=(r-\lambda )g$ and ${\sigma }^2{S}^2{f}^{\prime \prime }/2+(r-q)S{f}^{\prime }-\lambda f=0$. Substituting $f=S^{\alpha }$ gives the displayed eigenvalue. A stationary solution uses the two roots of ${\sigma }^2\alpha (\alpha -1)/2+(r-q)\alpha -r=0$. For positive volatility and $r>0$, their product is negative, so they are distinct real roots.

When the underlying stock pays dividends at rate $\theta S_t$, its gain is $d{S}_t+\theta S_{t}dt$. Under the risk-neutral measure its ex-dividend drift is $(\rho -\theta )S_t$. A sufficiently smooth terminal-claim value obeys $p_t+(\rho -\theta )x{p}_x+{\sigma }^2{x}^2{p}_{xx}/2-\rho p=0$. These are dividends on the stock, not a source term paid by the claim itself.

With constant continuous stock-dividend yield $d$, the risk-neutral measure gives stock drift $r-d$ but cash claims are discounted at $r$. The same terminal payoff in a no-dividend Black-Scholes model with interest rate $r-d$ has the identical stock-transition law but a different discount factor. Their prices therefore obey the displayed identity, under the usual payoff integrability conditions.

If the claim holder receives a cash-flow rate $k(S_t,t)$ before maturity, the self-financing hedge matches the claim's cum-dividend gain $dp+kdt$. Its stock holding is the option delta $p_x$ and its bank-account value is $p-x{p}_x$. Comparing Itô formula with portfolio gains gives the displayed equation, with terminal value equal to the terminal payoff. The positive source term $k$ represents dividends paid by the claim; dividends on the underlying stock would instead alter the stock's pricing drift. Risk-neutral pricing includes both the discounted terminal payoff and the integral of discounted interim payments.

In the Black-Scholes model, suppose a stock pays continuous dividend yield $q$, a claim pays cash flow $f(t)S_{t}dt$, and its terminal financial payoff is $c{S}_T$. If the discounted cash-flow integral is finite, its risk-neutral pricing value is the displayed expression. To derive it, substitute $V=S\varphi (t)$ into the Black-Scholes equation with claim dividends, including stock drift $(r-q)S$. This gives ${\varphi }^{\prime }-q\varphi =-f$, $\varphi (T)=c$, which an integrating factor solves. The interest rate cancels because all payments are proportional to the same stock.

A smooth value function satisfying the Black-Scholes equation admits a self-financing portfolio with the displayed delta hedge. An arbitrary decomposition $p=xg+Bh$ need not be self-financing. For example $p=0$, $g=1$, $h=-x/B$ solves the value equation but has portfolio gains $dS-\rho Sdt$, which are not zero when volatility is positive. For specified holdings, self-financing is equivalent to the value equation together with $g=p_x$.