The Greeks: What a Hedged Position Is Still Exposed To

An option's value moves for several distinct reasons: the underlying moved, time passed, the market's view of volatility changed, rates changed. A single profit number mixes all of them together and tells you nothing about which to act on.

The greeks are the partial derivatives that separate them. Each answers one question of the form: if this input changes by a small amount and nothing else does, how much does the position change?

That separation is what makes an option book manageable. A trader is rarely trying to be flat everything. They are usually trying to hold a deliberate exposure to one factor while neutralising the rest, and doing that requires the factors to be named and measured individually.

The five that matter are delta, gamma, theta, vega and rho, and only the first four are interesting most of the time.

Delta is the first derivative with respect to the underlying: how much the option's value changes per unit move in the price. A call's delta runs from near 0 far out of the money to near 1 deep in, passing near 0.5 at the money.

Gamma is the second derivative: how much delta itself changes per unit move. It is the curvature of the option's value.

Gamma is what makes delta-hedging an activity rather than a one-off. With zero gamma, hedge once and you are done. With positive gamma, every move changes the required hedge, so the position must be traded continually.

Gamma is largest at the money and near expiry, which is the same statement twice: it peaks where the outcome is most uncertain per unit of time remaining. Far from the strike or far from expiry, the option's value is nearly linear in the underlying, and delta barely moves.

Theta is the derivative with respect to time: how much value is lost per day, all else equal. For a long option it is negative, because time value decays to zero.

Gamma and theta are not independent risks. They are the two sides of one position, and the relationship is the actual trade.

Long an option means positive gamma and negative theta. Positive gamma means the delta hedge is profitable to run: as the price rises you are getting longer and sell into it, as it falls you get shorter and buy. You are systematically selling high and buying low, capturing money from movement. Negative theta is the rent you pay for that privilege.

Short an option is the exact mirror. You collect theta every day and your hedge loses money on every move, because you are forced to buy high and sell low.

So the question is never whether to have gamma. It is whether realised movement will exceed what you paid in theta, or fall short of what you collected.

The symmetry is exact and it is worth reading as a warning about the short side.

The long position pays a known, steady amount and receives an uncertain, occasionally large amount. The short position collects a known, steady amount and pays an uncertain, occasionally large amount. Their profit distributions are mirror images: one has many small losses and rare large gains, the other many small gains and rare large losses.

The short side therefore looks excellent by most summary statistics. It wins on most days, it has a high hit rate, and its average return is positive across quiet periods.

Those statistics are describing the shape of the payoff rather than the quality of the trade. The lesson Selection Bias and the Deflated Sharpe Ratio makes the general version of this point: a strategy with strongly negative skew may be selling tail risk rather than possessing an edge, and the two are indistinguishable until the tail arrives.

Vega is the change in value per one point change in implied volatility. It is not a Greek letter, which is a reliable trivia question and a hint that it was added to the list later than the others.

Vega matters because implied volatility is a market price, not a constant. An option can lose money with the underlying unchanged and no time elapsed, simply because the market repriced volatility downward.

This is the most common surprise for people who buy options ahead of a scheduled event. Implied volatility rises before an announcement because uncertainty is high, and collapses immediately after because the uncertainty is resolved. A buyer can be right about the direction of the move and still lose, because the vega loss exceeded the delta gain.

Vega is largest for at-the-money options with long maturities, which is the opposite concentration to gamma. A short-dated option has little vega and enormous gamma; a long-dated one has the reverse.

Under the model of the previous lesson the sensitivities have closed forms, and seeing the numbers together is more instructive than the formulas.

import math
from statistics import NormalDist

nd = NormalDist()
N, phi = nd.cdf, nd.pdf

def greeks(S, K, r, sigma, T):
    sq = sigma * math.sqrt(T)
    d1 = (math.log(S / K) + (r + sigma ** 2 / 2) * T) / sq
    d2 = d1 - sq
    return {
        "delta": N(d1),
        "gamma": phi(d1) / (S * sq),
        "vega":  S * phi(d1) * math.sqrt(T) / 100,      # per 1 vol point
        "theta": (-S * phi(d1) * sigma / (2 * math.sqrt(T))
                  - r * K * math.exp(-r * T) * N(d2)) / 365,   # per day
    }

greeks(100, 100, 0.04, 0.20, 1.00)   # gamma 0.0191, vega 0.381, theta -0.0161
greeks(100, 100, 0.04, 0.20, 0.02)   # gamma 0.1409, vega 0.056, theta -0.0828

Same strike, same spot, same volatility. Shortening the maturity from a year to about a week multiplies gamma by roughly seven, cuts vega to about a seventh, and multiplies daily decay by more than five.

That one comparison is why maturity choice is a decision about which risk you want, not about how long you want to be in the trade.

Each greek has a standard misreading, and each has cost people money.

Delta is not the probability of finishing in the money. It is close to the risk-neutral probability for an at-the-money option, and the resemblance is a coincidence of the formula rather than a definition. Delta is a hedge ratio: the quantity of underlying to hold. Treating it as a forecast imports the confusion about risk-neutral probabilities from the previous lesson.

Theta is not income. A short option position collecting theta has not earned anything until the position is closed. The daily decay is an accounting entry offset by an unrealised gamma liability that is invisible until the underlying moves.

The greeks are local. They are derivatives, valid for small changes. In a large move, delta is stale, gamma itself has changed, and the linear approximation the hedge relies on has broken. Precisely when risk is largest, the instruments for measuring it are least accurate.

One situation deserves separate treatment because the greeks stop being merely approximate and become actively misleading.

An option at the strike on its expiry day has enormous gamma. Delta swings between near 0 and near 1 on tiny moves in the underlying, so the required hedge flips between nothing and the full position, repeatedly, on noise.

Worse, the final outcome is binary and unknowable until the close. A trader who hedges to a delta of 0.5 will be wrong either way: the option settles either fully exercised or fully worthless, so they end the day either long or short a full position they did not intend.

This is pin risk, and it has no clean solution. The practical responses are all avoidance: close the position before expiry, or accept the exposure deliberately and size it so the wrong outcome is survivable.

It is a useful reminder of what the greeks are. They are a local linearisation of a nonlinear payoff, and they are least reliable exactly where the nonlinearity is sharpest.

A trading book is managed as a set of aggregate exposures rather than a list of positions. Total delta is usually held near zero, because directional risk can be removed cheaply and is rarely the intended bet. The others are held deliberately.

That is what an option trader is actually doing: expressing a view on volatility, or on the difference between implied and realised volatility, while using the underlying to strip out the direction they have no view on.

Which raises the question the last lesson answers. Implied volatility has been treated so far as a single number per underlying. It is not. It varies by strike and by maturity, in a shape the model of the previous lesson says should not exist.