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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

For $v,m\ge 0$, define $F(v,m)=\mathbb{E}[(e^{-v/2+\sqrt{v}Y}-m{)}^{+}]$, where $Y$ has the standard normal distribution. For $v,m>0$, completing the square in the standard normal density giveswhere $\Phi$ is the standard normal distribution function. The boundary values are $F(0,m)=(1-m{)}^{+}$ and $F(v,0)=1$. Multiplying by the current stock price produces a zero-interest Black-Scholes formula with total variance $v$ and relative strike $m$.

The vega of an European option is the derivative of its price with respect to spot volatility, holding its other parameters fixed. A dividend-free European call option under the Black-Scholes formula has $\nu =S_0\sqrt{T}\varphi (d_1)>0$. Differentiation proves this because $S_0\varphi (d_1)=K{e}^{-rT}\varphi (d_2)$ and ${\partial }_{\sigma }(d_1-d_2)=\sqrt{T}$. Positivity proves uniqueness of Black-Scholes implied volatility for prices strictly between the limiting call-price bounds.

Differentiate the Black-Scholes formula for the European put option with respect to its strike. The chain rule givesThe final two terms cancel by both identities in part i. Hence the put-price strike derivative is

For strike $K>0$ and maturity $T$, a European call option and a European put option have boundary conditionsThe Black-Scholes equation is solved backward from these conditions. At zero stock price, the zero state is absorbing in geometric Brownian motion, soAt the other boundary, with $\tau =T-t>0$,These conditions and a suitable growth bound select the financial solution and provide boundary data for numerical truncations. In the constant-coefficient model, under the risk-neutral measure,For $d_1=[\log (S/K)+(r+{\sigma }^2/2)\tau ]/(\sigma \sqrt{\tau })$ and $d_2=d_1-\sigma \sqrt{\tau }$, the exercise event is $Z>-d_2$. Completing the square in $e^{\sigma \sqrt{\tau }Z}$ times the normal density gives the truncated first moment and hencewhere $\Phi$ is the standard normal cumulative distribution function. These are the Black-Scholes formula values and satisfy the specified terminal and boundary conditions.

Put-call parity is the relationA call minus a put has terminal financial payoff $S_T-K$. A portfolio of one share minus a zero-coupon bond paying $K$ has the same financial payoff, so absence of arbitrage and law of one price equate their current values. The formula here assumes the same strike and maturity and no dividends; known dividends modify the stock-side value.

For positive spot, strike, volatility and maturity, the Black-Scholes formula iswhere $d_1=[\log (S_0/K)+(r+{\sigma }^2/2)T]/(\sigma \sqrt{T})$ and $d_2=d_1-\sigma \sqrt{T}$. Degenerate zero-volatility or zero-maturity cases are obtained by limits.

For positive $S_0,K,T,\sigma$, the Black-Scholes formula usesTo hedge the sold European call option, buy $\Phi (d_1)$ shares and borrow $K{e}^{-rT}\Phi (d_2)$ in cash. This long replicating strategy has initial value $S_0\Phi (d_1)-K{e}^{-rT}\Phi (d_2)$, financed by the option premium. The share holding must subsequently be rebalanced according to the option's option delta in a self-financing strategy.

As $\sigma \downarrow 0$, the Black-Scholes formula tends to $\boxed{(S_0-K{e}^{-rT}{)}^{+}}$. This also follows by discounting the deterministic zero-volatility payoff; at equality $S_0=K{e}^{-rT}$ the limiting value is zero.

If $S_0>K{e}^{-rT}$ and a call trades at $C<S_0-K{e}^{-rT}$, buy a call, short one share, and buy a bond paying $K$ at maturity. Invest the positive initial surplus $S_0-K{e}^{-rT}-C$. At maturity the call, share and bond together pay $(S_T-K{)}^{+}-S_T+K=(K-S_T{)}^{+}\ge 0$, in addition to the strictly positive invested surplus. If the lower bound is zero, violating it means $C<0$: buy the call and invest the cash received, obtaining a nonnegative option payoff and a strictly positive bank payoff. Both strategies are arbitrages.

Write $\varphi ={\Phi }^{\prime }$ for the standard normal density. Differentiating the Black-Scholes formula and using $S_0\varphi (d_1)=K{e}^{-rT}\varphi (d_2)$ givesThe last equality follows by differentiating $d_1-d_2=\sigma \sqrt{T}$. This derivative is the option's vega.

The Black-Scholes implied volatility of a traded call price $C$ is the volatility $\sigma$ which makes the Black-Scholes formula equal to $C$. Forthe price is a continuous strictly increasing function of $\sigma >0$, by its positive vega, with precisely these endpoint limits. The intermediate value theorem and strict monotonicity therefore give a unique positive implied volatility. The lower endpoint corresponds to the limiting value $\sigma =0$; the upper endpoint is reached only as $\sigma \to \infty$. Prices outside the bounds admit the arbitrages above and have no implied volatility.

A standard Brownian motion $W_t$ starts at zero, has continuous sample paths, and has independent increments $W_t-W_s\sim N(0,t-s)$ for $0\le s<t$. The Black-Scholes model assumes a frictionless market permitting continuous, self-financing trading and short selling, a bank account $B_t=e^{rt}$ at constant interest rate, and a non-dividend-paying assetwith constant coefficients and no arbitrage.

For an option price $C(S,t)$, Itô's formula gives diffusion coefficient $\sigma S{C}_S$. Holding $C_S$ units of the asset hedges this diffusion. The hedged portfolio has value $C-S{C}_S$ and instantaneous gain $(C_t+\frac{1}{2}{\sigma }^2{S}^2{C}_{SS})dt$, so no arbitrage requiresEquivalently, the risk-neutral measure makes $e^{-rt}{S}_t$ a martingale and gives the solution $C(S_0,0)=e^{-rT}{\mathbb{E}}_Q(S_T-K{)}^{+}$. Under that measure,Put $d_2=[\log (S_0/K)+(r-{\sigma }^2/2)T]/(\sigma \sqrt{T})$ and $d_1=d_2+\sigma \sqrt{T}$. Then ${\mathbb{P}}_Q(S_T>K)=\Phi (d_2)$, and completing the square in the normal integral gives ${\mathbb{E}}_Q[S_T{1}_{S_T>K}]=S_0{e}^{rT}\Phi (d_1)$. Therefore the Black-Scholes formula isHere $K>0$; a zero strike has value $S_0$ directly.

For the forward-start call option, condition at the strike-fixing time $t$. Given ${\mathcal{F}}_t$, the remaining asset-price ratio has the same lognormal law over $\tau =T-t$, independently of past increments. Thus its time-$t$ value isSince ${\mathbb{E}}_Q(e^{-rt}{S}_t)=S_0$, conditioning once more yieldsThe strike is random before $t$; inserting an unknown $S_t$ as a fixed strike at time zero would miss the conditioning step.

PutThe call payoff is $(S_T-K{)}^{+}$, and it is positive whenWriting $\varphi$ for the standard normal density, the given risk-neutral pricing formula becomesCompleting the square givesso the two tail integrals are $\Phi (d_{+})$ and $\Phi (d_{-})$, where $\Phi$ is the standard normal distribution function. Therefore the Black-Scholes formula is

The payoff identityreplicates the call-minus-put position by one stock and borrowing the present value of $K$. Hence put-call parity givesand therefore

Under $\mathbb{Q}$,DefineThe standard normal cumulative distribution function $\Phi$ then givesSubstitution into the discounted expected payoff yields the Black-Scholes formula