Hedging with options · RiskAtoms Academy

By the end you will be able to

1Protect a position with a protective put or a collar, and find its floor and cap.
2Weigh the cost of option protection, including the cost of keeping it in place all year.
3Compare hedging with options and with futures: what each one gives up.
4Choose a strike, trading the level of protection against the premium.

Why it matters

Selling futures protects a portfolio against a fall, but it also gives away every gain. A put protects against the fall and keeps the gains. Nothing is free: the question is what that protection costs, and whether it is worth it.

Core explanation

Insurance for a price

A futures or forward hedge (L2.6.02: Hedging with forwards and futures) is symmetric: it removes losses and gains alike. An option hedge is lopsided, like insurance. You pay a premium up front, and in return you are protected against bad outcomes while keeping the good ones. L1.7.07: Option payoffs and strategies drew the shapes; here we use them as hedges.

The protective put. Own the asset and buy a put on it. If the price ends below the strike K, the put pays K − S_T, so you can always sell at K. Your worst result, the floor, is the strike minus what you paid for the asset, minus the premium. Above the strike the put expires worthless and you keep every gain, less the premium.

Choosing the strike. The strike works like an insurance deductible. An out-of-the-money put, with its strike below today's price, is cheaper because you bear the first part of any fall yourself. Six-month puts on a holding worth 100, at 20% volatility:

Strike	Premium	% of holding	Floor (worst profit)
80	0.24	0.24%	−20.24
85	0.65	0.65%	−15.65
90	1.46	1.46%	−11.46
95	2.84	2.84%	−7.84
100	4.88	4.88%	−4.88

The collar. To cut the premium, also sell a call above the price. The call premium you receive pays for some or all of the put. In return, gains above the call strike go to the call buyer: you now have a floor and a cap. If the call strike is set so the two premiums match, it is a zero-cost collar. Zero cost is not free: you pay with your upside.

The cost over time. One put protects for one period. Keeping protection all year means buying new puts again and again. Rolling 3-month puts 10% below the price costs about 2.4% of the holding a year when volatility is 20%, and about 10.3% when it is 35%. Protection is dearest when markets are frightened, which is when people most want it. That is why many investors buy protection only for specific periods, use collars, or accept the risk and hold more capital instead (L2.6.01: Four responses to risk).

Futures, put or collar?

4 cards

A fund must hand back cash to clients in two months and wants the value fixed, gains or not.
An investor expects a strong rally but fears a crash before an election, and has budget for a premium.
A founder holds a large block of her company's shares, wants a floor, has no cash for premiums and would accept giving up gains above 20%.
An exporter will receive US dollars and wants a worst-case rate but also to benefit if the dollar strengthens.

The formal version

We keep the notation of L1.7.07: Option payoffs and strategies: $S_0$ is the price paid for the asset, $S_T$ the price at expiry, $K_p$ and $K_c$ the put and call strikes, and $p$ and $c$ their premiums. Profit at expiry, per unit:

\text{Protective put} = S_T - S_0 + \max(K_p - S_T,\,0) - p \qquad \text{floor} = K_p - S_0 - p

\text{Collar} = S_T - S_0 + \max(K_p - S_T,\,0) - \max(S_T - K_c,\,0) - p + c

\text{floor} = K_p - S_0 - p + c \qquad \text{cap} = K_c - S_0 - p + c

A zero-cost collar sets $K_c$ so that $c = p$ ; then the floor is simply $K_p - S_0$ and the cap $K_c - S_0$ . Selling futures at $F_0$ instead locks in $F_0 - S_0$ whatever happens.

Where the premiums come from

The premiums here are Black-Scholes prices (L1.7.09: What drives an option’s price) with a 3% continuously compounded rate, no dividends and one volatility for every strike. They are fair illustrative prices, not quotes: real index options below the price usually cost more than this.

Try it

The holding was bought at 100. Start with the 90 put and the zero-cost call strike. Drag the price at expiry down to 60: the share alone loses 40, the hedges stop at their floors. Then raise the volatility and watch the premiums, and the zero-cost call strike, move.

Try itShare, put, collar or futures?

Black-Scholes model →

Put strike90

Call strike (collar)115.6

Volatility20%

Months to expiry6

Price at expiry80

Put premium1.46

Call premium1.46

Collar net cost0.00

Put cost, % of holding1.46%

Profit at 80

Share alone−20.00

Protective put−11.46

Collar−10.00

Sold futures+1.51

Per unit bought at 100. Premiums are Black-Scholes prices with a 3% rate and no dividends. Real markets usually charge more for puts below the price than this model does.

Worked example

A pension fund protects €20m of shares

A pension fund holds €20m of a euro-area equity index fund, 200,000 units at 100. It wants protection for six months. A bank quotes options on the fund at Black-Scholes prices: volatility 20%, rate 3%.

Put, collar or futures?

Step 1 of 6

1
Buy the 90 put
Premium 1.4594 a unit, so 1.4594 × 200,000 = €291,874, or 1.46% of the holding.

No hedge wins in every row. The put costs the most in a flat market and pays off in a crash or a boom; the collar is free but caps the boom; futures remove all uncertainty, good and bad. The right choice depends on what the fund cannot afford to lose and what it is willing to give up.

Regulatory lens · looking ahead

Under Solvency II, an insurer that buys puts on its shares can reduce its equity risk capital, provided the hedge meets the rules for risk-mitigation techniques (M4.6: Solvency II and insurance regulation). Under IFRS 9 hedge accounting, a firm can treat the time value of an option as a cost of hedging, so changes in it do not swing profit as much (M4.7: Accounting for risk). Banks that sell these options manage the risk with Greeks such as delta (L2.5.04: Sensitivities), which return in M3.1: Market risk.

Sparks

SparkInsurance you make yourself

In the 1980s, “portfolio insurance” promised the effect of a protective put without buying one: a model told the fund to sell index futures as prices fell and buy them back as they rose. On 19 October 1987 many insurers were selling at once into a falling market. The US Presidential Task Force on Market Mechanisms (the Brady report) found that their selling added to the pressure that day (L1.1.10: A short history of financial crises).

SparkWhy real puts cost more than the model

Black-Scholes uses one volatility for every strike. Since the 1987 crash, equity index options have shown a skew: puts well below the price trade at higher implied volatilities than puts near it. Buyers pay extra for crash protection, so real collars usually need a lower call strike than the model in this lesson suggests.

Key takeaways

A protective put sets a floor at K − S₀ − p and keeps the upside, for a premium paid up front.
A lower strike makes the put cheaper, like a bigger deductible.
A collar sells a call to pay for the put: a floor and a cap. Zero-cost collars pay with upside.
Futures lock in the value with no premium, but give away all the gains.
Protection must be bought again and again, and costs most when volatility is high.

Reaction

Check what stuck

Glossary

Protective put: Owning an asset and buying a put on it, so its value cannot fall below the strike, less the premium paid.
Collar: Owning an asset, buying a put below its price and selling a call above it: a floor and a cap.
Zero-cost collar: A collar whose call strike is chosen so that the premium received for the call pays for the put.
Floor: The worst result a hedge allows, whatever happens to the price.
Cap: The best result a hedge allows: gains above it are given away.
Premium: The price paid today for an option, like an insurance premium.
Out of the money (put): A put whose strike is below the current price: it pays out only after a fall.
Volatility skew: The pattern in which options with strikes below the price cost more than a single-volatility model such as Black-Scholes suggests.

Deep Dives & Origins

Deep Dives

John C. Hull, Options, Futures, and Other Derivatives, 11th edition (Pearson, 2021), the chapter “Trading Strategies Involving Options”. Intermediate.
Sheldon Natenberg, Option Volatility and Pricing, 2nd edition (McGraw-Hill, 2015), the chapters on hedging and volatility skews. Intermediate.

Origins

John C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021.
Report of the Presidential Task Force on Market Mechanisms (the Brady report), US Government Printing Office, January 1988.
IASB, IFRS 9 Financial Instruments (2014), paragraph 6.5.15, time value of options.
Commission Delegated Regulation (EU) 2015/35 (Solvency II), the articles on risk-mitigation techniques.

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