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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 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 this terminal condition interpreted away from $S=K$. If the terminal law has no atom at $K$, its assigned value at that point does not affect expectation pricing; a degenerate or absorbed diffusion can instead require a separate strike-boundary analysis. 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.

For the arithmetic stock model with constant volatility, let $\sigma >0$, $\tau =T-t>0$, $a=K{e}^{-r\tau }$, andThus $\nu =\sigma \sqrt{(1-e^{-2r\tau })/(2r)}$ for $r\ne 0$, and $\nu =\sigma \sqrt{\tau }$ for $r=0$. With $\Phi ,\varphi$ the standard normal distribution function and standard normal density, the European call option price isIndeed, the discounted terminal price is conditionally Gaussian with mean $s$ and variance ${\nu }^2$. Integrating its positive part above the discounted strike price gives this expression. At maturity it reduces continuously to $(s-K{)}^{+}$.

Differentiation gives option delta $C_s=\Phi (d)$. Holding $C_s$ stock units and $(C-s{C}_s)/B$ bank-account units gives a self-financing portfolio replicating the financial payoff, under the usual stochastic-integrability conditions. Its nonnegative wealth is admissible, and the discounted-wealth supermartingale bound shows that this is the least replication capital among nonnegative-wealth strategies.

For positive volatility $\sigma >0$ and positive remaining maturity, the call price in an arithmetic stock model with interest has option delta $\Delta =\Phi (d)$, where $d$ is the standardized discounted moneyness. Since a finite argument of the standard normal distribution function gives a value strictly between zero and one,For a fixed spot $s\ne K$, the maturity limit is $1_{\{s>K\}}$, the derivative of the call financial payoff away from its kink. The payoff derivative is undefined at $s=K$. Under the nondegenerate Gaussian terminal law, the event $S_T=K$ has probability zero; it is this strike event that is exceptional. At zero interest use the same formula with normal scale $\sigma \sqrt{T-t}$.

A delta hedge holds ${\Delta }_t={\partial }_{s}V(t,S_t)$ units of the risky asset. In the Black-Scholes model, this choice cancels the claim's Brownian exposure.

The option delta is one of the Greeks (finance); hedging is the trading procedure that uses this sensitivity.

For strike $c>b$, write $C(s,c,\tau )$ for the ordinary European call option value. Before activation the down-and-in claim is $I(s,\tau )=(b/s{)}^{\alpha }C(b^2/s,c,\tau )$, where $\alpha =2\rho /{\sigma }^2-1$. Consequently its left-limit option delta at the barrier is $-\alpha C(b,c,\tau )/b-C_s(b,c,\tau )$, whereas after activation it is $C_s(b,c,\tau )$. Their difference is strictly positive for $\tau >0$: differentiating the killed normal transition density at its boundary gives the integral of a strictly positive function against the positive call payoff. The value is continuous, so changing the bank holding finances the stock rebalance.

Suppose $a^{\prime }$ is bounded and let $d{Z}_t=Z_t{a}^{\prime }(S_t)d{W}_t$, $Z_0=1$. The Novikov condition makes $Z$ a strictly positive true martingale. For ${\pi }_t=U(t,S_t)$ solving the call delta equation for a driftless local volatility diffusion, the product formula givesThe two drift terms cancel through quadratic covariation. Thus $Z\pi$ is a local martingale. If it is a true martingale on the closed horizon, its terminal value is $Z_T{1}_{\{S_T\ge K\}}$, soThe upper bound uses $\mathbb{E}[Z_T\mid {\mathcal{F}}_t]=Z_t$. Equivalently, the option delta is the terminal-exceedance probability under the measure with density $Z_T$.

The logarithmic coordinate requires the displayed chain-rule factors for option delta and option gamma. If only a value error bound $\Vert e_h{\Vert }_{\infty }=O(h^2)$ is known, central differentiation gives $D_x{e}_h=O(h)$ and $D_{xx}{e}_h=O(1)$. Thus first and second Greek convergence do not follow with the same order from a value bound alone. With a smooth spatial error expansion $e_h=h^{2}E+o(h^2)$ and corresponding derivative control, both Greeks can instead retain second-order convergence away from a nonsmooth payoff or boundary. Refinement and Greek-specific regularity are essential to distinguish these situations.

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 claim $\max (K,S_T)$ for $K>0$ combines a guaranteed cash floor with a European call option, equivalently a stock with a put. In the Black-Scholes model, for $\tau =T-t>0$, its price is $V=S_t\Phi (d_1)+K{e}^{-r\tau }\Phi (-d_2)$, where $d_1=[\log (S_t/K)+(r+{\sigma }^2/2)\tau ]/(\sigma \sqrt{\tau })$ and $d_2=d_1-\sigma \sqrt{\tau }$. Under the risk-neutral lognormal law, the truncated stock moment is $S_t{e}^{r\tau }\Phi (d_1)$ and the probability of ending below the floor is $\Phi (-d_2)$, proving the price. Its option delta is $\Phi (d_1)$, since $S_t\varphi (d_1)=K{e}^{-r\tau }\varphi (d_2)$ cancels the derivative terms. The replicating strategy holds that many stock units and $K{e}^{-r\tau }\Phi (-d_2)/B_t$ bank-account units.

Under the Black-Scholes model with interest rate $\rho$, volatility $\sigma$ and remaining time $\Delta$, risk-neutral pricing gives $V(S)=S[\log S+(\rho +{\sigma }^2/2)\Delta ]$. This follows by differentiating the lognormal exponential moment. Its option delta is $\log S+1+(\rho +{\sigma }^2/2)\Delta$ and its second stock derivative is $1/S$.

The second derivative of an option price with respect to its underlying price. It measures the change of option delta. A convex terminal payoff in the Black-Scholes model has nonnegative option gamma before maturity.

Differentiate the dividend PDE with respect to $S$. The diffusion term contributes ${\sigma }^{2}S{V}_{SS}+\frac{1}{2}{\sigma }^2{S}^2{V}_{SSS}$, and the stock-drift term contributes $(r-\delta )(V_S+S{V}_{SS})$. With $\Delta =V_S$, the coefficient of $\Delta$ is $(r-\delta )-r=-\delta$. ThereforeThe terminal condition is $\Delta (T,S)={\Psi }^{\prime }(S)$ wherever the contingent claim payoff is differentiable, with an appropriate weak or limiting interpretation at contingent claim payoff kinks. In the next part the squared positive-part contingent claim payoff is continuously differentiable, so its terminal option delta has no jump.

The contingent claim payoff derivative is ${\Psi }^{\prime }(s)=2(s-K{)}_{+}$. Differentiating the risk-neutral valuation is justified by the finite lognormal distribution, and $S_T$ is proportional to the starting stock value $s$. ThusFor a lognormal terminal stock, completing the square in its Gaussian exponent givesUse $p=2$ and $p=1$, and let $d_1=[\log (s/K)+(r+{\sigma }^2/2)\tau ]/(\sigma \sqrt{\tau })$. The result isAs $\tau \downarrow 0$ this tends to $2(s-K{)}_{+}$, including value zero at $s=K$, as required. The extra ${\sigma }^2$ in the option delta PDE's stock drift gives the same shifted Gaussian arguments. This is delta of a squared positive-part call.

For $V\in C^{1,2}$, Itô's formula givesMatch the Brownian coefficient with the self-financing portfolio by choosing the option delta $\varphi =V_s$ and bond value $\psi B=V-s{V}_s$. Equating drifts cancels the physical drift $\mu$ and givesThis is the pricing equation for a linear asset with additive noise. Its spatial domain is the whole real line, because the asset can be negative. In particular the diffusion term is ${\sigma }^2{V}_{ss}/2$, not ${\sigma }^2{s}^2{V}_{ss}/2$. The derivation is delta hedging and does not impose a boundary condition at $s=0$.

The discounted stock over the remaining horizon satisfiesThe Itô integral on the right is a centered normal random variable with variance $a_{\tau }=(1-e^{-2r\tau })/(2r)$. Therefore $\mathbb{Q}(S_T\ge 0\mid S_t=s)=\Phi (d)$, where $d=s/(\sigma \sqrt{a_{\tau }})$. The digital option price and its option delta areThe formula also shows $V(t,0)=e^{-r\tau }/2$ for every $t<T$. At maturity use the specified step payoff, including its chosen value one at zero; the terminal discontinuity is not removed by the earlier-time smoothing.

Assume $f\in C^{1,2}$ on positive prices before maturity and the usual local trading integrability. If its value is held in stock units ${\delta }_t$ and bank units ${\beta }_t$, thenfor a self-financing portfolio. The Itô formula diffusion coefficient is $\sigma S_t{f}_x$, so $\sigma S_t>0$ forces ${\delta }_t=f_x(S_t,t)$ and ${\beta }_t{B}_t=f-S_t{f}_x$. Equating drifts yieldsThe physical drift cancels, giving the Black-Scholes equationConversely, if the smooth function satisfies this equation, set $\delta =f_x$ and $\beta =(f-x{f}_x)/B$. Substituting into Itô formula gives exactly $df=\delta dS+\beta dB$, so these holdings are self-financing. This checks the holdings as well as the value equation; a partial differential equation solution does not justify arbitrary prescribed holdings. Appropriate admissibility and growth conditions are imposed when using it as an economic price.

For the logarithmic stock payoff, let $\tau =T-t$ and condition on $S_t=x$. Under the risk-neutral measure,The normal exponential moment $\mathbb{E}{e}^{uZ}=e^{u^2/2}$ and its derivative $\mathbb{E}[Z{e}^{uZ}]=u{e}^{u^2/2}$ giveThereforeThe option delta is the displayed stock holding. The bank account value is $f-S_t{f}_x=-S_t$, so ${\beta }_t=-S_t/B_t$. Directly, $f_{xx}=1/x$ and $f_t=-x(\rho +{\sigma }^2/2)$ verify the partial differential equation, while $f(x,T)=x\log x$ verifies the payoff. Lognormal moments give the required pricing integrability. Since $x\log x\ge -1/e$, the conditional price is bounded below over this finite horizon, so the resulting claim replication has the usual admissibility property.

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.

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.