Key takeaways
– A variance swap is a forward-style derivative that pays the difference between realized variance of an underlying asset and a pre‑agreed variance strike, multiplied by a notional amount.
– It is a pure play on volatility magnitude (variance) rather than direction; payouts scale with the square of price moves, so large jumps have outsized effects.
– Variance swaps can be replicated (and priced) using a static strip of European options across strikes; the fair strike is linked to the market price of those options.
– Main users: volatility speculators, portfolio hedgers, and volatility market makers/arbitrageurs.
– Key risks: jump risk, model/replication risk, counterparty and margin risk, sampling/annualization conventions.
1. What a variance swap is (plain explanation)
A variance swap is a bilateral contract that lets one party buy or sell the realized variance of an underlying (index, FX rate, interest rate, etc.) over a specified observation period. At maturity the buyer receives
payoff = Variance Notional × (Realized Variance − Variance Strike)
and the seller pays the opposite. The variance strike is set at inception so that the contract has zero value initially (i.e., it reflects the market’s expectation of future variance).
Because the payout depends on variance (the square of returns), the contract emphasizes large moves more than a contract that settles on standard deviation (volatility).
2. How realized variance is typically calculated
Realized variance is usually computed from sampled log returns r_t over N observation points and annualized. A common formula:
Realized Variance = (A / N) × Σ_{t=1..N} r_t^2
where r_t = ln(S_t / S_{t-1}), A is the annualization factor (e.g., 252 trading days per year), and N is the number of returns observed. Contracts specify sampling frequency, holidays, and treatment of stale data.
3. Payoff formula and units
– Payoff (cash settled) = Vn × (RV − Kvar)
where Vn = variance notional (currency per variance point), RV = realized variance (in decimal terms, e.g., 0.04 for 20%^2), and Kvar = agreed variance strike.
– If the buyer of variance is long and realized variance > strike, the buyer receives cash; if realized variance < strike, the buyer pays.
4. Why variance swaps differ from volatility swaps and options
– Volatility swap pays based on realized volatility (square root of variance). Variance swaps pay on variance directly. Because of convexity, variance swap payouts tend to be larger for the same level of realized volatility.
– Options provide exposure to volatility but carry directional (delta) exposure and depend on option expiries and strike structure. A pure variance exposure via options requires a strip of options across strikes plus dynamic hedging; variance swaps provide direct exposure.
5. Pricing and static replication (practical outline)
The fair variance strike can be replicated (and thus priced) by a static continuum of OTM puts and calls across strikes plus a position in the underlying; in continuous form:
Fair Variance = (2 / T) × [ ∫_0^F (P(K)/K^2) dK + ∫_F^∞ (C(K)/K^2) dK ]
where P(K), C(K) are put and call prices at strike K, F is forward price, and T is time to maturity. In practice, this formula is discretized using available option strikes. Key implications:
– Option market prices embed the market’s expectation of future variance.
– Liquidity of strikes and discrete sampling introduce approximation and model risk.
6. Example (numerical)
This analysis assumes that…
– Variance notional Vn = $1,000,000 per variance point,
– Strike Kvar = 0.04 (i.e., implied 20% annual volatility squared),
– Realized variance RV over the term = 0.0625 (i.e., 25% annual volatility squared).
Payoff to long variance = 1,000,000 × (0.0625 − 0.04) = 1,000,000 × 0.0225 = $22,500.
(If RV < Kvar, the long would pay the seller an analogous amount.)
7. Practical steps to trade or hedge with a variance swap
These steps are intended for institutional traders or sophisticated investors who understand OTC derivatives and counterparty processes.
Pre‑trade
1. Define objective and horizon: Are you speculating on an increase/decrease in volatility, hedging a portfolio’s volatility exposure, or arbitraging variance relative to implied volatility?
2. Quantify exposure: determine the desired variance notional and appropriate tenor to match your risk horizon or hedge period.
3. Market scan and liquidity check: obtain quotes for variance strikes and check option liquidity if replication is needed.
Structuring and execution
4. Agree contract specs: sampling frequency, annualization factor (A), observation start/end, holiday/stale data rules, and margin/margining triggers.
5. Negotiate notional and strike (or accept market‑quoted fair strike): the strike is usually set so that initial NPV = 0; you can trade at premiums/discounts if desired.
Hedging and risk management
6. Replication / hedging strategy: dealers typically hedge variance exposure by trading a strip of options and dynamically delta-hedging; understand the static replication used and ensure you can monitor replication P&L.
7. Monitor exposures and margin: track realized variance accumulation, mark-to-market, and margin calls. Large jumps can create sudden mark moves.
8. Closeout and settlement: at maturity the contract settles in cash based on the agreed realized variance convention.
8. Common use cases
– Hedging: an equity portfolio manager worried about an increase in realized volatility can buy variance (to receive cash if variance spikes).
– Speculation: a trader who expects realized volatility to rise above what is implied by the market buys variance.
– Relative value/arbitrage: exploiting differences between observed realized variance, implied variance from options, and forecasting models.
9. Risks and practical considerations
– Jump risk / tail risk: variance squares returns, so rare large moves have outsized impact.
– Replication/market‑model risk: discrete strikes, bid‑ask spreads, and limited option liquidity cause imperfect replication and pricing errors.
– Counterparty and credit risk: many variance swaps are OTC; collateral and credit terms matter.
– Margin and funding: some contracts require initial and variation margin; adverse moves can force cash calls.
– Sampling and settlement conventions: different contracts use different annualization factors and sampling frequencies; mismatch can cause basis risk.
– Regulatory and accounting/tax treatment: OTC derivative reporting, capital charges, and tax rules vary by jurisdiction.
10. Where to learn more (key references)
– Demeterfi, Derman, Kamal, and Zou (1999), “More Than You Ever Wanted To Know About Volatility Swaps” — classic quantitative exposition on pricing and replication.
– Carr & Madan, and Neuberger — foundational work on static replication of variance and volatility derivatives.
– Investopedia overview of variance swaps — good non‑technical introduction (source provided by user).
Closing summary
Variance swaps are powerful instruments to obtain a direct, traded exposure to the magnitude of price moves. They give a clean, single-number exposure to variance, but that simplicity brings concentrated tail sensitivity and operational/replication complexity. For anyone considering these instruments: define objectives and tenor carefully, understand the realized variance definition in the contract, check option market liquidity (if using replication), and be prepared for margin and large P&L swings from jumps.
Editor’s note: The following topics are reserved for upcoming updates and will be expanded with detailed examples and datasets.