Black-Scholes model

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]

Deterministic time-varying volatility enters the European Black-Scholes formula through its integrated square, not its arithmetic mean. For $\sigma (t)=\alpha (1+t)$ over one year, the arithmetic mean is $3\alpha /2$ but its square is only $9{\alpha }^2/4$, strictly below the actual integrated variance by ${\alpha }^2/12$ when $\alpha \ne 0$. Positive option vega therefore makes the arithmetic-mean-volatility call price too small. If the deterministic volatility path is known exactly in the complete frictionless model, the pricing difference can be locked in by buying the cheaper claim and selling its correctly computed replicating strategy.

For a fixed twice-differentiable terminal payoff with sufficient growth control, differentiate its lognormal distribution pricing integral. Gaussian integration by parts gives the displayed identities. Convexity implies $p_{xx}\ge 0$, so increasing spot volatility cannot lower the price. The interest rate sensitivity depends on the sign of the cash position $p-x{p}_x$. Calendar-time sensitivity is $p_t=\rho (p-x{p}_x)-\frac{1}{2}{\sigma }^2{x}^2{p}_{xx}$. Thus a convex payoff with a negative cash position has a nonincreasing price at fixed spot when $\rho \ge 0$. Negative rates invalidate that conclusion: $f(x)=x-K$ gives $p=x-K{e}^{-\rho (T-t)}$ and $p_t=-\rho K{e}^{-\rho (T-t)}>0$ when $\rho <0$.

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.

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 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.

Average constant-rate Black-Scholes model prices of a fixed terminal payoff over $\rho \sim N({\rho }_0,{\tau }^2)$. Tilting this Gaussian by the discount factor shifts its mean by $-{\tau }^{2}T$; combining the resulting log-price variance with stock variance gives the displayed effective parameters. Thus the average equals the Black-Scholes price with those parameters, whenever the expectation is well defined. This is an average of constant-parameter prices, rather than a specification of an adapted stochastic short-rate market.

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 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.

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.

For $S_t=S_0{e}^{\mu t+\sigma W_t}$, shifting Brownian drift bymakes $S_t=S_0{e}^{(r-{\sigma }^2/2)t+\sigma W_t^Q}$ under an equivalent martingale measure $Q$. The discounted stock is then a martingale.

For the payoff $S_T^p$, risk-neutral valuation givesIts delta hedge holds $p{V}_t/S_t$ units of the risky asset.

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.

Let $X_t=\mu t+\sigma W_t$ with $X_0=0$ and fixed horizon $T$. The reversed-increment process $X_T-X_{T-t}$, $0\le t\le T$, has the same path distribution as $X_t$, by stationary independent Gaussian increments. Taking its maximum givesFor $S_t=S_0{e}^{X_t}$, exponentiation therefore yields the ratio identityFunctions of these ratios have equal expectations whenever integrable. This does not by itself identify currency-valued payoffs such as $S_T-\min S$ and $\max S-S_0$: those depend on additional joint information. A change of numeraire or a separate joint-law calculation is needed when an extra stock-value multiplier occurs in a lookback claim.

For a non-dividend-paying stock, a European call with strike $K$ and remaining maturity $\tau$ has value

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 $m$.

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

Complementary digital call and put payoffs sum to one, so their time-$t$ values satisfy

The price $V(t,s)$ of a sufficiently regular European claim satisfies

Differentiating the displayed transformation shows that it preserves the Black-Scholes equation. The first derivative coefficient fixes the exponent $1-2r/{\sigma }^2$, which also makes the zero-order coefficient correct. Subtracting this reflected image of a European call option from the call itself enforces a zero boundary at $S=B$. For strike price $K\ge B$, the reflected maturity contingent claim payoff vanishes on $S>B$, giving the no-rebate down-and-out option formula. Prices using this reflection are conditional on the barrier not having been hit already.

If holding one unit of a commodity costs a constant cash amount $\delta dt$ for storage, its gain is $dS-\delta dt$. A delta hedge long a derivative and short $V_S$ commodity units has riskless drift $V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}+\delta V_S$ and value $V-S{V}_S$. Equating the drift to the bank return gives the displayed Black-Scholes equation. This is an additive cash carry term, unlike a proportional dividend or storage yield.

The corresponding pricing diffusion is $d\hat{S}=(r\hat{S}+\delta )dt+\sigma \hat{S}dW$. A fixed-strike forward has value $S+\delta (1-e^{-r\tau })/r-K{e}^{-r\tau }$, and put-call parity gives that same value for a matching call minus put. Replication buys the commodity and a bank reserve equal to the present value of its future storage payments. For $r=0$, the reserve is $\delta \tau$.

For the dividend-free Black-Scholes equation, the displayed transform preserves solutions when $B>0$ and $\sigma >0$. Put $x=\log (S/B)$ and $v(x,t)=V(B{e}^x,t)$. The equation becomesSubstitution shows that $\hat{v}(x,t)=e^{qx}v(-x,t)$ solves the same equation when $q=-2b/{\sigma }^2=1-2r/{\sigma }^2$: its first-derivative coefficient becomes $-{\sigma }^{2}q-b=b$, and the additional zeroth-order term ${\sigma }^2{q}^2/2+bq$ vanishes. Subtracting the reflected solution enforces a zero boundary value at $S=B$. Its use in a killed barrier option is restricted to the alive side of the barrier; the algebraic extension to the other side is generally not a valid price.

In the Black-Scholes equation with continuous stock dividends, separation of variables gives $g^{\prime }=(r-\lambda )g$ and ${\sigma }^2{S}^2{f}^{\prime \prime }/2+(r-q)S{f}^{\prime }-\lambda f=0$. Substituting $f=S^{\alpha }$ gives the displayed eigenvalue. A stationary solution uses the two roots of ${\sigma }^2\alpha (\alpha -1)/2+(r-q)\alpha -r=0$. For positive volatility and $r>0$, their product is negative, so they are distinct real roots.

When the underlying stock pays dividends at rate $\theta S_t$, its gain is $d{S}_t+\theta S_{t}dt$. Under the risk-neutral measure its ex-dividend drift is $(\rho -\theta )S_t$. A sufficiently smooth terminal-claim value obeys $p_t+(\rho -\theta )x{p}_x+{\sigma }^2{x}^2{p}_{xx}/2-\rho p=0$. These are dividends on the stock, not a source term paid by the claim itself.

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.

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 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.

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 $g=p_x$.