Greeks and differential identities

Definition Let $V(t, S)$ be the Black-Scholes price of a European option with fixed strike $K$ and maturity $T$, under the hypotheses of The Black Scholes formula and put call parity, viewed as a function of $t$, $S$ and the model inputs $\sigma$, $r$ and $q$. On the set $S > 0$, $K > 0$, $\sigma > 0$, $\tau = T - t > 0$ the price is infinitely differentiable, and its Greeks are the partial derivatives

each taken with all other inputs held fixed. Theta is a derivative in calendar time with the maturity date fixed. Rho moves the constant rate $r$ with $S$, $q$ and $\sigma$ fixed. Vega moves the annualized volatility in decimal units. The units are shares per option for delta, shares per currency unit for gamma, currency per unit of decimal volatility for vega, currency per year for theta and currency per unit of decimal rate for rho. So a one percentage point volatility change is about $0.01\,\nu$, one basis point of rate is about $0.0001\,\rho$, and one calendar day under an actual/365 count is about $\Theta/365$. A Greek computed by bumping, such as the centered difference $\bigl(V(x + h) - V(x - h)\bigr)/(2h)$ with a stated bump $h$, approximates the derivative with an error of order $h^2$ for a smooth price. A sensitivity that lets the volatility surface or the curves move with the bumped input is a different, total derivative and needs its own convention.

The Black-Scholes Greeks follow from one identity. With $d_1$ and $d_2$ as in the previous lesson,

Indeed $d_1^2 - d_2^2 = (d_1 - d_2)(d_1 + d_2) = 2\ln(S/K) + 2(r - q)\tau$, so $\varphi(d_2)/\varphi(d_1) = (S/K)e^{(r - q)\tau}$. For the call $C = S e^{-q\tau}\Phi(d_1) - K e^{-r\tau}\Phi(d_2)$ and any input $x$, the chain rule and this identity give

Since $d_1 - d_2 = \sigma\sqrt{\tau}$, the density term survives only in vega and theta.

Definition Under the hypotheses of the previous card, the deltas of the call and the put are

For the call, take $x = S$ in the cancellation formula. The factor $S e^{-q\tau}$ has derivative $e^{-q\tau}$, the factor $K e^{-r\tau}$ does not depend on $S$, and $d_1 - d_2$ does not depend on $S$, so the two density terms cancel. Differentiating put-call parity in $S$ gives $\Delta_C - \Delta_P = e^{-q\tau}$, hence the put delta. Consequently $0 < \Delta_C < e^{-q\tau}$ and $-e^{-q\tau} < \Delta_P < 0$. In the replication argument the portfolio that replicates one option holds $\Delta$ shares, so a long option is hedged by selling $\Delta$ shares, and for a small move $\delta$ of the stock $V(t, S + \delta) - V(t, S) \approx \Delta\,\delta$. The call delta $e^{-q\tau}\Phi(d_1)$ is a hedge ratio and differs from the pricing probability of exercise, which is $\Phi(d_2)$.

Definition The gamma of the call and of the put is

obtained by differentiating $\Delta_C$ with $\partial d_1/\partial S = 1/(S\sigma\sqrt{\tau})$. The put has the same gamma because $C - P$ is linear in $S$. The denominator is positive under the hypotheses $S > 0$, $\sigma > 0$ and $\tau > 0$, so $\Gamma > 0$ for long calls and puts. To second order,

As $\tau \downarrow 0$ the behavior depends on the position of the stock. Away from the strike $\Gamma \to 0$, while at the money with $r = q = 0$ one has $d_1 = \tfrac12\sigma\sqrt{\tau}$ and $\Gamma \approx 1/\bigl(S\sigma\sqrt{2\pi\tau}\bigr)$, which grows without bound. At expiry the payoff has a kink at $K$ and no classical second derivative there, so the formula, which needs $\tau > 0$, does not extend to that point.

Definition The vega of the call and of the put is

In the cancellation formula with $x = \sigma$ neither prefactor depends on $\sigma$ and $\partial(d_1 - d_2)/\partial\sigma = \sqrt{\tau}$, which gives the call vega, and the put has the same vega since $C - P$ does not depend on $\sigma$. Under the hypotheses $\nu > 0$, so each price is strictly increasing in $\sigma$. Comparing with gamma gives the vega-gamma identity

valid for calls and puts at every $S > 0$, $\sigma > 0$ and $\tau > 0$ in this model. The same identity holds for other European payoffs of $S_T$ in the Black-Scholes model whose prices are smooth, a fact stated here without proof. Vega is measured in currency per unit of decimal volatility, so a move from $0.20$ to $0.21$ is a change of $0.01$ in $\sigma$.

Definition With $T$ fixed, the thetas of the call and the put are

In the cancellation formula with $x = \tau$, the factor $S e^{-q\tau}$ has derivative $-qS e^{-q\tau}$, the factor $K e^{-r\tau}$ has derivative $-rK e^{-r\tau}$, and $\partial(d_1 - d_2)/\partial\tau = \sigma/(2\sqrt{\tau})$. This gives $\partial C/\partial\tau$, and $\Theta_C = -\partial C/\partial\tau$ reverses every sign. The put follows from parity, $\Theta_C - \Theta_P = qS e^{-q\tau} - rK e^{-r\tau}$. Both prices satisfy the Black-Scholes PDE, which in Greek notation reads

and substituting the formulas checks it, since the gamma term cancels the density term of theta and the remaining terms cancel by the price formula. Theta is in currency per year. Its sign is not fixed, because the dividend and interest terms can outweigh the negative density term.

Definition With $S$, $q$, $\sigma$ and $t$ held fixed, the rhos of the call and the put are

Here $\partial d_1/\partial r = \partial d_2/\partial r = \sqrt{\tau}/\sigma$, so the density terms cancel, and differentiating $-K e^{-r\tau}$ in $r$ gives $K\tau e^{-r\tau}$. Parity gives $\rho_C - \rho_P = K\tau e^{-r\tau}$. Rho is in currency per unit of decimal rate, so one basis point is about $0.0001\,\rho$. Because $S$ is held fixed, raising $r$ also raises the forward $F = S e^{(r - q)\tau}$. The sensitivity to the dividend yield, with $r$ fixed, is $\partial C/\partial q = -\tau S e^{-q\tau}\Phi(d_1)$. A curve bump followed by recalibration of other inputs gives a total derivative that can differ from this partial rho.