The Greeks and delta hedging explained

CM2 · Financial Engineering & Loss Reserving · 11 min read

What a Greek is

An option price depends on five things: the share price, the volatility, the time left, the interest rate and the strike. Four of the five move. A Greek is the partial derivative of the price with respect to one of them, holding the others fixed.

That is the whole definition, and holding on to it saves a great deal of memorising. Any Greek can be checked by nudging the input slightly, repricing, and dividing the change in price by the change in input. The worked example below does exactly that for all five, and every formula agrees with its numerical derivative to five or six figures. A Greek is not a separate theory; it is the price seen through a derivative.

Greeks exist because a trading book cannot be managed by its value alone. A desk short ten thousand options needs to know what happens if the share moves one rupee, if volatility rises a point, if nothing happens for a day. Those are four different questions and each has its own Greek.

Delta: sensitivity to the share price

Δ = ∂f/∂S
Δ_call = Φ(d₁) Δ_put = Φ(d₁) − 1

In words: how many rupees the option price moves when the share moves one rupee.

A call's delta runs from 0 to 1 and a put's from −1 to 0, and the two always differ by exactly 1 — which is put-call parity differentiated, since the right-hand side of parity has a slope of 1 in S. Delta is also the number of shares needed to replicate the option, which is how it arrived in the binomial model, and it is approximately the risk-neutral probability of finishing in the money, which is why a deep-in-the-money call behaves almost exactly like the share.

Gamma: how fast delta changes

Γ = ∂²f/∂S² = φ(d₁) / (Sσ√T)

In words: how much the delta itself moves when the share moves one rupee.

Gamma is the same for a call and a put on the same terms, and it is always positive for a long position in either — both are convex in the share price. It is largest for an at-the-money option close to expiry, where the delta is swinging between nearly 0 and nearly 1 over a small range of share prices, and it is near zero for options far in or far out of the money, whose deltas are pinned at 1 or 0.

Gamma is the Greek that decides what hedging costs, and it is worth being precise about why. A delta hedge is a straight line tangent to a curve. Gamma is the curvature, so it measures how quickly the line stops fitting. A high-gamma position must be rehedged often, and every rehedge is a trade in the direction the market has just moved — buying after a rise, selling after a fall. For the writer of an option, who is short gamma, that is buying high and selling low, repeatedly, and it is the real cost of running the book.

Vega: sensitivity to volatility

ν = ∂f/∂σ = Sφ(d₁)√T

In words: how much the option price moves when volatility rises.

Vega is positive for a long call and a long put alike, and the same for both: more uncertainty is worth more to anyone holding optionality, whichever direction they are betting on. It grows with the square root of the time remaining, so long-dated options are far more exposed to a change in volatility than short-dated ones.

Two practical notes. Vega is usually quoted per percentage point of volatility, so the formula's answer is divided by 100. And volatility is the one input that is not observable, so vega measures exposure to the thing you are least sure about — which is why it gets more attention than its position in the list suggests.

Theta: the cost of waiting

Θ = ∂f/∂t = −Sφ(d₁)σ/(2√T) − rKe^−rTΦ(d₂)

In words: how much the option loses as a day passes with nothing else changing.

Theta is negative for a long call: time runs out, and the chance of the share moving enough shrinks with it. The holder of an option pays for time by holding it, and the writer collects. Theta is usually reported per calendar day, so the annual figure is divided by 365.

Theta and gamma are two sides of one thing, and noticing that explains most of what an option desk does. The writer of an option collects theta every day and pays for gamma every time the share moves. A quiet market is profitable for the writer; a jumpy one is not; and the volatility at which the two exactly balance is, in effect, the volatility the option was sold at. That is what it means to say an option is priced on its volatility.

Rho: sensitivity to the interest rate

ρ = ∂f/∂r = KTe^−rTΦ(d₂)

In words: how much the option price moves when the risk-free rate rises.

Rho is positive for a call and negative for a put. The reason is clearest from parity: holding a call rather than the share defers paying the strike, and a higher rate makes deferring more valuable. Rho is the least watched of the five on short-dated equity options, where it is small, and the most important of them on long-dated contracts and anything interest rate related — which for an actuary is most things.

Hedging, and what each hedge leaves behind

A portfolio is delta-neutral when its total delta is zero, so a small move in the share leaves its value unchanged. For a writer of N calls that means holding N × Δ shares.

What remains after a delta hedge is the curvature, and its size is predictable:

change in value ≈ Δ × (change in S) + ½Γ × (change in S)²

With the delta term hedged away, the leftover is the gamma term — and because it depends on the square of the move it has the same sign whichever way the share went. A delta-hedged option writer therefore loses on a large move up or down, which is the single most useful thing to be able to say about delta hedging, and the worked example demonstrates it in both directions with nearly identical losses.

The further hedges follow the same pattern. Adding a second option to cancel the gamma makes the position gamma-neutral, which cannot be done with shares alone because a share has zero gamma. Cancelling vega needs another option too. Each additional Greek hedged needs another instrument, and each hedge costs something to put on, which is why in practice a desk hedges delta continuously, gamma and vega loosely, and rho hardly at all.

Where marks are lost

Quoting a Greek without its units. Vega per 1.00 of volatility is a hundred times vega per percentage point; theta per year is 365 times theta per day. A question will say which it wants, and an unlabelled number cannot be marked.

Getting the sign of theta wrong, or of rho on a put. Long options lose value as time passes, so theta is negative. A put's rho is negative. Work the sign out from what the position is rather than remembering it.

Thinking gamma differs between a call and a put. It is identical, as is vega. Only delta, theta and rho differ, and delta differs by exactly 1.

Believing a delta hedge is a hedge. It protects against small moves only, and it has to be reset as the share moves. An answer that says "the position is now hedged" without the words "for small movements" or "until it is rebalanced" has missed the point of the question.

Using φ where Φ belongs. Gamma, vega and the first half of theta carry the normal density φ; delta, rho and the second half of theta carry the distribution function Φ. They are different functions of the same argument and mixing them gives plausible nonsense.

Questions students ask

Why can delta hedging not be perfect?

Because the delta changes continuously and a hedge can only be reset at discrete moments. Between resets the position carries the gamma exposure, and the loss over each interval is roughly ½Γ(ΔS)². Hedging more often reduces it but multiplies the transaction costs, and the trade-off between those two is a real and unavoidable cost of writing options. The Black-Scholes derivation assumes continuous, costless rebalancing precisely because that is the only way the hedge is exact.

Is being short gamma always bad?

No, and that is the point. The option writer is paid for it in theta. If the share moves less than the volatility in the price implied, the theta collected exceeds the gamma losses and the writer profits. Short gamma is a bet that the market will be quieter than the option was priced for, and it pays off most of the time and badly occasionally — which is the risk profile that makes option writing a professional activity rather than a free lunch.

Which Greek matters most?

Delta is managed most actively, because it is the largest exposure and the cheapest to hedge. Gamma usually determines whether the book makes money, because it decides what the delta hedging costs. Vega matters most to anything long-dated, which for actuarial work means guarantees. There is no single answer and a question asking for one is asking which position you are describing.

Do the Greeks change with dividends?

Yes, and the adjustment is the same one as in the price. For a continuous dividend yield q, replace S by Se^−qT throughout, which scales delta and gamma and adds a new Greek for sensitivity to q. The structure is unchanged, which is the useful part: a dividend is handled once, in the price, and every derivative of the price inherits it.

A worked example

A non-dividend-paying share trades at ₹1,200. A three-month European call is struck at ₹1,250. Volatility is 28% a year and the continuously compounded risk-free rate is 7%.

The share and all the parameters are invented for this example.

Step 1: the price

ln(1200/1250) = −0.0408220
(r + ½σ²)T = (0.07 + 0.0392) × 0.25 = 0.0273000
σ√T = 0.28 × 0.5 = 0.1400000

d₁ = (−0.0408220 + 0.0273000) / 0.14 = −0.0965857
d₂ = −0.0965857 − 0.14 = −0.2365857

Φ(d₁) = 0.4615277, Φ(d₂) = 0.4064892, φ(d₁) = 0.3970858
Ke^−rT = 1250 × e^−0.0175 = 1228.315295

c = 1200 × 0.4615277 − 1228.315295 × 0.4064892 = ₹54.5363

The put follows from parity: 54.5363 + 1228.3153 − 1200 = ₹82.8516.

Step 2: all five Greeks

Greek	Formula	Value	Quoted as
Delta	Φ(d₁)	0.4615277	0.4615 per ₹1 on the share
Gamma	φ(d₁)/(Sσ√T)	0.00236361	0.002364 per ₹1
Vega	Sφ(d₁)√T	238.251473	₹2.3825 per percentage point
Theta	see below	−168.371605	−₹0.4613 per day
Rho	KTe^−rTΦ(d₂)	124.824214	₹1.2482 per percentage point

Theta in full:

−1200 × 0.3970858 × 0.28 / (2 × 0.5) − 0.07 × 1228.315295 × 0.4064892
= −133.4208 − 34.9508 = −168.3716 a year

The check: every Greek against the price itself

Nudge one input, reprice, divide. None of these uses the formulae above.

Greek	From the formula	From a finite difference
Delta	0.4615277	0.4615274
Gamma	0.00236361	0.00236441
Vega	238.251473	238.251122
Theta	−168.371605	−168.371391
Rho	124.824214	124.824150

All five agree to five or six significant figures. The residual differences are the finite-difference step, not an error: a central difference has an error of order the step squared, and gamma, which needs a larger step to stay numerically stable, is correspondingly the least precise of the five. If a formula and its numerical derivative disagree in the first figure, the formula has been copied wrongly.

Step 3: hedge a book of 1,000 written calls

Short 1,000 calls is a liability of 1,000 × 54.5363 = ₹54,536 and a delta of −1,000 × 0.4615277 = −461.53. To be delta-neutral, buy 461.53 shares, costing 461.53 × 1,200 = ₹5,53,833.

Step 4: the share moves ₹25, and the hedge fails both ways

	Share rises to 1,225	Share falls to 1,175
Profit on the short calls	−₹12,274	+₹10,799
Profit on the 461.53 shares	+₹11,538	−₹11,538
Net	−₹736	−₹739
Gamma prediction	−₹739	−₹739
New delta	0.5202158	0.4024669
Shares to trade to re-hedge	buy 58.69	sell 59.06

Read the net row twice. The hedged book loses about ₹737 whether the share rose or fell, and the two losses are almost identical. That is not bad luck; it is the structure. The delta term cancelled by construction, and what is left is −½ × 1,000 × Γ × 25², which depends on the square of the move:

−0.5 × 1,000 × 0.00236361 × 625 = −₹738.63

against the actual −₹736.00 and −₹738.97. The small asymmetry is the third-order term, which the quadratic approximation ignores.

Now look at the last row. After the move the hedge is wrong by about 59 shares, and the required trade is to buy after a rise and sell after a fall. The writer of an option is forced to trade in the direction the market has already gone, every time. Do that repeatedly and the losses accumulate, and that accumulated cost is what being short gamma means in cash.

Step 5: what the writer collects for it

One day passing, with the share unchanged, is worth

−1,000 × (−168.371605) / 365 = +₹461.29 to the writer

Check it by repricing with one day less: the call falls from ₹54.536332 to ₹54.074159, a drop of ₹0.462173, or ₹462.17 across 1,000 options. The theta formula said ₹461.29. The two differ by less than a rupee, the difference being that theta is the instantaneous rate and a day is not instantaneous.

So the trade-off is now in numbers. The writer earns about ₹461 a day from the passage of time and loses about ₹739 for every ₹25 move in the share. A ₹25 move is about 2.1%, and 28% annual volatility implies a typical daily move of 28/√365 = 1.47%, or roughly ₹17.60 — which would cost about ₹366. Against ₹461 of theta, the writer is ahead on an average day and behind on a volatile one. That comparison, done with the position's own numbers, is what an option book is managed on.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

The Greeks: delta, gamma, vega, theta and rho, and how a hedge decays