FINC3017 Chap.11 Volatility as an Asset, Options & Variance Swaps
Volatility as an Asset, Options & Variance Swaps
Volatility as an Asset, Options & Variance Swaps (Week 11) treats volatility itself as something you can trade. Volatility is mean-reverting, carries a negative long-run return (it behaves like insurance), and is strongly negatively correlated with equities.
Implied volatility lives under the risk-neutral (Q) measure while realised volatility is physical (P), and the gap — the volatility risk premium — is positive on average and accrues to the seller. You meet the VIX as model-free implied volatility, the Black-Scholes Greeks (especially Vega), strategies like the straddle, and the variance swap whose fair strike equals the risk-neutral expected variance.
What this chapter covers
- 01
Volatility as a traded asset: mean-reverting, negative long-run return, negatively correlated with equities
- 02
P (physical) vs Q (risk-neutral) measure; implied vol is a Q quantity
- 03
Volatility risk premium VRP = σ^Q − σ^P > 0 on average, earned by the seller
- 04
VIX / A-VIX as model-free 30-day implied vol; daily move ≈ VIX/√252
- 05
Black-Scholes pricing and implied volatility; the volatility skew
- 06
The Greeks: Delta, Vega (primary vol exposure), Theta, Gamma
- 07
Strategies: the straddle (long vol, Vega > 0, Delta ≈ 0) and butterfly (short vol)
- 08
Variance swap: fair strike σ_K² = E^Q[σ_R²], replicated by a strip of options (the VIX method)
Variance-swap strike, VIX daily move, and who profits
- 2 marks(a) The fair variance-swap strike is the squared volatility: σ_K² = 0.20² = 0.04 (in variance units).
- 2 marks(b) The expected daily move is the annual vol scaled by trading days: 20%/√252 = 20/15.875 ≈ 1.26% per day.
- 2 marks(c) Compare realised variance to the strike. Realised vol 24% gives realised variance 0.24² = 0.0576, which exceeds the strike 0.04.
- 1 mark(c cont.) The variance-swap buyer receives (realised variance − strike), which is positive here, so the BUYER (long variance) profits because realised volatility came in above the implied level.
Key terms
- Volatility risk premium (VRP)
- The systematic gap between option-implied (risk-neutral, Q) volatility and subsequently realised (physical, P) volatility, VRP = σ^Q − σ^P, which is positive on average. It is the compensation the seller of volatility (insurance) earns for bearing the risk of volatility spikes.
- VIX
- A model-free index of 30-day implied volatility on the equity market, computed from a strip of out-of-the-money option prices. Quoted as an annualised percentage, it spikes in market stress and is converted to a one-day move by dividing by √252.
- Vega
- The sensitivity of an option's price to a change in volatility, ∂C/∂σ — the Greek that measures direct volatility exposure. A long straddle has positive Vega, profiting when implied volatility rises, which is why it is the natural way to trade volatility itself.
- Straddle
- A long call plus a long put at the same strike, giving a roughly delta-neutral, positive-Vega position with negative Theta. It profits from a large move in either direction (or a rise in implied vol) and loses if the underlying stays still and time decays the options.
- Variance swap
- A forward contract on realised variance whose payoff is (realised variance − fair strike) × notional. Its fair strike equals the risk-neutral expected variance σ_K² = E^Q[σ_R²] and is replicated by a 1/K²-weighted strip of options — the basis of the VIX calculation.
Volatility as an Asset, Options & Variance Swaps FAQ
Why does volatility behave like insurance?
Volatility tends to spike exactly when markets fall, so a long-volatility position pays off in crashes — like an insurance policy. Investors will pay up for that protection, which is why implied volatility sits above realised volatility on average (the volatility risk premium) and why being long volatility has a negative expected long-run return: you are paying the insurance premium.
What is the difference between the P and Q measures here?
The physical measure P describes the volatility that actually occurs in the data (realised volatility), while the risk-neutral measure Q is embedded in option prices (implied volatility). Because investors are risk-averse and fear volatility spikes, the Q-implied volatility is systematically higher than the P-realised volatility, and that wedge is the volatility risk premium.
How do I convert a VIX level to an expected daily move?
Divide the annualised VIX by the square root of the number of trading days in a year, about 252. So a VIX of 20% implies a daily move of roughly 20%/√252 ≈ 1.26%. This works because volatility scales with the square root of time, the same √n rule used to annualise risk earlier in the course.
Exam move
Keep the variance-versus-volatility distinction front of mind — always square before comparing in a variance swap — and memorise the VIX/√252 daily-move rule. Understand the volatility-as-insurance story (implied > realised, seller earns the VRP) and what each Greek measures, since Vega and the straddle are recurring conceptual targets in this distinctive part of the course.
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