Tim Olson
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69 lines
4.1 KiB
Markdown
69 lines
4.1 KiB
Markdown
---
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description: "Variance swaps are derivative contracts whose payoff is proportional to the difference between realized variance and a preset variance strike, allowing pure volatility bets without the need for continuous Delta-hedging."
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tags: [volatility, variance-swap, realized-variance, delta-neutral, derivatives]
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---
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# Volatility Trading with Variance Swaps
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**Section**: 7.6 | **Asset Class**: Volatility | **Type**: Volatility / Derivatives
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## Overview
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Variance swaps allow traders to bet on future realized variance without the operational burden of continuous Delta-hedging required by options-based volatility strategies. The payoff at maturity is proportional to the difference between the realized variance of the underlying and a preset variance strike K, scaled by a variance notional N.
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## Construction / Mechanics
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**Payoff at maturity T** (for the long side):
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```
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P(T) = N × (v(T) - K) (434)
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```
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**Realized variance** over the period:
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```
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v(T) = (F/T) Σ_{t=1}^T R(t)² (435)
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```
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**Log-return** at time t:
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```
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R(t) = ln[S(t) / S(t-1)] (436)
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```
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where:
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- t = 0, 1, ..., T: sample points (e.g., trading days)
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- S(t): underlying price at time t
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- F: annualization factor (F = 252 for daily data)
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- N: variance notional (preset, in dollar terms)
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- K: variance strike (preset at contract inception; determines the fair value entry price)
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- Note: the mean return is NOT subtracted in v(T), so the denominator is T (not T-1)
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**Long variance swap**: benefits when realized variance v(T) > K (realized vol was higher than expected).
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**Short variance swap**: benefits when realized variance v(T) < K (realized vol was lower than expected).
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**Fair value of K**: at inception, K is set such that the swap has zero initial value; it equals the market's expected future variance (often proxied by the square of the VIX for equity index variance swaps).
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## Payoff / Return Profile
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- **Long variance swap** profits when realized variance exceeds the variance strike K: v(T) > K.
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- Typical use: hedge against a volatility spike; gains are convex in the volatility move (since variance is the square of vol).
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- **Short variance swap** profits when realized variance is below K: v(T) < K.
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- Typical use: harvest the volatility risk premium (implied vol > realized vol on average); earns N × (K - v(T)) in normal markets.
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- The payoff is in units of variance (not volatility); a move from 10% to 20% vol generates 4× the payoff of a move from 10% to 15% vol.
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## Key Parameters / Signals
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- K: variance strike (the break-even point)
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- N: variance notional (dollar value per unit of variance)
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- v(T): realized variance (the key realized outcome)
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- F: annualization factor (252 for daily, 52 for weekly, etc.)
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- T: number of observation periods
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- v(T) - K: the net P&L per unit of notional
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## Variations
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- **Volatility swap**: payoff based on sqrt(v(T)) - K_vol (realized volatility minus vol strike); less convex than variance swap; harder to replicate.
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- **Conditional variance swap**: accumulates variance only on days when the underlying is within a specified range.
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- **Corridor variance swap**: similar, accumulates variance only when the underlying is above (or below) a barrier.
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## Notes
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- The key advantage over options-based volatility strategies: no Delta-hedging required, so the trader takes pure variance exposure with no directional risk (assuming no P&L drift from Delta).
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- The convexity of the variance payoff (variance = vol²) means long variance swaps gain more from large vol spikes than short options positions gain from vol increases.
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- Short variance swaps capture the volatility risk premium (K > expected realized variance) but have unlimited downside: if realized variance is very high (e.g., a market crash), N × (v(T) - K) can be very large.
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- Variance swaps on equity indexes are liquid OTC instruments; single-stock variance swaps are less common.
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- The mean is not subtracted in v(T) (Eq. 435), which is standard market convention; subtracting the mean would change the denominator to T-1.
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