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In the Black-Scholes model, a digital call option and digital put option with remaining maturity $\tau$ have valueswhere $d_{-}$ is defined in the Black-Scholes formula. For $t<T$, the digital-call delta hedge is

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 $g=p_x$.

In the cash-numéraire market $d{S}_t=a(S_t)d{W}_t$, let a smooth call value solve $V_t+\frac{1}{2}{a}^2{V}_{SS}=0$. Its delta hedge is $U=V_S$. Differentiating the pricing equation giveswith the terminal value at the strike understood up to the null probability of the diffusion hitting that exact terminal value. By Itô formula, $dU(t,S_t)=-a{a}^{\prime }{U}_{S}dt+a{U}_{S}d{W}_t$, while $dV(t,S_t)=U(t,S_t)d{S}_t$. Thus holding $U(t,S_t)$ shares and cash $V-SU$ replicates the terminal call. The drift in the option delta equation comes from differentiating the spatially varying diffusion coefficient, not from a drift in the original stock.

If $V$ satisfies the pricing equation for a local volatility model, hold $h_t^S=V_S(t,S_t)$ stocks and $h_t^B=[V(t,S_t)-S_t{V}_S(t,S_t)]/B_t$ units of the bank account. Their value is $V(t,S_t)$. The pricing equation for a local volatility model and Itô formula giveThus this is a self-financing strategy, with terminal value $g(S_T)$. It is a delta hedge, and nonnegative $V$ makes its wealth an admissible trading strategy. The stock and bank wealth amounts are $S_t{V}_S$ and $V-S_t{V}_S$, rather than the asset-unit holdings.

In the zero-interest local volatility model $d{S}_t=\sqrt{S_t}d{W}_t^Q$, the risk-neutral pricing value of the European contingent claim $e^{v{S}_T}$, for $v>0$ and $v(T-t)<2$, isThe compound Poisson transition law of a driftless square-root diffusion proves finiteness and this conditional-expectation formula. Alternatively, substituting $V(t,S)=e^{a(t)S+b(t)}$ into the pricing equation for a local volatility model gives $a^{\prime }+a^2/2=0$ and $b^{\prime }=0$, with terminal values $a(T)=v,b(T)=0$. Hence $a(t)=v/[1-v(T-t)/2]$ and $b(t)=0$. The conditional-expectation calculation verifies that this formal solution is the price of the unbounded claim. The delta hedge holds $a(t)V$ stocks and $(1-a(t)S_t)V/B_t$ bank units.

The Greeks are sensitivities of a financial claim value to model inputs, expressed through partial derivatives. The option delta measures sensitivity to the underlying price; a delta hedge uses that sensitivity to cancel first-order price exposure.

The option delta is the derivative of an option value with respect to the underlying asset price. In a smooth complete diffusion model, it is also the number of underlying units in the replicating delta hedge.

Assume a frictionless market with no taxes or transaction costs, unrestricted short selling and borrowing/lending at the same constant rate $r$, continuous trading, and absence of arbitrage. The non-dividend-paying stock has geometric Brownian motion dynamics $dS=\mu Sdt+\sigma SdW$ with constant $\sigma >0$, and the bank account satisfies $dB=rBdt$. A contingent claim with no interim cash flows has sufficiently regular value $f(t,S)$, meaning $f\in C^{1,2}$ before maturity; terminal financial payoff kinks can be handled by the smooth value for earlier times. Trading strategies are adapted, self-financing portfolios, and admissible trading strategies so that doubling strategies are excluded.

The Itô formula givesChoose $\Delta =f_S$ shares and put the remaining value $f-S{f}_S$ in the bank account. Its self-financing gain isMatching the contingent claim's gain to this replicating gain cancels the random increment and equates the drifts:This is the Black-Scholes equation. The argument uses the self-financing gain $\Delta dS$, not a product differential that incorrectly treats a changing $\Delta$ as free cash. Rebalancing purchases are financed from the bank account. Conversely, a smooth solution with suitable growth and boundary conditions defines a delta hedge whose value replicates the contingent claim; law of one price identifies this value with the contingent claim price. Here $S$ is the current stock price, $t$ is time, $\sigma$ is proportional volatility, $r$ is the risk-free rate, and $f$ is the contingent claim price. The physical expected return $\mu$ cancels because exposure to price risk has been hedged.

Let $T=t_0$, $\tau =T-t$, and let $N$ have the standard normal distribution. For a twice differentiable payoff with enough growth control for the following expectations and derivatives, risk-neutral pricing in the dividend-free Black-Scholes model givesThe conditional discounted payoff is a martingale under the risk-neutral measure. Applying the Itô formula to $e^{-\rho s}V(S_s,s)$ and setting its drift to zero therefore gives the Black-Scholes equationThe delta hedge holds $\Delta =V_x$ shares; the remaining bond value is $\beta =V-x\Delta$, or $h=\beta /e^{-\rho (T-t)}$ units of the zero-coupon bond. Substitution in the Itô formula and the displayed equation gives $dV=\Delta dS+hdB$, proving replication and self-financing.

Differentiating the pricing expectation in the current stock price gives its option delta and option gamma:Thus a nondecreasing payoff has a nondecreasing price; a nonincreasing payoff has a nonincreasing price. A convex payoff has $\Gamma \ge 0$, and a concave payoff has $\Gamma \le 0$. This is convexity preservation in Black-Scholes pricing.

The Black-Scholes parameter sensitivities follow directly from the same expectation. Differentiation in the interest rate gives the option rhoFor spot volatility, differentiation first givesThe Gaussian integration by parts identity $\mathbb{E}[N\psi (N)]=\mathbb{E}[{\psi }^{\prime }(N)]$, applied to $\psi (N)=f^{\prime }(xM)xM$, cancels the term containing $f^{\prime }$ and gives the option vegaConsequently increasing spot volatility increases a convex payoff's value and decreases a concave payoff's value, with weak inequalities in affine or degenerate cases. A short bond position gives positive option rho, while a long bond position gives negative option rho. The physical stock drift $\mu$ does not enter these prices: the risk-neutral measure replaces it by $\rho$.

The option theta is the calendar-time derivative at fixed current stock price. The Black-Scholes equation givesFor $\rho \ge 0$, a convex payoff with $\beta <0$ therefore has $V_t\le 0$, while a concave payoff with $\beta >0$ has $V_t\ge 0$. The signs are strict when $\rho >0$. At $\rho =0$, an affine payoff can give equality. Remaining-maturity sensitivity is $V_T=-V_t$ when the payoff is independent of $T$. All these time comparisons hold at fixed current stock price.

The stated time-monotonicity conclusion needs the nonnegative-rate condition. For example, the convex affine payoff $f(x)=x-K$ hasThus with negative interest rates it provides an explicit counterexample even though the replica is short in bonds.

A self-financing portfolio changes its holdings using only its existing wealth. If it holds $g_t$ shares and $h_t$ units of a traded bond or bank account $B_t$, its value $V_t=g_t{S}_t+h_t{B}_t$ obeys the gain identityThe changes in holdings do not add or withdraw capital. Set $T=t_0$, $B_t=e^{-\rho (T-t)}$, so $d{B}_t=\rho B_{t}dt$, and let $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, with $\sigma >0$. Write $V_t=p(S_t,t)$ and suppress the arguments of the smooth holdings $g,h$.

Differentiate $p=xg+B_{t}h$:The Itô formula minus the proposed portfolio gains givesUniqueness of the local-martingale and finite-variation decomposition shows that self-financing requires $p_x=g$. Since $S_t$ has full support on $(0,\infty )$ for each $t>0$, smoothness gives the corresponding functional identity for every positive $x$, and continuity extends it to the interval endpoints. ThusDifferentiating this identity once more in $x$ gives $g_x+x{g}_{xx}+B_t{h}_{xx}=0$, so $p_{xx}=g_x$. Substitute both identities into the drift condition to obtainConversely these equations give $p_x=g$, $p_{xx}=g_x$ and zero drift and diffusion in the displayed gain difference. They therefore prove the claimed self-financing conditions for smooth stock and bond holdings in both directions.

Using $B_{t}h=p-xg$ in the gain identity and setting $g=p_x$, self-financing impliesThis is the Black-Scholes equation, independent of the physical drift $\mu$. Conversely, if a smooth $p$ satisfies it, define its delta hedge byThe preceding Itô formula verifies that these holdings are self-financing.

For prescribed arbitrary holdings, the value equation alone is insufficient. The corrected equivalence is the Black-Scholes value equation needs delta-compatible holdings: a specified portfolio is self-financing if and only if its value solves the equation and $g=p_x$. Indeed $p=0$, $g=1$, $h=-x/B_t$ satisfies the value equation but its proposed gains are $d{S}_t-\rho S_{t}dt$, which have nonzero Brownian part while $dp=0$. Thus the printed final equivalence is valid as an existence assertion for the delta hedge, not for every decomposition of the same value. No change to the valid two holdings equations is needed.

With continuous underlying stock dividends, the stock's gain is $d{S}_t+\theta S_{t}dt$, so self-financing becomes $d{V}_t=g_t(d{S}_t+\theta S_{t}dt)+h_{t}d{B}_t$. Under the risk-neutral measure the ex-dividend stock drift is $(\rho -\theta )S_t$. Matching its diffusion still gives $g=p_x$, and matching its gain drift gives the Black-Scholes equation with continuous stock dividendsOnly the coefficient of $x{p}_x$ changes. The $-\rho p$ discount term remains, because the claim is financed at the bank rate; there is no separate source term for a dividend paid by the claim itself. The dividends here are paid by the underlying stock.

Let ${\varphi }_t(y)=(2\pi t{)}^{-1/2}{e}^{-y^2/(2t)}$ be the centered normal probability density with variance $t$, and take $t>0$. For a zero-drift Brownian motion $B$, reflect the path after its first hit of $a>0$. The Brownian reflection principle identifies endpoints $y<a$ on paths that have hit $a$ with endpoints $2a-y>a$. Thus the endpoint density on paths staying below $a$ is ${\varphi }_t(y)-{\varphi }_t(2a-y)$ for $y<a$.

To introduce drift $\nu$, use the Girsanov theorem with likelihood $e^{\nu B_t-{\nu }^{2}t/2}$. The coordinate path then has the law of $W_s^{\nu }=B_s+\nu s$. Since the likelihood depends only on the endpoint, the joint event in question has probabilityCompleting the two squares givesConsequently the joint endpoint and maximum law for drifted Brownian motion isThe endpoint and the maximum have no relevant atom at the boundary, so the same integral includes $x=a$.

For the down-and-in claim, use non-dividend-paying Black-Scholes model dynamics under the risk-neutral measure. Put $T=t_0$ andLet ${\tau }_b$ be the first time the stock reaches $b$. If $Y_T\le \ell$, continuity ensures ${\tau }_b\le T$, so these terminal values contribute the payoff $f(S_T)1_{S_T\le b}$. For $y>\ell$, lower-barrier reflection followed by the same exponential tilt gives the endpoint density on paths that have hit the barrier:Indeed at zero drift the reflected density is ${\varphi }_T(y-2\ell )$, and multiplying by $e^{\nu y-{\nu }^{2}T/2}$ gives the displayed expression.

Thus the remaining, undiscounted payoff expectation isSet $z=y-2\ell$ andThen $e^{2\nu \ell }={\kappa }^{-\nu /\sigma }$, $S_0{e}^{\sigma y}=S_0{e}^{\sigma z}/\kappa$, and $z>-\ell$ is equivalent to $S_0{e}^{\sigma z}>\kappa b$. The integration variable $z$ has the ordinary terminal log-price density. Adding the first contribution therefore proves the static terminal-payoff representation of a down-and-in claim:This holds for nonnegative measurable payoffs, and for signed payoffs whenever the expectations are absolutely integrable. It is an equality of prices, not equality of pathwise terminal payments.

For the call, write $C(s,c,\tau )$ for the ordinary European call option value, with remaining lifetime $\tau >0$ and strike $c>b$. Before activation take current spot $s>b$. The first term in $g$ is zero because the call pays nothing below $b$. The second term reduces to ${\kappa }^{-\nu /\sigma -1}(x-\kappa c{)}^{+}$, since $\kappa c>\kappa b$. The homogeneity $C(s,\kappa c,\tau )=\kappa C(s/\kappa ,c,\tau )$ therefore gives the inactive down-and-in claim valueAt $s=b$ its value is $C(b,c,\tau )$, as required by activation. Its pre-activation option delta, however, has boundary limitwhereas the active contract's delta hedge holds $C_s(b,c,\tau )$ shares. The stock holding consequently jumps byThis jump is strictly positive, even for negative interest rates. To verify that there is no exceptional cancellation, put $z=\log (s/b)/\sigma$ and $y=\log (S_T/b)/\sigma$. The density for log-price paths killed at the lower barrier isThe down-and-out claim is the ordinary call minus $I$. Differentiate its value at $z=0+$: the bracket vanishes there and its derivative is $2y{\varphi }_{\tau }(y)/\tau$. Consequently its derivative with respect to the spot at $b$ isThis derivative is exactly the displayed delta jump. Thus on reaching the barrier before maturity the number of shares changes discontinuously. The portfolio value remains continuous; the bank position changes to finance the share adjustment. Hitting precisely at maturity has probability zero and does not alter this conclusion. This is the delta jump at activation of a down-and-in call.

The option delta is a derivative in $S$, not in $x$. With $d=e^{-r(T-t)}$, the logarithmic chain rule givesThus use ${\Delta }_i=d{e}^{-x_i}{D}_x{u}_i$ for the delta hedge, and ${\Gamma }_i=d{e}^{-2x_i}(D_{xx}{u}_i-D_x{u}_i)$ for option gamma. These formulas also identify the finite-difference option Greeks correctly on the equally spaced log-price grid.

There are two distinct accuracy statements. A smooth exact function has central first- and second-derivative truncation errors of order $h^2$. But the computed values themselves have error $e_i=O(h^2)$ under the coupled refinement. Without further control on its spatial variation,Therefore the value error bound alone guarantees only first-order delta accuracy and does not guarantee convergent gamma. For example grid error $e_i=h^2\sin (\pi i/2)$ has a central first difference of order $h$ and a second difference of order one, despite second-order value accuracy.

If instead the scheme has a smooth error expansion $e_i=h^{2}E(x_i,t)+o(h^2)$ with corresponding derivative bounds, both Greeks can retain second-order accuracy at smooth interior points. Hence reliable gamma is possible with additional regularity and a verified Greek refinement, but cannot be inferred merely from the value convergence in part (e). This qualification is important because the stated bounded-volatility assumption supplies no such error-derivative control.

Take a non-dividend-paying stock with physical dynamics $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$ and continuous-time bank account $B_t=e^{\rho t}$. Write $T=t_0$ and assume the candidate claim price $p(x,t)$ is $C^{2,1}$ before maturity, with enough growth control for the hedge to be admissible. The terminal condition is $p(x,T)=f(x)$.

The Itô formula gives the claim's price changeA self-financing portfolio worth $p$ with $\Delta$ shares has bank-account value $p-\Delta S_t$. Its gain isTo replicate the claim, the Brownian coefficients must agree, forcing the delta hedge $\Delta =p_x$. Equating the remaining drift coefficients cancels the physical drift $\mu$ and yieldsThe cancellation explains why the Black-Scholes equation contains the interest rate, not the physical stock drift. Conversely, if an admissible solution of this equation is given, holdings $p_x$ in the stock and $(p-x{p}_x)/B_t$ in the bank account have value $p$ and gains exactly $dp$. They are therefore a self-financing portfolio replicating the terminal payoff. Absence of arbitrage forces the claim to have this price. Merely decomposing the current value into arbitrary holdings would not justify self-financing; the gain identity and delta hedge are essential.

Now suppose the claim holder receives the cash rate $k(S_t,t)$ before $T$. The price $p$ is the ex-dividend value, so its total gain is $dp+kdt$. The replicating portfolio pays the same amounts out of its wealth, with no external injections. Thus its gain identity isAgain the Brownian coefficients force $\Delta =p_x$. Comparing drifts now gives the Black-Scholes equation with claim dividends:In particular the cash-flow source has a positive sign in the left-hand side. It is not an underlying-stock dividend yield and does not replace the drift $\rho x$ by a dividend-adjusted drift.

For another justification, under the risk-neutral measure $Q$, discount the claim's cum-dividend gain. The Itô formula shows thatexactly when the displayed equation holds. Subject to the usual integrability making this a true martingale, conditional expectation at maturity givesPositive interim payments therefore add positive value, which also checks the source sign. For instance, a constant payment rate $k_0$ and zero terminal payoff have price $k_0(1-e^{-\rho (T-t)})/\rho$, interpreted as $k_0(T-t)$ at $\rho =0$; substitution satisfies $p_t-\rho p+k_0=0$.

Let $T=t_0$, $\tau =T-t$, $x=S_t>0$, and $\sigma >0$. Use a dividend-free Black-Scholes model with constant interest rate $\rho$. The terminal payoff is assumed independent of the parameters being varied. In addition to twice differentiability, impose enough integrability to differentiate the pricing formula; for example, $f,f^{\prime },f^{\prime \prime }$ of polynomial growth suffice. Twice differentiability alone is not enough: the smooth payoff $f(x)=e^{x^2}$ has infinite expectation under every nondegenerate lognormal distribution.

Under the risk-neutral measure $Q$, future stock values conditional on $S_t=x$ have the formClaim replication, or equivalently the martingale property of discounted value in the complete market, givesThe replicating strategy holds the option delta $p_x$ shares and bank-account value $\beta =p-x{p}_x$. No physical drift parameter appears: the Girsanov theorem replaces the physical drift by $\rho$ under the pricing measure, and the replication argument cancels it directly. The risk-neutral pricing operator is linear in the payoff and preserves pointwise payoff inequalities.

Here is a verification of the Black-Scholes equation using the integral itself. Differentiation in spot givesDifferentiating with respect to $\tau$ givesFor $G(Z)=S_T{f}^{\prime }(S_T)$, the Gaussian integration by parts identity $\mathbb{E}[ZG]=\mathbb{E}[G^{\prime }]$ yieldsSubstitution cancels the $-{\sigma }^2/2$ term and provesSince $p_t=-p_{\tau }$, this is precisely the Black-Scholes equation. As $\tau \downarrow 0$, $M\to 1$ and growth control permits passage to the limit, giving $p(x,T)=f(x)$. The delta hedge then supplies the admissible replicating gains under the same regularity assumptions.

The Black-Scholes parameter sensitivities describe the dependence on the model parameters without assuming a particular payoff. If $f$ is increasing, then $p_x\ge 0$. If $f$ is convex, then $p_{xx}\ge 0$, so its price is convex in spot. The option gamma $p_{xx}$ measures the response of the option delta to spot changes.

For the interest rate, the distribution of the future stock also changes with $\rho$, so looking only at discounting gives the wrong general answer. Differentiation gives the option rhoThus a short bank-account position gives positive rate sensitivity, while a positive bank-account position gives negative rate sensitivity. For a cash payoff the result is negative; for the payoff $f(x)=x$ it is zero.

For spot volatility, differentiate the lognormal multiplier and then use Gaussian integration by parts:Consequently a convex payoff has nonnegative option vega. This monotonicity can also be understood by a mean-preserving increase of dispersion in the discounted stock. Strict positivity requires nonzero curvature seen with positive probability; an affine payoff has zero option vega.

Time enters through the remaining maturity $\tau$. Holding spot fixed, the Black-Scholes equation givesUnder the conventional assumption $\rho \ge 0$, a convex payoff and a short bond position $\beta <0$ imply $p_t\le 0$. The inequality is strict if $\rho >0$, or if $p_{xx}>0$ with positive volatility. Thus the price at fixed stock value is nonincreasing in calendar time, and nondecreasing in maturity, under these hypotheses. It is not a claim that the realized stochastic price path decreases: that path still has Brownian exposure $\sigma S_t{p}_{x}d{W}_t^Q$.

The printed conclusion needs the nonnegative-rate qualification; it is false for arbitrary negative rates. For $K>0$, the twice-differentiable convex payoff $f(x)=x-K$ hasThis is an exact counterexample, not a failure of the pricing formula. Also, with $\rho =0$ and an affine payoff, “decreasing” can only mean nonincreasing. Without payoff monotonicity or the specified bond-position sign, neither the spot nor rate nor time sensitivity has a universal sign. Any additional payoff parameter, such as a strike, is differentiated through $f$; pointwise payoff ordering, rather than a generic parameter rule, determines its price effect.

Consider a binomial market with dates $0,\dots ,n$, bank account $B_k=b^k$ for $b>0$, and stock price $S_{k+1}=S_{k}u$ or $S_{k}d$, where $0<d<u$. Both moves have positive physical probability, and the information at date $k$ is the history of the first $k$ moves. Absence of arbitrage requires and is implied by $d<b<u$. If $b\le d$, borrowing cash to buy stock has nonnegative payoff with a strict gain in one state; if $b\ge u$, the reverse trade does. In the interior range, the risk-neutral probability of an up move isIt makes ${\mathbb{E}}_Q[S_{k+1}\mid {\mathcal{F}}_k]=b{S}_k$, so $S_k/B_k$ is a martingale. This proves absence of arbitrage by pricing discounted gains under the equivalent measure.

At a node with stock price $s$, let the desired next-period portfolio values be $V_u,V_d$. Solve the two replication equations $\Delta su+\beta B_{k+1}=V_u$ and $\Delta sd+\beta B_{k+1}=V_d$. The replicating portfolio in a binomial market hasIts current value isRebalance at the next node using exactly the value supplied by the previous holdings, so the strategy is self-financing. Backward induction from any terminal contingent claim $H$ gives its hedge and the unique priceThus the finite binomial market is a complete market. The construction works for path-dependent claims on the full history tree, even when their values do not recombine merely as functions of the current stock price. For $H=f(S_n)$, independence under $Q$ gives the explicit time-zero priceThe stock hedge is a difference quotient of successor values, the discrete analogue of the delta hedge.

For the Cox-Ross-Rubinstein diffusion approximation, fix calendar maturity $T$, take $h=T/n$, and choose $u=e^{\sigma \sqrt{h}}$, $d=e^{-\sigma \sqrt{h}}$, and the one-step bank factor $b=e^{\rho h}$. For sufficiently fine mesh, $d<b<u$. The risk-neutral probability expands asEach stock log-increment is $\pm \sigma \sqrt{h}$. Its mean under $Q$ is $(\rho -{\sigma }^2/2)h+O(h^2)$ and its variance is ${\sigma }^{2}h+O(h^2)$. Independent increments, their vanishing maximum size, and the Donsker invariance principle for this triangular array give the path limitExponentiation gives the geometric Brownian motion of the Black-Scholes model, and the bank account tends to $e^{\rho t}$.

The physical model is matched by taking $p_h=(e^{\mu h}-d)/(u-d)$. Its corresponding limit has log drift $\mu -{\sigma }^2/2$, so $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t^P$. The likelihood ratios from part (ii) have the matching continuous-time limit. With $\lambda =(\mu -\rho )/\sigma$, a Taylor expansion of their two possible log-multipliers giveswhere ${\epsilon }_k$ is $+1$ or $-1$. Thus the limiting Radon-Nikodym derivative is the Girsanov theorem density $\exp (-\lambda W_T^P-{\lambda }^{2}T/2)$, turning the physical stock drift into $\rho$.

The binomial risk-neutral pricing formula converges to $e^{-\rho T}{\mathbb{E}}_Q[f(S_T)]$ for bounded continuous terminal payoffs by convergence in distribution. For ordinary call and put payoffs, uniform stock second-moment bounds give the required uniform integrability and hence convergence despite unboundedness. Bounded payoffs with discontinuities only at events of zero limiting probability are handled by the same weak-convergence criterion. Corresponding continuous path functionals can be treated at the process level. Arbitrary discontinuous path claims do not follow from weak convergence without further conditions. In the smooth limiting model, the discrete hedging difference quotient becomes the delta hedge $V_x$, and its value satisfies the Black-Scholes equation.

Write $T=t_0$ and $B_t=e^{-\rho (T-t)}$ for the risk-free asset paying one at $T$. Under the physical measure, $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, with $\sigma >0$, while $d{B}_t=\rho B_{t}dt$. A self-financing portfolio with predictable stock and bond holdings $g_t,h_t$ has value $V_t=g_t{S}_t+h_t{B}_t$ and gainsRebalancing therefore adds no outside money and withdraws none. This gains equation, rather than merely the value decomposition, defines self-financing.

Let $p(x,t)=xg(x,t)+B_{t}h(x,t)$. The Itô formula givesThe prescribed portfolio gains instead have diffusion coefficient $\sigma S_{t}g$ and drift $\mu S_{t}g+\rho B_{t}h$. Equality of diffusion coefficients requires $p_x=g$, equivalentlyUnder this condition, $p_{xx}=g_x$. Equality of drifts is then $p_t+\frac{1}{2}{\sigma }^2{x}^2{g}_x=\rho B_{t}h$. Since $p_t=x{g}_t+B_t{h}_t+\rho B_{t}h$, this becomesThese conditions are also sufficient by the same Itô formula. Necessity initially holds along the stock process. Its positive transition density on $(0,\infty )$ for every positive time, together with continuity of the coefficient functions, extends the identities to all $x>0$ and then to the time endpoints by continuity.