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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

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.

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.

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 mesh $h$, the risk-neutral up probability is $(e^{\rho h}-d)/(u-d)$. The log-price increment has mean $(\rho -{\sigma }^2/2)h+O(h^2)$ and variance ${\sigma }^{2}h+O(h^2)$. The Donsker invariance principle gives convergence of the log-price process to Brownian motion with this drift, hence of prices to the Black-Scholes model. Payoff-price convergence additionally uses continuity at almost every limiting path and uniform integrability for unbounded payoffs.

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 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.

For exponential utility $(1-e^{-ax})/a$ on unrestricted terminal portfolio wealth in the Black-Scholes model, the optimal dollar holding in the stock is the displayed deterministic amount. Set $\vartheta =(\mu -\rho )/\sigma$ and let $W^Q=W+\vartheta t$ be the Brownian motion under the risk-neutral measure. The terminal optimizer is $X^{\ast }=w_0{e}^{\rho T}+\vartheta W_T^Q/a$. Its marginal utility is proportional to the state-price density, and the concave supporting-tangent inequality proves global optimality under the state-price budget constraint. The expected utility at the optimum is $(1-\exp (-a{w}_0{e}^{\rho T}-{\vartheta }^{2}T/2))/a$. A nonnegative-wealth constraint changes this optimizer.

For terminal logarithmic utility $a\log V_T$ and running logarithmic utility $b\log c_t$, with $a,b,w_0>0$, constant Black-Scholes model coefficients give $\lambda =(a+bT)/w_0$ and $V_t=[a+b(T-t)]/(\lambda {\xi }_t)$, where $\xi$ is the state-price density. The optimal terminal wealth and consumption are $a/(\lambda {\xi }_T)$ and $b/(\lambda {\xi }_t)$. Their discounted budget adds to $(a+bT)/\lambda =w_0$. The concave supporting-tangent inequality, integrated over time, proves optimality; the Itô formula gives the displayed dollar stock holding. The time-dependent consumption-to-wealth ratio reflects both remaining consumption opportunities and the terminal utility weight.

For constant relative risk aversion utility with parameter $\gamma >0$, terminal weight $\eta >0$ and utility discount rate $\delta$, in a Black-Scholes model put $\lambda =(\mu -r)/\sigma$, $k=[\delta +(\gamma -1)(r+{\lambda }^2/(2\gamma ))]/\gamma$ and $b(\tau )={\int }_0^{\tau }{e}^{-ks}ds+{\eta }^{1/\gamma }{e}^{-k\tau }$. The displayed optimal consumption and dollar stock position follow by inverse marginal utility and the state-price budget constraint. For $\gamma \ne 1$ the value is $b(T-t{)}^{\gamma }{W}_t^{1-\gamma }/(1-\gamma )$ with discount measured from $t$. Finite horizons do not require $k>0$. The concave supporting-tangent inequality and replication of the optimal budget prove global optimality, rather than only a formal differential-equation solution.

An Asian option whose averaging uses the geometric mean. In the constant-coefficient Black-Scholes model, the logarithm of the continuous geometric average is Gaussian with mean $\log S_0+(r-{\sigma }^2/2)T/2$ and variance ${\sigma }^{2}T/3$, because $\mathrm{Var}({\int }_0^T{W}_{t}dt)=T^3/3$. Its forward mean is $S_0{e}^{rT/2-{\sigma }^{2}T/12}$. The discounted lognormal call formula gives its price. It is strictly cheaper than the terminal European call when $\sigma ,K,T>0$ and $r\ge -{\sigma }^2/6$, since both its forward mean and log-variance are no larger, with strictly smaller variance. No unconditional comparison holds for arbitrary negative interest rates: for $r<-{\sigma }^2/6$, sufficiently small positive strikes reverse the inequality.

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.

Implied volatility is the volatility input of an option-pricing model that reproduces an observed option price. Black-Scholes implied volatility specializes this inversion to the Black-Scholes model.

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.

The derivative of a claim price with respect to the interest rate, holding the payoff and other model parameters fixed. For a dividend-free Black-Scholes model with remaining time $\tau$, it is $\tau (x{p}_x-p)$, so its sign is opposite the bank-account value in the replicating strategy.

The derivative of an option's value with respect to calendar time at fixed current underlying price and other parameters. In the dividend-free Black-Scholes model, the Black-Scholes equation givesThe first term is the interest earned on the bond value of the replicating portfolio, while the second uses option gamma. A convex payoff with a short bond position has $\Theta \le 0$ when $\rho \ge 0$; a concave payoff with a long bond position has $\Theta \ge 0$. Negative interest rates need not obey these signs: the affine payoff $x-K$ has $\Theta =-\rho K{e}^{-\rho (T-t)}$. For a fixed payoff function, remaining-maturity sensitivity is $-\Theta$.