Closed-form prices and their partial derivatives — $\Delta$, $\Gamma$, $\Theta$, $\mathcal{V}$, $\rho$ — and how a trader actually hedges with them.
Solving the Black–Scholes PDE under the European call's terminal condition $V(S,T) = \max(S - K, 0)$ gives the famous formula:
$$\boxed{\;C(S,t) = S\,N(d_1) - K e^{-r\tau}\,N(d_2)\;}$$
where $\tau = T - t$ is the time to maturity and
$$d_1 = \frac{\log(S/K) + (r + \tfrac{1}{2}\sigma^2)\,\tau}{\sigma\sqrt{\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{\tau}.$$
$N(\cdot)$ is the standard normal cumulative distribution function. The corresponding put price follows from put–call parity:
$$P(S,t) = C(S,t) - S + K\,e^{-r\tau} = K\,e^{-r\tau} N(-d_2) - S\,N(-d_1).$$
$S\,N(d_1)$ is the present value of receiving the share only when in the money; $K e^{-r\tau} N(d_2)$ is the present value of paying the strike only when in the money. The difference is the value of the option to exchange them.
$N(d_2)$ is the risk-neutral probability that $S_T > K$ (the call ends in the money). $N(d_1)$ has a similar interpretation under a measure where the stock is the numeraire. Their difference reflects the variance of $\log S$.
$S\, n(d_1) = K e^{-r\tau}\, n(d_2)$, where $n(\cdot)$ is the standard normal PDF. This collapses many of the apparent uglinesses in the Greeks below into clean single-PDF terms.
A digital call pays $1$ if $S_T > K$ and zero otherwise. Its present value is exactly the risk-neutral probability of ending in the money, discounted:
$$C_{\text{dig}}(S,t) = e^{-r\tau}\,N(d_2).$$
The matching digital put pays $1$ if $S_T < K$:
$$P_{\text{dig}}(S,t) = e^{-r\tau}\,N(-d_2).$$
Digital + complementary digital = $e^{-r\tau}$ (you're certain to receive £1, just possibly via the other gate).
The payoff of a digital is a step function in $S_T$. Near $t = T$ and $S \approx K$, the value function develops an arbitrarily steep slope — tiny moves in $S$ cause large value swings. Gamma blows up. These contracts are notoriously hard to hedge near expiry.
A vanilla European call is a continuum of digitals, since $\max(S-K, 0) = \int_K^\infty \mathbf{1}_{S > k}\,dk$. So $C$ is a "stack" of $C_{\text{dig}}$ over $k$. This is the seed of static-replication arguments for exotics.
The first Greek — the share-equivalence of an option, and the hedge ratio of deck 06.
$$\Delta_{\text{call}} = N(d_1), \qquad \Delta_{\text{put}} = N(d_1) - 1 = -N(-d_1).$$
For a call: $\Delta \in (0, 1)$. Deep out of the money it's near $0$; deep in the money it's near $1$. At $S = K$ at the money, $\Delta \approx \tfrac{1}{2}$ — the option behaves like half a share.
To hedge a long call: short $\Delta$ shares. To hedge a long put ($\Delta < 0$): long $|\Delta|$ shares. After hedging, the portfolio's value is to first order insensitive to $S$ — the residual exposure is second-order (gamma).
Delta is additive over a book: $\Delta_{\text{book}} = \sum_i n_i\, \Delta_i$. A trader watches the book's net delta and trades shares to bring it back to zero each rebalance. Most of risk management starts here.
$N(d_1)$ is the probability that $S_T > K$ in a measure where $S$ is the numeraire. Practitioners often quote a call's "delta" interchangeably with its "in-the-money probability" — close but not identical.
Second derivative of the price with respect to spot:
$$\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{n(d_1)}{S\,\sigma\sqrt{\tau}}.$$
Same formula for calls and puts (the difference between them is the linear $S - K e^{-r\tau}$, which has zero gamma).
Gamma is always positive for long options. It peaks at the money and falls off to zero far from the strike or as $\tau \to 0$ — though right at the strike near expiry it spikes sharply.
A long option position has positive gamma: when $S$ moves, your delta moves in your favour. You re-hedge by buying low and selling high — mechanically. The expected gain over a small interval is
$$\tfrac{1}{2}\,\Gamma\,\sigma^2 S^2\,\delta t.$$
This is the Itô term from deck 05, made concrete.
Short options have negative gamma: you re-hedge by buying high and selling low. Each oscillation of $S$ costs you. Over a small interval the expected loss is the same $\tfrac{1}{2}\,|\Gamma|\,\sigma^2 S^2\,\delta t$ — precisely compensated by the theta you collect (next slide).
Sensitivity to time. With $\tau = T - t$ a call's theta is
$$\Theta_{\text{call}} = -\frac{S\,n(d_1)\,\sigma}{2\sqrt{\tau}} - r K e^{-r\tau} N(d_2).$$
Theta is typically negative for long options — the value bleeds away as time passes, all else equal. (Deep-in-the-money European puts can have positive theta because the discounting effect dominates — an oddity of European-only contracts.)
Theta is most negative at the money near expiry, where there's the most time value to lose and where the gamma is largest. The two effects are not coincidental.
Rewriting Black–Scholes:
$$\Theta + r\,S\,\Delta + \tfrac{1}{2}\sigma^2 S^2\,\Gamma = r\,V.$$
For a delta-hedged portfolio ($\Delta = V_S$ cancels with the share leg) this reduces to $\Theta + \tfrac{1}{2}\sigma^2 S^2\,\Gamma = r(V - SV_S)$ — the gamma you collect exactly pays the theta you bleed, after rate costs. This identity is the entire game.
Traders usually quote theta per calendar day: $\Theta/365$. So "a theta of $-0.04$" means the option loses $4$ p of value per day, all else held fixed.
The most important Greek that isn't a Greek letter. Sensitivity of price to the input volatility $\sigma$:
$$\mathcal{V} = \frac{\partial V}{\partial \sigma} = S\,\sqrt{\tau}\,n(d_1).$$
Same for calls and puts. Always positive for long options. Like gamma, peaks at the money. Unlike gamma, increases with time to maturity.
Vega is usually quoted per volatility point: $\mathcal{V}/100$. "Vega of $0.20$" means the option gains $20$ p in value if $\sigma$ rises from, say, $20\%$ to $21\%$.
The Black–Scholes derivation assumed $\sigma$ was a constant. In reality the market re-prices implied volatility every day, and vega is the P&L impact of those re-pricings. Vega risk dwarfs gamma risk for many books.
From the explicit formulas:
$$\mathcal{V} = S^2\,\sigma\,\tau\,\Gamma.$$
So vega and gamma are not independent — you can't hedge one without affecting the other. Multi-Greek hedging therefore needs multiple instruments.
The supporting cast. None of them dominate day-to-day P&L for short-dated options, but each matters somewhere.
| Greek | Definition | What it captures |
|---|---|---|
| Rho ($\rho$) | $\partial V/\partial r$ | Rate sensitivity. Matters for long-dated FX and rate-product options. |
| Speed | $\partial^3 V/\partial S^3$ | How fast gamma changes with spot. Big near digital-like payoffs. |
| Charm | $\partial \Delta/\partial t$ | How delta moves with time. Important for delta hedgers as expiry approaches. |
| Colour | $\partial \Gamma/\partial t$ | How gamma changes with time. |
| Vanna | $\partial^2 V/\partial S \partial \sigma$ | How delta changes with vol; sells hedge ratios on the smile. |
| Volga | $\partial^2 V/\partial \sigma^2$ | Convexity in vol — the heart of vega–gamma trading. |
Closed forms exist for all of these under BS — tedious but mechanical. In a real risk system, every product reports all six.
Vanilla calls have smooth Greeks. Barriers, digitals, double-no-touch — these can have spiked second-order Greeks near barriers or strikes, and those spikes drive the hedging error and the bid–offer.
The BS formula maps $(S, K, T, r, \sigma) \to V$. Given a market price $V^*$, invert numerically for $\sigma$:
$$V_{\text{BS}}(S, K, T, r, \sigma_{\text{imp}}) = V^*.$$
$\sigma_{\text{imp}}$ is the implied volatility. It is not what $\sigma$ "is"; it is the number that, plugged into a known-wrong formula, gives a known-right price.
If BS were true, $\sigma_{\text{imp}}$ would be a constant across strikes and maturities. It isn't. The market quotes higher implied vol for OTM puts (and OTM calls) than for at-the-money options — the smile, or, in equities, the asymmetric skew.
Real returns have heavier tails than lognormal. The market correctly prices the bigger probability of extreme moves; BS, blind to that, requires a higher $\sigma$ to reproduce the same price for OTM options.
Crashes are bigger than rallies. Demand for downside protection (OTM puts) bids up their prices, lifting their implied vol relative to OTM calls.
Deck 08 in this series digs into the smile in earnest — sticky-strike vs sticky-delta, local vol, and stochastic vol models that reproduce the observed surface.
BS assumed continuous trading. Real traders rebalance at discrete times. What goes wrong?
If you rebalance every $\delta t$, the cumulative hedging error over $[0,T]$ is approximately normal with standard deviation
$$\mathrm{SD}(\text{error}) \approx \frac{1}{\sqrt{2}}\,\sigma^2 S^2 \, |\Gamma|\, \sqrt{\delta t \cdot T}.$$
Halving $\delta t$ reduces the error only by a factor of $\sqrt{2}$ — not by 2. Transaction costs work the other way (more rebalances = more costs). The trader picks an interior optimum.
Hold another option to neutralise gamma. The first option's gamma is offset by an opposite gamma in a hedge option; only one option per book of options need be traded for delta.
Trade vol-sensitive instruments (other vanillas, variance swaps) to neutralise vega. Most large books are run vega-neutral first, then delta-neutral.
Some exotics (e.g. some barriers) admit a static replication by a basket of vanillas held to expiry — no rebalancing. The trader vastly prefers static; dynamic hedging has slippage.
For $n$ equal rebalances over $[0,T]$, the variance of the hedging error scales as $1/n$. Doubling $n$ halves the variance but doubles the costs — classical bias–variance optimisation pitched as a trading problem.
Three small subplots: $V$ vs $S$, $\Delta$ vs $S$, $\Gamma$ vs $S$. The current $S$ is marked on each. Slide $K, T, \sigma, r$ and toggle call/put. The metric grid below reports the trader-quoted Greeks at the current $S$.
Watch gamma peak at the money and rise sharply as $T \to 0$. Watch the delta curve sharpen from a smooth S-shape (long-dated) to a step (near expiry). Push $\sigma$ up and gamma flattens out across more strikes. Toggle to a put and see delta sit in $(-1, 0)$.