Equity Derivatives —
Options, Greeks & vol.

A call option gives the holder the right, but not the obligation, to buy an underlying asset at a fixed price (the strike) on or before expiry. A put option gives the right, but not the obligation, to sell the underlying at the strike.

The buyer pays an option premium upfront to the seller (the writer). This premium is the maximum the buyer can ever lose — they simply do not exercise if the option is not in their favour. The seller receives the premium but bears the obligation: for a short call, losses are theoretically unlimited if the underlying rises; for a short put, losses are substantial if the underlying falls toward zero.

The premium itself has two components: intrinsic value (the immediate exercise value — max(0, spot − strike) for a call) plus time value (the additional amount paid for the possibility of further favourable movement before expiry).

Options can be European (exercisable only at expiry) or American (exercisable at any point up to expiry). Most OTC equity and index options are European; listed single-stock options are often American.

Intrinsic value is the immediate exercise value. For a call: max(0, spot − strike). If spot is 110 and strike is 100, intrinsic value is 10. If spot is 90, intrinsic is zero — you would not exercise at a loss. For a put: max(0, strike − spot).

Time value is everything above intrinsic value — the premium the market pays for the possibility of the underlying moving further in your favour before expiry. Even an out-of-the-money option with zero intrinsic value has time value as long as there is time remaining and uncertainty about future prices.

Time value is highest when the option is at-the-money (spot ≈ strike). This is because the outcome is most uncertain — the underlying is right at the strike, and small moves in either direction determine whether the option expires in or out of the money. Deep ITM and OTM options have less time value because the outcome is already more predictable.

As expiry approaches, time value erodes — this is theta decay. At expiry, an option is worth exactly its intrinsic value and nothing more.

Long call: Buy the right to purchase the underlying at the strike. Profit when the underlying rises above strike + premium. Max loss: premium paid. Max gain: unlimited. Bullish, defined risk.

Short call: Sell the right for someone else to buy from you at the strike. Max gain: premium received only. Max loss: unlimited as the underlying rises — you are obligated to sell at the strike even if the market has moved far above it. Used to generate income when you expect the underlying to stay flat or fall.

Long put: Buy the right to sell the underlying at the strike. Profit when the underlying falls below strike − premium. Max loss: premium paid. Max gain: substantial (underlying can fall to zero, so max gain is approximately strike − premium). Bearish, defined risk.

Short put: Sell the right for someone else to sell to you at the strike. Max gain: premium received. Max loss: substantial if the underlying falls sharply — you are forced to buy at the strike. Used when you want to acquire the underlying at a lower effective price, or to generate income in a range-bound market.

Core principle: buyers always have defined maximum losses. Sellers have defined maximum gains but bear the risk of large losses.

Protective put: Long the underlying stock + long a put. The put acts as insurance — you retain upside if the stock rises, but the put limits losses if it falls below the strike. Cost: the put premium reduces your net return. Used by investors who want downside protection without selling their position.

Covered call: Long the underlying stock + short a call. You collect the call premium, enhancing income if the stock stays flat or rises modestly. The trade-off: if the stock rises sharply above the call strike, you forfeit that gain (your stock gets called away at the strike). Used to generate income from a position expected to be range-bound.

Collar: Long the underlying + long an OTM put + short an OTM call. The collar caps both upside and downside. The premium received from the short call offsets the cost of the long put — if strikes are chosen correctly, it is structured as a zero-cost collar where the two premiums cancel. Gives downside protection at no net premium cost, in exchange for capping upside above the call strike.

Collars are widely used by corporate executives with large concentrated equity positions who need protection but cannot sell the stock outright (for tax or regulatory reasons).

A barrier option is an option whose existence or payoff depends on whether the underlying reaches a specified barrier level during the option's life.

• Knock-in: The option only activates if the underlying hits the barrier. Before that, it does not exist. A down-and-in put activates only if the underlying falls to the barrier — cheaper than a standard put because activation is conditional on the adverse scenario you are hedging.
• Knock-out: The option exists from inception but is cancelled if the underlying reaches the barrier. A down-and-out call gives upside participation but expires worthless if the stock falls to the barrier — also cheaper than a standard call, reflecting reduced probability of full payoff.

Barrier options are cheaper than vanilla options because the payoff is conditional on barrier events — the seller bears less risk. They are widely used in structured products sold to retail and institutional clients, allowing manufacturers to offer attractive headline participation rates while managing cost through the barrier feature.

The key risk for buyers: the barrier can be breached in a volatile market precisely when the hedge is most needed, extinguishing the position at the worst possible moment.

Black-Scholes prices European options from five inputs:

• Spot price (S): Current price of the underlying. Higher spot increases call value, decreases put value.
• Strike price (K): The agreed exercise price. Higher strike decreases call value; increases put value.
• Time to expiry (T): More time means more uncertainty — both calls and puts worth more. At expiry, time value collapses to zero.
• Volatility (σ): The assumed standard deviation of underlying returns. Higher vol increases both calls and puts — it widens the distribution of possible outcomes. The only input that must be estimated rather than observed directly, which is why vol is where all the trading judgment sits.
• Cost of carry (r − q): Risk-free rate (r) minus dividend yield (q). Affects the forward price of the underlying. Dividends reduce the forward price below spot because they represent cash paid out before expiry.

Volatility is the crucial input. The other four are observable; vol is a judgment. This is why "trading options" is really "trading vol" — the question is always whether implied vol is cheap or expensive relative to what you expect to realise.

Four key assumptions fail in practice:

• Normal distribution: Black-Scholes assumes log-normally distributed returns. Real equity returns show fat tails and negative skew — large crashes occur far more often than the normal distribution predicts. This is why OTM puts trade at higher implied vol than the model suggests (the market prices crash risk explicitly).
• Continuous random walk with no jumps: Prices are assumed to move continuously. In reality, stocks gap on earnings, regulatory news, or macro shocks — discontinuous moves Black-Scholes cannot price correctly.
• Constant volatility: The model assumes vol is fixed. In practice, vol changes constantly and different strikes and maturities on the same underlying trade at different implied vols — the vol surface. If vol were constant, all options on the same underlying would trade at the same implied vol. They do not.
• No transaction costs, perfect liquidity: The model assumes frictionless continuous delta hedging. Actual rehedging involves transaction costs and market impact, especially for large positions or illiquid underlyings.

Despite all this, Black-Scholes remains the industry standard — not because it is accurate but because it is a shared quoting convention. Traders quote in implied vol using Black-Scholes, then apply their own adjustments to decide if that vol is rich or cheap.

Historical (realised) volatility is the standard deviation of past returns on the underlying — backward-looking, calculated from market data. It tells you how volatile the asset actually was.

Implied volatility is the vol assumption embedded in a current option's market price. You run Black-Scholes backwards: take the observed dollar premium and solve for the vol level that would produce that price. The result is the market's forward-looking consensus on expected future uncertainty over the option's life.

If implied vol is 30% and historical vol has been 20%, the market is paying a premium for expected future uncertainty — perhaps ahead of an earnings announcement or central bank decision. If implied vol is 18% and you expect the stock to move in line with history at 20%, options look cheap: you might buy straddles to go long vol.

On average, implied vol tends to sit 1–2 percentage points above subsequent realised vol — a persistent vol risk premium. Option sellers earn this premium for bearing uncertainty. This premium is why systematic option-selling strategies (selling OTM options, collecting premium) appear consistently profitable in calm markets — but suffer sharp losses when realised vol suddenly spikes above what was implied.

If Black-Scholes held perfectly, all options on the same underlying with the same expiry would trade at the same implied vol — there is only one vol input. In practice they do not. The implied vol varies across both strike and maturity: this three-dimensional relationship is the vol surface.

Across strikes at a given maturity, the pattern in equity markets is a downward-sloping skew: OTM puts (low strikes) trade at significantly higher implied vol than ATM options, which in turn trade at higher vol than OTM calls (high strikes). This is negative or left skew.

Why it exists:

• Crash demand: Institutional investors systematically buy OTM puts to hedge portfolio downside. This persistent buying bids up put implied vol relative to calls.
• Fat tails in realised returns: Equity market crashes are sharper and faster than rallies. The market prices this non-normal distribution into options, making OTM puts genuinely more valuable than Black-Scholes would suggest.
• Post-1987 repricing: Before the 1987 crash, the vol surface was flatter. After Black Monday — which Black-Scholes failed to price anywhere near correctly — markets permanently repriced downside options to reflect the real crash probability. The skew has been a structural feature ever since.

The risk reversal (implied vol of OTM call minus OTM put at the same delta) quantifies skew. A very negative risk reversal signals high demand for downside protection — rising fear.

The VIX (CBOE Volatility Index), introduced in 1993, measures the market's implied vol expectation for the S&P 500 over the next 30 days. It is calculated from the prices of a wide range of S&P 500 options across strikes — not just ATM — and is updated in real time throughout the trading day.

The VIX is negatively correlated with equity markets: when stocks fall, the VIX spikes as investors rush to buy protective puts. When markets are calm and rising, the VIX tends to decline gradually. Historical reference levels:

• 10–15: Very low — market complacency. Common in long, slow bull markets.
• 15–25: Normal range in moderately uncertain markets.
• 25–35: Elevated — meaningful market stress.
• 35+: Crisis territory. VIX reached ~80 during the COVID crash (March 2020) and ~80 during the 2008 financial crisis. The August 2024 yen carry unwind briefly spiked VIX to ~65 intraday before reversing.

Know the current VIX level before any interview. VIX futures and options are traded instruments — long VIX positions serve as portfolio insurance because they tend to gain value exactly when equity portfolios are losing. The VSTOXX is the European equivalent for the Euro Stoxx 50.

The Greeks are partial derivatives of the option pricing model — each measures how the option's value changes when one input changes, holding everything else constant. In practice, they are the daily working language of an equity derivatives trader: knowing your delta, gamma, vega, and theta exposures is how you manage the risk in your book.

• Delta (Δ): Change in option value for a $1 move in the underlying. Also the hedge ratio — how many shares you need to be long or short to offset the option's directional exposure.
• Gamma (Γ): Change in delta for a $1 move in the underlying. Measures how quickly your delta hedge needs to be rebalanced as the market moves. High gamma = delta changes fast = frequent rehedging.
• Theta (Θ): Change in option value per day as time passes, all else constant. Always negative for long options — you lose time value every day you hold.
• Vega (V): Change in option value for a 1% change in implied vol. Both calls and puts gain value when vol rises — buying options is being long vol.
• Rho (ρ): Change in option value for a 1% change in interest rates. Least important for short-dated options; matters for longer maturities.

Delta measures the change in option value for a $1 change in the underlying, all else constant. It ranges from 0 to 1 for calls and −1 to 0 for puts.

Practical interpretation: an ATM call has delta ≈ 0.5 — roughly a 50% chance of expiring in the money, and the option gains $0.50 for every $1 rise in the stock. A deep ITM call has delta approaching 1 — it behaves almost like owning the stock. A deep OTM call has delta near 0 — very unlikely to be exercised.

Delta-hedging means taking an offsetting position in the underlying to make the portfolio delta-neutral — insensitive to small moves in the underlying. If a trader sells a call with delta 0.6 on 1,000 shares, they buy 600 shares to hedge. As the stock moves and delta changes, the hedge must be continuously adjusted — this is dynamic hedging. If delta rises to 0.7, the trader must buy 100 more shares to remain neutral.

Delta-hedging eliminates directional exposure but leaves gamma, theta, and vega risk — which become the active sources of P&L. A delta-hedged long option position profits when large realised moves (gamma) or vol rises (vega) outpace the daily time decay cost (theta).

Gamma is the rate of change of delta — it tells you how quickly your delta hedge needs to be adjusted as the underlying moves. Graphically, it is the curvature (convexity) in the option price curve relative to the underlying.

Long options always have positive gamma: as the underlying rises, a long call's delta increases toward 1 (you become more sensitive to further upside). As the underlying falls, delta decreases toward 0 (limiting further losses). This is the convexity benefit of owning options — your position automatically becomes more bullish when the market moves your way and less exposed when it moves against you.

Gamma is highest when the option is at-the-money — spot is right at the strike, so small moves dramatically change the probability of finishing in or out of the money. Delta can swing sharply in this region. Gamma is also higher for shorter-dated options: near expiry, an ATM option's delta can jump from near 0 to near 1 on a tiny move, requiring large rapid rehedging.

The gamma-theta trade-off is the central tension in options: long gamma (positive convexity) comes at the cost of negative theta (time decay). You pay for convexity through daily erosion of time value. Short gamma earns theta daily but suffers losses when the market moves sharply. Every options position is a bet on this trade-off.

Theta measures the change in an option's value as one day elapses, with all other inputs held constant. It represents time decay — the daily erosion of time value as the option approaches expiry.

Theta is negative for long options: every day you hold a long call or put, it is worth slightly less — like paying daily rent for the right to hold the option. An option with theta of −$0.05 loses $0.05 of value per day regardless of where the stock moves.

Theta is positive for short options: the seller earns time decay daily. If the option expires OTM, the seller keeps the entire premium — this is why selling options in calm, range-bound markets appears attractive.

Theta accelerates near expiry: the decay is nonlinear, with the final 30 days before expiry seeing much faster erosion than the first 30 days of a long-dated option's life. This is why buyers of short-dated options face significant time pressure, and why options market makers pay close attention to option books as expiry approaches.

Theta and gamma are inversely related: the positions with highest positive gamma (long ATM options near expiry) also have the most negative theta. Convexity costs money every day.

Vega measures the change in an option's value for a 1 vol point (1%) increase in implied volatility. A vega of $0.20 means the option gains $0.20 per share for every 1-point rise in implied vol, and loses $0.20 for every 1-point fall.

Vega is positive for both calls and puts — rising implied vol benefits all long option positions because it widens the distribution of possible outcomes, increasing the probability of large favourable moves. Vega is highest for ATM options and longer-dated options — there is more time over which vol can affect the outcome.

Being "long vol" means owning options — you profit if implied vol rises. Being "short vol" means selling options — you profit if implied vol falls or stays flat. Most equity derivatives trading ultimately comes down to a vol view: is implied vol cheap (buy options) or expensive (sell options)?

Practical example: if a stock's ATM implied vol is 30% but it has only realised 20% vol historically, options look expensive. A vol seller would short options, delta-hedge daily, and hope that realised vol stays below 30% — collecting the premium as time decay. The risk: if the stock suddenly moves sharply, realised vol spikes above 30% and the short vol position loses money.

Rho measures the change in an option's value for a 1% change in the risk-free interest rate. Calls have positive rho (higher rates increase call value); puts have negative rho.

The intuition: higher interest rates increase the forward price of the underlying, because you earn more by holding cash and entering a forward than by buying the stock outright. A higher forward benefits call owners (their right to buy at the fixed strike becomes more valuable relative to the rising forward) and hurts put owners.

Rho is typically the least important Greek in practice — interest rates move far more slowly than spot, vol, or time, and their impact on short-dated options is small. For a 1-month option, rho is nearly irrelevant. For a 2-year option, it becomes more meaningful as rate changes compound over the option's life.

In the 2025–26 environment with active central bank easing cycles, rho matters somewhat more than it did in the near-zero rate era for longer-dated options — but it still ranks well below delta, gamma, theta, and vega in day-to-day importance for most equity derivatives books.

A vega of $0.20 means: for every 1 vol point (1%) increase in implied volatility, the option increases in value by $0.20 per share. For a 1 vol point decrease, it loses $0.20 per share.

Worked example: you hold a long call on 10,000 shares with vega of $0.20. Implied vol rises from 25% to 27% — a 2 vol point increase. Your option position gains:

10,000 shares × $0.20/vol point × 2 vol points = $4,000 gain

At the book level, aggregate vega exposure is expressed in dollars per vol point. A book with net vega of −$50,000/vol point loses $50,000 if implied vol rises by 1 point (because the book is net short options). Managing this aggregate vega — deciding how long or short vol you want to be — is central to running an equity derivatives book.

Vega is not constant: it changes as spot moves (vanna) and as vol moves (volga). These second-order effects matter for managing large or complex option books, particularly around skew and vol-of-vol exposure.

Memorise this table — it comes up directly in superday interviews and is the foundation for discussing any option position:

Key patterns to remember rather than rote-learning each cell:

• Long any option → positive gamma and vega, negative theta. You own convexity but pay time decay daily.
• Short any option → negative gamma and vega, positive theta. You earn time decay but bear jump risk.
• Delta is the only Greek that differs by option type: long call is positive delta (benefits from price rise), long put is negative delta (benefits from price fall).
• Gamma and vega always share the same sign. Theta always has the opposite sign to gamma. This is the gamma-theta trade-off made concrete.

A long straddle combines a long call and a long put on the same underlying, same strike (typically ATM), same expiry. You pay premium on both legs. The payoff: you profit if the underlying moves significantly in either direction. If it stays flat, both options decay and you lose the combined premium.

The straddle is a pure vol trade with no directional bias. In Greek terms: long gamma, long vega, short theta. You benefit from large realised moves (positive gamma), from rising implied vol (positive vega), and pay time decay daily (negative theta).

Ideal use case: before a known binary event where the direction is uncertain but the magnitude of the move could be large — earnings announcements, regulatory approvals, central bank decisions. If implied vol going into the event is low relative to the expected move, straddles offer cheap event-driven exposure.

A short straddle is the opposite: sell ATM call and put, collect combined premium, profit if the underlying stays range-bound. Short gamma, short vega, long theta — earns daily time decay but loses sharply if the market moves significantly in either direction.

A bull spread (call spread): buy a call at a lower strike, sell a call at a higher strike — same underlying, same expiry. You pay a net premium (the long call costs more than the short call generates). Profit profile: you benefit if the underlying rises, but gains are capped at the upper strike. Max gain: spread between strikes minus net premium. Max loss: net premium paid.

Use case: moderately bullish — you expect the stock to rise to a certain level but not far above. The sold upper call caps your upside but significantly reduces the entry cost versus buying a call outright. Capital-efficient for a directional view with a specific target.

A bear spread (put spread): buy a put at a higher strike, sell a put at a lower strike. Profits if the underlying falls, capped at the lower strike. Max loss: net premium paid. Use case: moderately bearish with defined risk and lower entry cost than buying a put outright.

Both spreads reduce option cost in exchange for capping maximum profit. They are most appropriate when you have a specific price target in mind rather than expecting an unlimited move.

A strangle is similar to a straddle — long call and long put on the same underlying and expiry — but the strikes are different. The call strike is set above current spot; the put strike is set below. Both options are OTM at initiation.

Because both options are OTM, the strangle costs less than a straddle. The trade-off: you need a larger move in the underlying before the position profits — you must recover the combined premium paid, and you start further from both strikes.

The strangle is still long gamma and vega, short theta — the same directional vol trade as the straddle, just cheaper entry and requiring a bigger move. Suitable when you expect high vol but want to reduce premium outlay.

A short strangle sells both OTM options: wider profitability range than a short straddle (you need a larger move to lose money), same unlimited risk profile if the market moves sharply beyond either strike. One of the most common systematic short-vol strategies — popular in calm markets but dangerous in stress.

A forward is a bilateral OTC agreement to buy or sell an asset at a fixed price on a specified future date. Both parties are obligated to transact at the agreed price regardless of where the market is at delivery — unlike an option, there is no optionality. The forward price is set using cost-of-carry: Forward = Spot × (1 + r − q)^T.

Key differences from futures:

• Standardisation: Futures are standardised contracts (fixed sizes, fixed quarterly expiry dates) traded on exchanges. Forwards are custom bilateral agreements — any size, any date.
• Clearing and counterparty risk: Futures are centrally cleared through a clearing house with margin requirements. Forwards carry bilateral counterparty risk — if your counterparty defaults before settlement, you may not receive the agreed price.
• Daily settlement: Futures mark to market daily — gains and losses are settled in cash every day through margin calls. Forwards settle only at maturity.
• Liquidity: Exchange-traded futures are more transparent and often more liquid for standardised products. Forwards offer flexibility for customised terms.

For equity derivatives: S&P 500 futures (CME), FTSE futures (ICE), and Euro Stoxx 50 futures are the primary instruments for hedging or gaining index exposure. Single-stock forwards are typically OTC.

Contango: Futures prices are higher than the current spot price — the futures curve slopes upward with maturity. The normal state for most markets where there is a positive cost of carry (financing the asset, storage costs for commodities). For equity index futures, contango occurs when the risk-free rate exceeds the dividend yield — currently typical for most indices given positive rates.

Backwardation: Futures prices are below spot. The curve slopes downward. Occurs when: there is strong immediate demand for the physical asset (commodity supply squeezes); the dividend yield exceeds the risk-free rate (for equity indices — uncommon but observed during the zero-rate era when some high-yield indices had futures trading below spot); or the market expects spot prices to fall.

Basis is the spot price minus the futures price at any given moment. As delivery approaches, the futures price must converge to spot — basis goes to zero at expiry, regardless of whether the market was in contango or backwardation. Basis risk is the risk that this convergence does not happen at the expected rate, which matters for hedgers using futures.

A convertible bond is a corporate bond that gives the holder the right to convert it into a predetermined number of shares — the conversion ratio. The current market value of those shares is the parity or conversion value.

The embedded option is a call option on the company's equity. The holder benefits if the stock rises above the conversion price — they can convert into shares worth more than the bond's face value. The call cannot be stripped out and traded separately; it is exercised only through conversion of the bond itself.

The convertible has two value floors:

• Bond floor: What the instrument is worth as a pure fixed-income security — present value of coupons plus par, ignoring the conversion option. The minimum value assuming the company remains solvent.
• Parity value: Conversion ratio × current share price. If shares are worth more than par, this becomes the binding floor.

In practice, the CB trades at a premium to both floors, reflecting the option value. Typically around 75% of a convertible bond's value is bond value and 25% is the embedded call option. Issuers benefit: they pay a lower coupon than on a straight bond (investors accept less because of the option). Investors benefit: downside protection via the bond floor, upside via the equity option.

The fundamental difference comes down to asymmetry and defined risk. A forward or future has symmetric risk: agree to buy oil at $80, and if it trades at $60 at delivery you are obligated to buy at $80 and lose $20. Your downside is open-ended depending on how far the market moves against you.

An option gives you the right but not the obligation. If the market moves against you, you simply do not exercise. Maximum loss is always the premium paid — defined and known from the outset. This is what makes options attractive for hedgers who want protection without unlimited liability exposure.

In Greek terms, options provide convexity (positive gamma) that forwards lack. A long option position benefits more than linearly from large favourable moves — as spot moves your way, delta increases toward 1 and you participate fully. Against you, delta decreases toward 0 and losses are capped at premium. A forward has linear P&L with no convexity.

The honest comparison: forwards are simpler and more capital-efficient for pure hedging where you just need to lock in a rate and do not care about retaining optionality. Options are superior when you want defined risk, convexity, or the ability to participate if the market moves in your favour. You pay for that asymmetry through the premium and through theta decay.

Skew refers to the difference in implied vol between options with the same expiry but different strikes. In equity markets, OTM puts consistently trade at higher implied vol than OTM calls — creating a downward-sloping implied vol curve from low to high strikes. This is negative (left) skew.

The structural reasons: institutional investors systematically buy OTM puts for portfolio protection, bidding up their implied vol. Equity returns exhibit genuine fat left tails — crashes are sharper than rallies — so the market correctly prices higher probabilities of large downside moves than Black-Scholes assumes. After 1987, this crash premium became a permanent structural feature.

The risk reversal (implied vol of OTM call minus implied vol of OTM put at the same delta distance) is the standard market measure of skew. A risk reversal of −3 means the OTM put trades 3 vol points richer than the equivalent OTM call. More negative risk reversals signal rising demand for downside protection — rising fear. Watching the risk reversal is a cleaner gauge of market anxiety than spot prices alone, because it reflects what investors are actually paying for insurance.

Implied vol and equity prices are persistently negatively correlated: when equity markets fall sharply, implied vol rises — often dramatically. When markets grind higher, implied vol tends to decline slowly. This asymmetry is sometimes described as "vol spikes down, grinds up."

The mechanisms driving this: when markets sell off, investors rush to buy protective puts, bidding up premiums and therefore implied vol. Liquidity providers widen bid-ask spreads in stress, reducing the supply of cheap options and pushing vol higher. The leverage effect also operates: falling equity prices increase corporate financial leverage (debt/equity ratios rise), genuinely increasing future uncertainty and therefore warranting higher vol.

The magnitude can be substantial: a 5% S&P decline might push VIX from 15 to 22. A 20% decline in a genuine crisis can push VIX above 50. Flash crashes — like August 2024 — see extreme intraday VIX spikes that partially reverse as markets stabilise.

This negative correlation is why long vol positions (long straddles, long VIX futures) serve as portfolio hedges: they gain value precisely when equity portfolios are losing value, providing the negative correlation benefit that bonds used to provide more reliably before the rate regime shifted.

Delta-hedging as a trading strategy means buying options and then continuously hedging the delta exposure by trading the underlying — aiming to isolate the pure vol exposure (vega and gamma) while eliminating the directional bet.

The P&L generated from a delta-hedged long option position depends on the difference between the implied vol you paid when buying the option and the realised vol that actually occurs over the option's life:

• If realised vol exceeds implied vol you paid: you profit. Each delta-hedge rebalance captures a small gamma scalp — you buy low and sell high as the market oscillates, and the cumulative P&L from these rebalances exceeds the theta you paid.
• If realised vol is below implied vol: you lose money. Theta costs you daily, but the gamma scalps from hedging are insufficient to offset because the market is not moving enough to generate large rebalancing gains.

This is why the fundamental options question is always: is implied vol cheap or expensive relative to the vol you expect to realise? Delta-hedged options trading is, at its core, a bet on the difference between implied and realised vol — nothing more, nothing less.

Both instruments allow investors to trade volatility directly as an asset class, without the need for continuous delta-hedging.

A vol swap is a forward contract where the payoff at expiry depends on the difference between a fixed vol level (the strike, agreed at initiation) and the realised vol over the life of the contract. If realised vol exceeds the fixed strike, the buyer profits; if realised vol is lower, the seller profits. Vol swaps allow pure vol exposure without the complexity of managing delta hedges.

A variance swap is similar but the payoff depends on the difference between realised variance (vol squared) and a fixed variance strike, rather than vol itself. The distinction matters because variance is convex in vol — owning a variance swap has more positive convexity than a vol swap. When vol spikes unexpectedly, a variance swap benefits more than a vol swap on the same notional, because the payoff grows with vol squared rather than linearly.

Variance swaps are more liquid and more common in institutional markets because they can be replicated more cleanly using a portfolio of options across all strikes (the replication is exact in theory, unlike vol swaps). Long dated variance swaps have historically been one of the highest-correlation hedges with equity market selloffs — they spike sharply in crises.

A forward-starting option has both its start date and expiry date set in the future. The option does not begin its life immediately — it starts at a specified future date, with the strike typically set ATM at that future start date rather than at inception.

The key attraction: a forward-starting option gives you vega exposure without gamma exposure for the period between now and the option's start date. If you want to buy vol exposure to a period starting in 3 months and lasting 3 months, a forward-starting option lets you buy exactly that vol — without paying for short-term gamma between now and the start date, and without the theta decay of that gamma period.

This makes forward-starting options useful for:

• Isolating vol exposure to a specific future time window (e.g., around a known event 3 months away).
• Building calendar trades — simultaneously buying forward-starting vol for a future period while selling current vol, expressing a view that future vol will be higher than current implied vol for that period.
• Structuring products that reference future vol levels rather than current ones.

The no-theta-payment benefit during the stub period is the core commercial attraction — you get the vol exposure you want without paying for interim time decay.