Options and volatility · 10 lessons

Revision sheet

Every Options and volatility lesson’s essentials, formula and takeaway on one page. Each heading opens the full lesson.

O1Option payoffs

Calls and puts: A is the right, not the duty, to buy at a fixed price, the , until expiry; a is the right to sell. The buyer pays the seller a up front.
Payoff and breakeven: At expiry a call pays the amount the price is above the strike and a put the amount below, or nothing. A buyer breaks even at the strike plus the premium for a call, minus it for a put: a 100 call bought for 5 needs 105.
Buyer and seller: The buyer can lose only the premium. The seller keeps at most the premium and can lose far more: a short call’s loss has no limit, and a short put can lose the strike less the premium.
Moneyness: A call is in the money when the price is above the strike and out of the money when it is below; a put is the reverse. With a share at 100, the 95 call is in the money and the 105 call out of it.
Intrinsic and time value: is what the option would pay if it expired now; is the rest, and falls to zero by expiry. A 95 call at 7.50, share at 100, is 5 intrinsic and 2.50 time value.

Payoff and breakeven at expiry

Call = max(S − K, 0); put = max(K − S, 0); breakeven = K ± premium

S: price of the underlying at expiry
K: the strike
±: + for a call, − for a put; the same for buyer and seller

Remember: Breakeven = strike ± premium. Buyers can lose only the premium; sellers keep at most the premium, and a short call’s loss has no limit.

O2What drives an option’s price

Six inputs: An option’s price depends on the share price, the strike, time to expiry, volatility, interest rates and dividends. A higher share price lifts calls and lowers puts; a higher strike does the reverse.
Volatility: How much a price swings, as a yearly %. Bigger swings make calls and puts dearer: the payoff can grow but the loss stops at the premium. Options are quoted in , the volatility their price implies.
Time: More time makes an option dearer, but value grows with the square root of time: four times as long costs about twice as much. A one-year at-the-money option at 20% volatility is worth about 8% of the share price.
Rates and dividends: Higher interest rates lift calls and lower puts, since the strike changes hands later. A bigger dividend does the reverse, since the share price drops by the dividend when it is paid. Both effects are small for short-dated options.
Early exercise: options can be exercised on any day, European ones only at expiry. Exercising a call early on a share with no dividends throws away time value; a deep in-the-money put can be worth exercising.

The at-the-money shortcut

C ≈ 0.4 × S × σ × √T

S: share price
σ: volatility a year, as a decimal (20% = 0.20)
T: years to expiry
0.4: about 1 ÷ √(2π); Black–Scholes gives 7.97 where the rule gives 8

Remember: At the money ≈ 0.4 × S × σ × √T. Volatility and time make every option dearer; spot, strike, rates and dividends push calls and puts in opposite directions.

O3Put–call parity and synthetics

A call minus a put is a forward: A bought call and a sold put at the same strike and expiry buy the share at the strike whatever happens. So the pair behaves like owning the share, except that the strike is paid later instead of the price now.
Put–call parity: The call minus the put equals the share price less the strike’s . A 100 strike six months away at 4% is worth 98.04 today, so with the share at 100 the call costs 1.96 more than the put.
Rates and dividends: Higher rates shrink the strike’s present value, so calls get dearer against puts. A dividend before expiry goes to the shareholder, not the option holder, so it makes calls cheaper and puts dearer.
Synthetics: Any one of call, put, share and cash can be built from the other three: a . A bought call plus a sold put is long the share; the share plus a bought put is a call; the share plus a sold call is a sold put.
Conversions and reversals: When the call is dear against parity, a sells it, buys the put and buys the share, locking in the gap. A reversal does the opposite for a cheap call. Traders run them all day, so gaps rarely last.

Put–call parity

C − P = S − PV(dividends) − PV(K); PV(K) = K ÷ (1 + r × T)

C, P: call and put prices, same strike and expiry
S: spot price of the stock
K: the strike
r: interest rate a year, simple
T: years to expiry

Remember: C − P = S − PV(K). A call minus a put at the same strike is a forward, so any three of call, put, stock and cash price the fourth.

O4Delta and hedging

Delta: The change in an option’s value for a 1-point move in the share. A call’s runs from 0 to 1 and a put’s from 0 to −1; an at-the-money call is about 0.5, so it moves about half as much as the share.
Delta and moneyness: An option far in the money moves almost one for one with the share, because it is almost sure to pay; one far out of the money barely moves. Near expiry, deltas head to these extremes.
Position delta: Each contract covers 100 shares, so 30 bought calls at 0.55 act like 1,650 shares: a £1.20 rise makes about £1,980. Sold options flip the sign, and adding every line gives the book’s delta.
Delta hedging: Holding the opposite delta in shares, so small moves cancel: 20 bought calls at 0.45 are hedged by selling 900 shares. Delta shifts as the share moves, at a rate called , so the hedge needs topping up.
Hedging bought and sold options: Hedging bought options means selling after rises and buying after falls. Hedging sold options is the reverse, so dealers who have sold many options can push the market further the way it is already going.

Delta hedge

Shares to trade = −Δ × contracts × 100; P&L ≈ Δ × ΔS

Δ: delta of one option: 0 to 1 for calls, 0 to −1 for puts (sign flipped if sold)
100: shares per contract
ΔS: change in the share price

Remember: Hedge = −Δ × contracts × multiplier. Delta is the hedge ratio: hold the opposite delta in stock and small moves cancel, then rebalance as delta moves.

O5Gamma, theta and the hedger’s P&L

Gamma: How much delta changes for a 1-point move in the share. Bought options have positive and sold options negative; it is highest at the money with little time left, where delta can jump from near 0 to near 1.
Theta: The value an option loses as a day passes, other things equal: negative for bought options, positive for sold ones. speeds up in the final weeks for options at the money.
Gamma against theta: A hedged bought option earns from big moves either way, locked in by re-hedging (), and pays theta daily. With gamma of 200 shares per £1 and theta of −£300, a £2 move makes £100 and a £1 move loses £200.
Breakeven move: The daily move at which gamma gains pay the theta: about £1.73 for gamma of 200 and theta of −£300. It is the move the option’s volatility priced, so the hedged buyer wins if the share moves more.
Dealers’ gamma: Dealers who are long gamma sell rallies and buy dips, which calms the market and near expiry can a share to a strike. Dealers who are short gamma sell falls and buy rallies, which can make moves bigger.

Delta-hedged P&L over a day

P&L ≈ ½ × Γ × (ΔS)² + Θ; breakeven move = √(2 × |Θ| ÷ Γ)

Γ: gamma: the change in delta per 1-point move (in shares, for a book)
ΔS: the day’s move in the underlying
Θ: theta: the day’s time decay (negative for long options)

Remember: Hedged P&L ≈ ½ × Γ × (ΔS)² + Θ. Long gamma earns from big moves and pays theta every day; it wins when the market moves more than the option priced.

O6Volatility: implied, realised and the risk premium

Realised volatility: How much a price actually moved: the standard deviation of daily returns, scaled to a year by about 16 (√252 trading days). Typical daily moves of 1.6% make of about 25%.
Implied volatility and the premium: The volatility built into option prices. It has tended to sit above the volatility that followed, because buyers pay for crash insurance: that gap, the , is why sellers usually win.
Rule of 16: A typical daily move is about yearly volatility ÷ 16: 16% means about 1% a day. A of 20 (the 30-day implied volatility of S&P 500 options) prices daily index moves of about 1.25%.
Vega: The change in an option’s value for a 1-point rise in implied volatility: positive for bought options, largest at the money and for long expiries. A book with of −£230 a point makes £920 if implied falls 4 points.
Index against single-stock volatility: Shares in an index partly cancel, so index volatility is lower: stocks at 30% with a of 0.25 make an index at about 15%. In a sell-off correlation jumps and index volatility rises towards the stocks’.

Volatility in numbers

Daily move ≈ σ ÷ 16; vega P&L = vega × Δσ

σ: volatility a year, in %
16: about √252, the trading days in a year
Δσ: change in implied volatility, in points

Remember: Daily move ≈ volatility ÷ 16. Implied volatility is the price, realised is what happens, and a hedged option wins or loses by the gap.

O7Skew and term structure

The smile: Options on one underlying trade at different implied volatilities by strike, and plotted against strike they form a curve. With an index at 100, the 90 put might trade at 24%, the 100 strike at 19% and the 110 call at 16%.
Put skew and call skew: In share indices, low-strike puts carry the highest , called : markets fall faster than they rise and investors keep buying protection. Some commodities, such as natural gas, show call skew.
Risk reversal and butterfly: The is the 25-delta call’s volatility minus the 25-delta put’s: −4 means puts trade 4 points over calls. The shows how far both wings sit above the at-the-money volatility.
Term structure: The is at-the-money volatility by expiry. It slopes upwards in calm markets and inverts in a sell-off, when near-dated volatility jumps most; a known event lifts the first expiry after it.
Volatility surface: Implied volatility by strike and expiry together: a smile for every expiry. Desks value their whole options book from the every day and watch how it shifts as markets move.

Quoting the smile

Risk reversal = σ(25Δ call) − σ(25Δ put); butterfly = ½[σ(25Δ call) + σ(25Δ put)] − σ(ATM)

σ(25Δ call): implied volatility of the 25-delta call
σ(25Δ put): implied volatility of the 25-delta put
σ(ATM): at-the-money implied volatility

Remember: Risk reversal = call vol − put vol. The smile prices the tail the market fears, and the term structure shows when: calm curves slope up, stressed ones invert.

O8Strategies

Choosing a strategy: Three questions set the strategy: which way the price will go, whether it will move more or less than options price, and how much you can lose. Bought options cap the loss at the premium; sold ones can lose far more.
Vertical spreads: A buys one option and sells another of the same type at a further strike: cheaper, with capped gains. A 100/110 bull call spread costing 3.50 breaks even at 103.50 and makes at most 6.50.
Collars and covered calls: A holds the share, buys a put below and sells a call above to pay for it, capping both loss and gain; with matched premiums it costs nothing. Selling the call alone against shares you own is a covered call.
Straddles and strangles: A buys the at-the-money call and put and pays on a big move either way: at a 100 strike, costing 8, it needs the price below 92 or above 108. A uses out-of-the-money strikes: cheaper, but it needs a bigger move.
Butterflies and calendars: A long butterfly buys the 90 and 110 calls and sells two 100 calls: a cheap bet that the price ends near 100. A sells a near-dated option and buys a later one: short gamma, long vega.

Straddle breakevens

Breakevens = K ± (C + P); at the money, C + P ≈ 0.8 × σ × √T × S

K: the strike
C + P: total premium paid for the call and the put
σ, T, S: volatility a year (as a decimal), years to expiry and the share price

Remember: Straddle ≈ 0.8 × σ × √T × S. Pick a strategy by direction, movement and risk; premium ÷ strike is the move a straddle buyer needs.

O9Volatility products and flows

VIX futures and the roll: The can’t be bought, only futures on it. In calm markets they sit above it, in , and drift down as expiry nears, so funds that keep rolling them lose value; in a sell-off the curve flips to .
Variance swaps: A pays on squared against a strike, so a spike gains more than a lull loses: on £100,000 vega notional struck at 20, realised 25 pays £562,500 and 15 costs £437,500.
Short volatility: positions, such as sold options or short VIX futures, make small gains most days and large losses rarely. On 5 February 2018 a spike cut XIV, a short-VIX product, by 80% or more in a day.
0DTE options: expire on the day they trade; S&P 500 index options have had an expiry every trading day since 2022. With hours left, at-the-money options swing hard on small moves and lose value fast.
Dispersion: A sells volatility on an index and buys volatility on the index’s shares. It wins when the shares move a lot but separately, and loses when they fall together and correlation jumps.

Variance swap P&L

P&L ≈ N_vega × (σ² − K²) ÷ (2K)

N_vega: vega notional: roughly the P&L per volatility point near the strike
σ: realised volatility over the swap’s life, in points
K: the strike, in volatility points

Remember: Var swap P&L ≈ N_vega × (σ² − K²) ÷ 2K. Short volatility earns small and often and loses large and rarely; contango makes long-VIX products pay to wait.

O10Options across asset classes

Currency options: Every currency option is a call on one currency and a put on the other, so traders name both: a EUR call / USD put is the right to buy euros with dollars. Struck at 1.05, it is in the money with EUR/USD at 1.08.
Caps and floors: A pays a borrower whenever a floating rate such as fixes above its strike; a floor pays when the rate fixes below. Selling a floor to pay for a cap makes an interest rate collar.
Swaptions: A is the right to enter an later at a rate agreed now. A payer (pay fixed) gains when rise, a receiver when they fall; a payer on gains as spreads widen.
What a swaption pays: The rate gap for every year of the swap, in today’s money: a payer struck at 4%, with the swap rate at 4.6%, an of 4.38 and £100m, pays about £2.63m.
Barrier options: A switches on or off at a set level: a knock-out dies if the price touches it; a knock-in comes alive only if it does. Knock-outs cost less, but the protection can vanish in the spike it was bought for.

Rates options at expiry

Payer swaption = max(S − K, 0) × annuity × N; caplet = max(L − K, 0) × N × days ÷ 365

S: the swap rate at expiry
K: the strike rate
annuity: present value of 1 a year over the swap’s life, in years
N: notional
L: the floating rate for the period, such as compounded SONIA (ACT/365)

Remember: Payer = max(S − K, 0) × annuity × N. Rates, FX, credit and commodity options follow the same rules as equity options; only the underlying and the naming change.