Lesson 6.4: Volatility trading and hedging
Learn how delta hedging makes an option a volatility bet, why realised against implied volatility drives profit, how often to hedge and how variance swaps work.
By the end of this lesson you will be able to explain how a delta-hedged option position makes or loses money, work out the profit of one day of hedging from gamma and the size of the move, say why the result depends on whether realised volatility is above or below the implied volatility paid, see how often hedging reduces the spread of results and what it costs, describe a variance swap and compute its payoff, and hedge a book against gamma and vega as well as delta. You will show that a long six-month at-the-money option hedged daily has an average profit of about +1.40 when the market then realises 25 per cent volatility, about 0.00 at 20 per cent and about −1.41 at 15 per cent, that hedging 126 times instead of 4 cuts the standard deviation of the outcome from 2.27 to 0.44, that a variance swap with a vega notional of ₹1 lakh per point and a strike of 20 pays ₹4.4 lakh when volatility realises at 24, and that a gamma and vega position of −50 and −300 needs 1,062.5 and 250 contracts of two hedging options. You should have read Lessons 6.2 and 6.3. The mathematics is the Greeks, squares and a small system of two equations. This course is education about how markets and research work, not investment advice.
Lesson 6.2 showed that you can neutralise an option's delta by trading the underlying, and that what remains is exposure to gamma, theta and vega. Lesson 6.3 showed that the market's implied volatility usually sits above what the share later realises. This lesson puts the two together. A trader who buys or sells options and hedges the price direction is no longer betting on where the share goes. They are betting on how much it moves, and the winner is whoever correctly judged the volatility. That is volatility trading.
How delta hedging turns an option into a volatility bet
Suppose you buy a call and sell short enough of the underlying to make the position delta-neutral. A moment later the price has moved. Your delta has changed, since gamma is positive, so you rebalance: if the price rose, your option's delta rose, so you sell more of the underlying, higher up; if it fell, you buy back some, lower down. You are selling after rises and buying after falls, buying low and selling high on every wiggle. That is the gain from being long gamma.
Over a short time dt, with the share moving by ΔS, the profit of the hedged position, after allowing for the option's time decay, is approximately
hedged profit ≈ ½ × Γ × S² × [ (ΔS ÷ S)² − σimplied² × dt ]
The first term is the gamma gain from the actual move, which grows with its square. The second is the time decay, and, as Lesson 6.2 showed, it equals the gamma gain from a move of exactly the size the market priced: σimplied × √dt of the price. So you gain when the squared return is larger than the implied variance for that period, and lose when it is smaller. Over the life of the option the profit is the sum of these, so it depends on the difference between realised variance and implied variance.
Worked example 1. The six-month at-the-money call of Lesson 6.2 has Γ = 0.0271 and S = 100. Its implied volatility is 20 per cent, so a day's implied move is 20 ÷ √365 = 1.05 per cent, or 1.05 in price. The time-decay charge for a day is 0.5 × 0.0271 × 100² × 0.04 ÷ 365 = 0.0149.
If the share moves 2.0 in a day, the gamma gain is 0.5 × 0.0271 × 2.0² = 0.0542. The net profit is 0.0542 − 0.0149 = +0.039.
If it moves 0.5, the gamma gain is 0.5 × 0.0271 × 0.25 = 0.0034, and the net is 0.0034 − 0.0149 = −0.012.
If it moves 1.05, the gamma gain is 0.0149 and the net is zero.
A long option holder is paying a small fixed daily fee and collecting a large payment on days of large moves. Which kind of year it turns out to be depends on the realised volatility.
What realised against implied volatility does: a simulation
To see this clearly, we simulated the strategy. A trader buys one six-month at-the-money call on a share at 100, with a strike of 100 and 6 per cent interest, at an implied volatility of 20 per cent, and delta-hedges it every day until expiry. The share's returns are then drawn with a realised volatility of 15, 20 or 25 per cent. The drift of the share is set to the risk-free rate so that only the volatility matters. We ran 4,000 paths for each case. The profit is measured at expiry, including the financing of the hedge.
Realised volatility · Mean profit per option · Standard deviation of profit15% · −1.41 · 0.57
20% (equal to implied) · −0.01 · 0.44
25% · +1.40 · 0.80
When the realised volatility equals the implied volatility, the long option holder breaks even, on average. When the share is more volatile than the price assumed, the holder gains about 1.4 per option, roughly 20 per cent of the premium of 7.16. When it is calmer, the holder loses about the same amount. The seller of the option, hedging in the same way, has exactly the opposite result. That is the simulation counterpart of Lesson 6.2's claim that gamma and theta are two sides of one trade.
A rule of thumb for the size of the result comes from the formula above, using the option's initial gamma: the expected profit is about ½ × Γ × S² × (σrealised² − σimplied²) × T. For the example it gives 0.5 × 0.0271 × 10,000 × (0.0625 − 0.04) × 0.5 = 1.52 at 25 per cent and −1.19 at 15 per cent, close to the simulated 1.40 and −1.41. It is an approximation, since gamma changes over the life of the option and with the price.
The profit also depends on where the variance is earned. Gamma is largest when the share is near the strike and falls away as it moves far from it, so a volatile market that spends its time away from the strike gives a smaller gamma gain than the same volatility near the strike. The theory says that the profit of a delta-hedged option is the sum, over each day, of the dollar gamma (½ × Γ × S²) times the realised variance minus the implied variance of that day. The result is path-dependent, and that is why the standard deviations in the table are not small.
Use the calculator to estimate the profit from your own numbers.
How often should you hedge?
A hedge made once a day leaves a gap between rebalances in which the position is exposed to gamma. In the limit of continuous hedging, the result is exactly the realised-against-implied-variance payoff, with no error. In practice the error shrinks with the number of hedges but never vanishes. We ran the same simulation at realised volatility equal to implied, 20 per cent, for different numbers of hedges over the six months.
Hedges in six months · About how often · Mean profit · Standard deviation of profit4 · every six weeks · +0.03 · 2.27
13 · every two weeks · −0.02 · 1.32
26 · weekly · 0.00 · 0.94
63 · every other day · −0.01 · 0.64
126 · daily · 0.00 · 0.44
The average is zero in every row, as it should be when realised equals implied. What changes is the spread of outcomes, which falls roughly in proportion to one over the square root of the number of hedges: going from 4 to 126 hedges, a factor of about 32, cuts the standard deviation by a factor of about 5, from 2.27 to 0.44. Each extra hedge costs money, though: the spread on the shares, fees and market impact (Lesson 4.4). So there is a trade-off between risk and cost. Professionals do not hedge on a timetable. They hedge when the delta has moved by more than a set amount, a no-trade band, which saves cost in quiet markets and acts quickly in fast ones. Models such as Whalley and Wilmott's give the optimal width of the band.
Ways to trade volatility
Delta-hedged options. Buy or sell a straddle or a strangle, or a single option, and hedge the delta. A buyer is long gamma and long vega, and pays theta. A seller collects theta and is short gamma and short vega.
Gamma scalping. A long-gamma trader hedges often and treats the profit from rebalancing as their income, which has to exceed the theta they pay.
Spreads across strikes and expiries. A calendar spread buys one expiry and sells another to trade the term structure. A risk reversal trades the skew (Lesson 6.3).
Variance and volatility swaps. Contracts on realised volatility, described next.
Volatility index products. Futures and options on a volatility index, where they exist, let a trader take a view on the level of implied volatility without a position in single options.
Dispersion. Sell index options and buy options on the index's stocks. The strategy bets that the stocks' individual volatility will be high compared with what is implied by the index, which includes the diversification between them.
Variance swaps
A variance swap is a contract that pays the difference between the realised variance of an asset over a period and a fixed strike variance agreed at the start, times a notional. Its payoff avoids the complexity of delta-hedged options, since it is a pure bet on realised volatility and has no path dependence on where the price was. It is convenient to quote the notional as a vega notional: the profit or loss for a one-point change in volatility near the strike. The variance notional is then
variance notional = vega notional ÷ (2 × strike volatility)
and the payoff is variance notional × (realised volatility² − strike volatility²), with volatilities in points.
Worked example 2. A trader buys a variance swap with a vega notional of ₹1 lakh per volatility point and a strike of 20. The variance notional is 100,000 ÷ (2 × 20) = ₹2,500 per squared point. If realised volatility turns out to be 24, the payoff is 2,500 × (24² − 20²) = 2,500 × 176 = ₹4.4 lakh. If it is 16, the payoff is 2,500 × (256 − 400) = −₹3.6 lakh. Notice that the gain from four points above the strike, ₹4.4 lakh, is more than the loss from four points below it, ₹3.6 lakh: a variance swap is convex in volatility, so the buyer gains more from a spike than they lose from a lull, and the strike is set a little above the expected volatility to compensate the seller. A variance swap can be replicated by holding a static portfolio of options at all strikes with weights that fall with the square of the strike, which is how dealers price and hedge it.
Hedging more than delta
A book of options has exposure to gamma and vega as well as delta, and a trader may want to remove them. The underlying has a delta of 1 and no gamma or vega, so it can hedge delta only. To hedge gamma and vega you must trade other options. Each hedging instrument contributes some gamma and some vega, and the trader solves for the quantities that cancel the book's exposures.
Worked example 3. A book has a gamma of −50 and a vega of −300, in the units of the Greeks of the instruments below. Two hedging options are available: option A with a gamma of 0.04 and a vega of 0.20 per contract, and option B with a gamma of 0.03 and a vega of 0.35. Find the quantities a of A and b of B so that the total is zero:
gamma: −50 + 0.04a + 0.03b = 0
vega: −300 + 0.20a + 0.35b = 0
From the first, a = (50 − 0.03b) ÷ 0.04. Substituting into the second: 0.20 × (50 − 0.03b) ÷ 0.04 + 0.35b = 300, which is 250 − 0.15b + 0.35b = 300, so b = 250, and a = (50 − 7.5) ÷ 0.04 = 1,062.5 contracts. Buying these two options makes the book gamma-neutral and vega-neutral. The new options also change the delta, so the trader then adjusts the underlying position to make the whole book delta-neutral again. Real books do this across many strikes and expiries with a risk system, and they limit the exposures that they leave unhedged.
Since the Greeks are only local measures, a trader also checks the book against scenarios: a table of profit and loss for combinations of price moves, for instance −10 to +10 per cent, and volatility moves, for instance −5 to +10 points. A position that looks safe in delta, gamma and vega may lose a great deal in a large move, because the Greeks themselves change.
The risks of selling volatility
Because implied volatility has on average exceeded what was realised, selling it earns a premium most of the time. That makes it popular and dangerous, for the same reasons as Lessons 4.2 and 5.2:
Short gamma in a crash. A large fall is a fast move, with rising volatility and widening spreads, just when a short-gamma trader has to buy back to hedge at worse and worse prices. The losses can exceed many years of premium.
Short vega in a volatility spike. Implied volatility can double in days. A short option position loses on both the price move and the rise in volatility.
Margin calls. Exchanges raise margin when volatility rises, and a position that was affordable becomes a forced sale.
Selection bias in track records. A short-volatility record of several quiet years has a high Sharpe ratio and says little about the tail, as the example of Lesson 4.2 showed.
Crowding. If many traders hold the same short positions, a spike forces them all to buy options back at once.
The sensible way to trade volatility, long or short, is to size by the worst realistic loss, not by the average profit, to hold some protection against a tail and to watch the exposures in a scenario table.
The code below runs the delta-hedging simulation for different realised volatilities and hedge frequencies, and computes the variance swap payoff, the one-day hedging arithmetic and the two-option hedge.
python
import numpy as np
from math import exp, sqrt, log
from scipy.stats import norm
def bscalldelta(S, K, r, sigma, T):
d1 = (np.log(S / K) + (r + 0.5 sigma 2) T) / (sigma np.sqrt(T))
d2 = d1 - sigma np.sqrt(T)
return S norm.cdf(d1) - K np.exp(-r T) norm.cdf(d2), norm.cdf(d1)
def hedgedprofit(seed, realised, nhedges, paths=4000, S0=100.0, K=100.0, r=0.06, implied=0.20, T=0.5):
"""Buy one call at the implied volatility, delta-hedge nhedges times, and report the profit at expiry."""
rng = np.random.defaultrng(seed)
dt = T / nhedges
S = np.full(paths, S0)
premium, delta = bscalldelta(S0, K, r, implied, T)
delta = np.full(paths, delta)
cash = np.full(paths, -premium) + delta S # pay the premium, sell delta shares short
for i in range(1, nhedges + 1):
z = rng.standardnormal(paths)
S = S np.exp((r - 0.5 realised 2) dt + realised sqrt(dt) z)
cash = cash exp(r dt) # the cash earns interest
if i < nhedges:
, newdelta = bscalldelta(S, K, r, implied, T - i dt)
cash += (newdelta - delta) S # sell more shares if delta rose
delta = newdelta
return np.maximum(S - K, 0) + cash - delta S # option payoff, cash, buy back the shares
print("realised volatility -> average profit and spread (daily hedging, 126 hedges)")
for realised in (0.15, 0.20, 0.25):
pnl = hedgedprofit(1, realised, 126)
print(f" {realised:.0%}: mean {pnl.mean():6.2f} | sd {pnl.std():.2f}")
print("hedges -> spread of the profit when realised equals implied")
for n in (4, 13, 26, 63, 126):
pnl = hedgedprofit(2, 0.20, n)
print(f" {n:>3}: mean {pnl.mean():6.2f} | sd {pnl.std():.2f}")
the rule of thumb from the initial gamma
S0, sigma, T, r = 100.0, 0.20, 0.5, 0.06
d1 = (log(1) + (r + 0.5 sigma 2) T) / (sigma sqrt(T))
gamma = norm.pdf(d1) / (S0 sigma sqrt(T))
for realised in (0.25, 0.15):
print(f"rule of thumb at realised {realised:.0%}:", round(0.5 gamma S0 2 (realised 2 - sigma 2) T, 2))
one day of hedging
decay = 0.5 gamma S0 2 sigma 2 / 365
print("daily time-decay charge", round(decay, 4))
for move in (2.0, 0.5, 1.05):
print(f" a move of {move}: gamma gain {0.5 gamma move 2:.4f} | net {0.5 gamma move 2 - decay:+.4f}")
variance swap
veganotional, strike = 100000, 20
variancenotional = veganotional / (2 strike)
for realised in (24, 16):
print("variance swap, realised", realised, "payoff", variancenotional (realised 2 - strike 2))
hedging gamma and vega with two options
book = np.array([50.0, 300.0]) # minus the book's gamma and vega
options = np.array([[0.04, 0.03], [0.20, 0.35]]) # rows: gamma, vega; columns: option A, option B
print("contracts of A and B:", np.linalg.solve(options, book))
Common mistakes
Calling a delta-hedged position riskless. It is free of small price moves, not of volatility, and its profit has a spread.
Judging a volatility trade by price direction. The result depends on realised against implied variance, whichever way the share went.
Hedging on a fixed timetable only. Hedging when delta has moved by a set amount handles both quiet and fast markets better.
Ignoring hedging costs. More hedges lower the spread of results and raise costs, and for a thin stock costs can exceed the gamma profit.
Selling options for the premium without a stress limit. The loss in a crash can be many times the premium collected.
Confusing vega with a bet that is easy to size. A volatility spike can be violent, and the margin rises with it.
Forgetting convexity of variance contracts. A strike set near the expected volatility already prices the asymmetry.
Hedging gamma and vega without re-hedging delta. The new options change the delta.
Relying on Greeks alone. They are local, so check scenario tables for large moves.
Key takeaways
A delta-hedged option position is a bet on volatility, not direction: the profit over a short time is about ½ΓS² × [(ΔS ÷ S)² − σimplied² dt]. You gain when the realised squared return exceeds the implied variance.
One day's example: gamma 0.0271 at a price of 100 with an implied 20 per cent volatility has a time-decay charge of 0.0149. A move of 2.0 gives +0.039, a move of 0.5 gives −0.012, and a move of 1.05 breaks even.
In simulation, a long six-month at-the-money call hedged daily averaged +1.40 at realised volatility of 25 per cent, −0.01 at 20 per cent and −1.41 at 15 per cent. The rule of thumb ½ΓS²(σr² − σi²)T gave +1.52 and −1.19.
The spread of results falls roughly with the square root of the number of hedges: 2.27 with four hedges to 0.44 with 126. Each hedge costs money, so traders hedge on a no-trade band.
Volatility can be traded with delta-hedged options, gamma scalping, spreads, variance and volatility swaps, volatility index products and dispersion trades.
A variance swap pays variance notional × (realised² − strike²), with the variance notional equal to the vega notional ÷ (2 × strike). With ₹1 lakh per point and a strike of 20, a realised 24 pays ₹4.4 lakh and a realised 16 pays −₹3.6 lakh: the payoff is convex.
To hedge gamma and vega, solve for the quantities of other options: −50 and −300 needed 1,062.5 and 250 contracts, followed by a new delta hedge. Check scenario tables for large moves.
Selling volatility earns a premium most of the time and suffers rare large losses from short gamma, short vega, margin calls and crowding. Size by the worst realistic loss.
Check yourself
1. A delta-hedged long option has a gamma of 0.05 at a share price of 200. The implied volatility is 25 per cent. What is the daily time-decay charge, using 365 days?
0.5 × 0.05 × 200² × 0.0625 ÷ 365 = 0.5 × 0.05 × 40,000 × 0.0625 ÷ 365 = 62.5 ÷ 365 = 0.171.
2. With that gamma, what daily move breaks even?
The implied daily move is 25 ÷ √365 = 1.31 per cent of 200 = 2.62. Check: 0.5 × 0.05 × 2.62² = 0.172, matching the charge.
3. A trader sold options and hedged them daily. Realised volatility came in 4 points below implied. Did they gain or lose, and why?
They gained, since the seller of a delta-hedged option profits when realised variance is below implied variance: the gamma loss on the smaller moves was less than the theta collected.
4. A variance swap has a vega notional of ₹50,000 and a strike of 25. What is the payoff if volatility realises at 30?
Variance notional = 50,000 ÷ 50 = 1,000. Payoff = 1,000 × (900 − 625) = ₹2.75 lakh.
5. Why does halving the hedging interval not halve the spread of the hedging error?
The spread falls with the square root of the number of hedges, so doubling the number of hedges reduces it by a factor of 1.41, not 2, while the cost of hedging doubles.
Try this
Set aside forty-five minutes. You need Python and the code in this lesson.
1. Run the code. Check that you see profits of about −1.41, −0.01 and +1.40 and the declining spread with the number of hedges.
2. Change the implied volatility to 25 per cent and rerun with realised volatility of 20 per cent. Does the buyer or the seller gain, and by how much? Compare with the rule of thumb.
3. Add a proportional cost of 5 basis points on every share traded in the hedge. How does the mean profit change for 126 hedges and for 13 hedges? Which is better after costs?
4. Replace the daily schedule with a no-trade band: rebalance only when the delta has moved by more than 0.05. How many hedges per path do you make, and what is the spread?
5. Simulate the same strategy with a share that has a crash, for example a drop of 20 per cent on one day. What happens to a seller who is hedged daily?
6. Use the variance swap simulator to find the realised volatility at which a swap struck at 18 pays zero, and the payoffs at 12 and 30.
7. Write a short note: one volatility position, its delta, gamma and vega, the scenario in which it loses most and how you would limit that loss.
A good result is a table of hedged profits against realised volatility and hedge frequency, a clear statement of what the position wins and loses on and a stress scenario for its worst day. This completes the module on derivatives and volatility. The next module goes under the hood: Brownian motion, Itô's lemma, and why the Black–Scholes formula has the form it does.
Read more
Peter Carr and Liuren Wu's papers on variance swaps and Kresimir Demeterfi, Emanuel Derman, Michael Kamal and Joseph Zou's 1999 note "More Than You Ever Wanted to Know About Volatility Swaps" explain how to replicate and price them. Euan Sinclair's Volatility Trading and Sheldon Natenberg's Option Volatility and Pricing are practitioner texts on delta hedging and gamma. Nassim Taleb's Dynamic Hedging covers the management of option books, and Alan Whalley and Paul Wilmott's 1997 paper "An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs" in Mathematical Finance derives the no-trade band. Jim Gatheral's The Volatility Surface discusses variance swaps and the risk premium.
Tags: Quant Trader Volatility trading Delta hedging Variance swap Gamma scalping