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For a stock with constant continuous dividend yield $\delta$, holding one fixed share and depositing each dividend in a bank account at rate $r$ gives the displayed wealth process. Since $d{A}_t=r{A}_{t}dt+\delta S_{t}dt$, this is a self-financing portfolio: its only cash inflow to the bank is the dividend earned by its stock holding. A claim paying $n{X}_T$ is replicated by $n$ copies and costs $n{X}_t$, including the known accrued-dividend balance at time $t$.

A different claim paying only $n{S}_T$ costs $n{S}_t{e}^{-\delta (T-t)}$ in the Black-Scholes model with dividends. It is replicated by $n{e}^{-\delta (T-t)}$ shares with dividends reinvested in new shares: the share count grows at rate $\delta$ and reaches $n$ at expiry. Thus banking dividends and reinvesting them produce distinct terminal payoffs and distinct current values, even though both strategies are self-financing.

A cash-at-maturity upper-barrier digital option pays a fixed amount if the stock touches an upper barrier by maturity. In the Black-Scholes model, write $a=\log (c/S_0)/\sigma$ and $b=-(r-{\sigma }^2/2)/\sigma$. Its unit-payoff price is $e^{-rT}$ times the linear-boundary Brownian first-passage distribution at $T$. This differs from paying at the hitting time, when the discount factor is random, and from an ordinary terminal-price digital option.

A contingent claim paying one at maturity if a specified price path has remained below an upper barrier, and zero otherwise. This survival payoff differs from a terminal digital put option, which tests only the terminal price. In the dividend-free Black-Scholes model, its value is the discount factor times the risk-neutral probability of avoiding the barrier, obtainable from the linear-boundary Brownian first-passage distribution.

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

For a standard European call option, the Black-Scholes implied volatility is the value $\hat{\sigma }$ solving $C_{\mathrm{BS}}(\hat{\sigma })=C_{\mathrm{observed}}$, with spot, strike, remaining maturity, interest rate and dividend convention held fixed. In the non-dividend model with $S,K,\tau >0$,Positive option vega makes the call price continuous and strictly increasing in $\sigma >0$. Thus there is exactly one finite positive Black-Scholes implied volatility when the observed call price lies strictly between these bounds. The lower boundary corresponds to zero volatility; the upper boundary is approached only as volatility tends to infinity. Outside the bounds there is no solution in this model.

Monotonicity makes bracketed root finding a direct numerical inversion. This is a model-implied parameter; calls with different strikes can imply different values even though the basic Black-Scholes model assumes one constant volatility. For more general claim payoffs, uniqueness requires a separate monotonicity argument.

The Black-Scholes model is a frictionless market with a positive bank account $d{B}_t=r{B}_{t}dt$ and a non-dividend-paying stock with constant volatility $\sigma >0$ and initial price $S_0>0$. Under its risk-neutral measure,Thus the conditional stock price has a lognormal distribution. A physical-measure drift $\mu$ may differ from $r$; the pricing dynamics express the martingale condition on discounted traded prices, not a forecast of actual returns.

In the usual natural Brownian filtration, nonzero volatility permits claim replication by trading the stock and bank. For an integrable replicable terminal financial payoff $g(S_T)$, the price is $e^{-r(T-t)}{\mathbb{E}}_Q[g(S_T)\mid {\mathcal{F}}_t]$. The delta hedge holds the stock-price derivative of this value, with the remainder in the bank. The Black-Scholes formula evaluates this expectation for a European call option.^[ref]

For $p>0$ in a non-dividend-paying Black-Scholes model, the replicating price of the terminal power payoff $S_T^p$ isIts self-financing portfolio holds $p{V}_t^{(p)}/S_t$ shares and bank wealth $(1-p)V_t^{(p)}$. Itô's formula gives physical drift $p\mu -(p-1)r$ and volatility $p\sigma$ for this replicating asset; its risk-neutral drift is $r$.

Since $V_T^{(p)}=S_T^p$, the payoff $(S_T^p-K{)}^{+}$ is a European call option on this replicating asset. The Black-Scholes formula gives the displayed price with $\tau =T-t$ and $d_2=d_1-p\sigma \sqrt{\tau }$. Using the raw process $S_t^p$ as a non-dividend-paying underlying would miss its time-dependent replication factor.

This compound European contingent claim gives the right to buy a later-expiring put option at time $T_1$ for $K_1$. Its payoff is $[p(T_1,S_{T_1};K_2,T_2)-K_1{]}^{+}$, with $T_1<T_2$. In the dividend-free Black-Scholes model with positive volatility, the underlying put price is continuous and strictly decreasing in positive spot. It approaches $K_2{e}^{-r(T_2-T_1)}$ as spot tends to zero and approaches zero as spot tends to infinity.

Thus a unique positive exercise threshold $s_{\ast }$ exists exactly when $0<K_1<K_2{e}^{-r(T_2-T_1)}$. Exercise is favorable for $S_{T_1}<s_{\ast }$, where the put value exceeds its purchase cost. If the cost is at least the upper limiting put value, exercise is never favorable. At zero exercise cost, the compound claim has exactly the underlying put's price by the tower property. These endpoint cases make the meaning of a nondegenerate threshold explicit.

A barrier claim paying a fixed cash amount at the first time the stock reaches a higher level, provided this happens before expiry. In the Black-Scholes model, take $a=\log (c/S_0)/\sigma$ and $b=({\sigma }^2/2-\rho )/\sigma$. Its unit-cash price is ${\mathbb{E}}_Q[e^{-\rho T_{a,b}}{1}_{\{T_{a,b}\le T\}}]$, given by truncated discounted Brownian first passage. The effective square root $\sqrt{b^2+2\rho }=|\rho +{\sigma }^2/2|/\sigma$ is real for every real interest rate.

At its choice time this claim permits purchase of either of two later-expiring European options for an exercise cost $K$, or abandonment. Its payoff is $\max (p-K,c-K,0)$. For common strike $K_3$ and remaining time $\tau$, put-call parity places the branch crossing at $D=K_3{e}^{-r\tau }$. In the Black-Scholes model, both branch values there equal $D[2\Phi (\sigma \sqrt{\tau }/2)-1]$. An exercise cost above this minimum creates a central interval in which neither choice is worth exercising.

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.

For $I_t={\int }_0^t\log S_{u}du$ and $G_T=\exp (T^{-1}{I}_T)$ in the Black-Scholes model, the conditional law of $\log G_T$ under the risk-neutral measure has a normal distribution with mean $m_t$ and variance $q_t$. The future Brownian integral equals ${\int }_t^T(T-u)d{W}_u$, so the Itô isometry gives the variance. For strike $K>0$ and $t<T$, the call price is $e^{-\rho (T-t)}[e^{m_t+q_t/2}\Phi (d_1)-K\Phi (d_2)]$, where $d_2=(m_t-\log K)/\sqrt{q_t}$ and $d_1=d_2+\sqrt{q_t}$. This follows by completing the square in the truncated normal exponential moment; the known past integral must be included in the mean.

Let $A_t=\exp (t^{-1}{\int }_0^t\log S_{u}du)$ for a positive geometric Brownian motion in the non-dividend-paying Black-Scholes model, and put $\tau =T-t$. Under the risk-neutral measure, the future log-average isThe final integral is a centred normal variable independent of the past, with variance ${\tau }^3/3$. Applying the moment-generating function of a normal distribution proves the displayed identity for every real $p$. Multiplication by $e^{-r\tau }$ prices the payoff $A_T^p$. The historical average must remain a state variable; terminal stock price alone does not determine this payoff.

A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.

In the Black-Scholes model the terminal stock price conditional on current price $x$ is $xM$ for a positive lognormal multiplier $M$. Differentiation under the risk-neutral pricing expectation gives the displayed option gamma. Its sign follows that of $f^{\prime \prime }$, so convexity and concavity of the terminal payoff are preserved. An analogous first-derivative identity preserves monotonicity. With appropriate growth conditions, Gaussian integration by parts also gives $V_{\sigma }=\sigma (T-t)x^2{V}_{xx}$, fixing the sign of option vega for these payoff classes.

For a time step $h>0$, choose the undiscounted stock multipliers and money-market multiplier asFor sufficiently small $h$, $D_h<R_h<U_h$ and the risk-neutral up probability is $q_h=(R_h-D_h)/(U_h-D_h)$. A one-step log-stock increment takes values $\pm \sigma \sqrt{h}$; expanding $q_h$ gives mean $(\rho -{\sigma }^2/2)h+O(h^2)$ and variance ${\sigma }^{2}h+O(h^2)$. The Donsker invariance principle, in its triangular-array form, gives the risk-neutral log-price limitThus prices converge to the geometric Brownian motion used in the Black-Scholes model.^[ref]

The parent model's discounted multipliers are instead $u_h=U_h/R_h$ and $d_h=D_h/R_h$. Their log-price drift loses the $\rho$ term, as required for the discounted price to be a martingale. Convergence of claim prices also needs the payoff functional to be continuous at almost every limiting path, together with Uniform integrability when payoffs are unbounded. A process limit alone does not justify taking expectations of every payoff.

For contingent claim payoff $(S_T-K{)}_{+}^2$ in the non-dividend Black-Scholes model, differentiating the discounted expected value gives $\Delta =2e^{-r\tau }{S}^{-1}\mathbb{E}[(S_T^2-K{S}_T)1_{\{S_T\ge K\}}]$. Completing the square in the Gaussian exponent computes each truncated lognormal moment and yields the displayed formula, where $d_1=(\log (S/K)+(r+{\sigma }^2/2)\tau )/(\sigma \sqrt{\tau })$. Its terminal limit is $2(S-K{)}_{+}$.

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.

A digital call option that pays its terminal indicator only if a lower barrier was avoided. When its strike equals the lower barrier, survival already implies the terminal indicator is one. It is then a no-touch option, with alive price $C_d(s,t;B)-(s/B{)}^{1-2r/{\sigma }^2}{C}_d(B^2/s,t;B)$ in the dividend-free Black-Scholes model.

A down-and-out European call pays the usual European call option payoff only if a lower barrier has not been hit before maturity. In the zero-interest, zero-dividend Black-Scholes model, for $0<B\le K<S$, its value is $C(S,K)-(S/B)C(B^2/S,K)$. This follows from reflection of the drifted logarithmic price killed at the barrier.