Replicating Variance Swaps with a Strip of Vanilla Options
Summary
The document explains the intuition behind representing a variance swap hedge as a weighted portfolio of vanilla options across strikes. The integral represents a continuous strip: at each strike, the portfolio holds an amount determined by a weighting function. In the theoretical construction, the weight is proportional to the inverse square of strike, producing the familiar one-over-strike-squared rule.
The response clarifies that this continuous portfolio is an idealization. A practical hedge uses a finite set of options over a truncated strike range, approximating the integral with a weighted sum. The portfolio is designed to retain sensitivity to volatility while reducing sensitivity to movements in the underlying asset, which is the purpose of seeking nonzero, constant vega as the asset price changes. The explanation gives conceptual guidance rather than implementation details; actual hedges must account for discrete strikes and the limits of the available option market.
Key ideas
- The integral describes a continuous strip of options, with a weight assigned at every strike.
- A finite set of options over a limited strike range can approximate the theoretical portfolio.
- The inverse-square strike weighting is associated with the variance swap replication construction.
- Reducing dependence on the underlying price while retaining volatility exposure motivates the hedge.
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# Hedge variance swapping by vanilla option(constant vega portfolio against underlying asset)
# Hedge variance swapping by vanilla option(constant vega portfolio against underlying asset)
One book said `hedging variance swaps` $$I= \sqrt{\dfrac{1}{t}\int^t_0\sigma^2(S,t)}d t$$ by `vanilla option`，say value $V(S,E;\sigma)$(`Black-Scholes` fomula) where $S$ is `underlying asset`, $E$ is the `strike price`. Then he constructed a `portfolio` with value $$P=\int^{\infty}_0f(E)V(S,E;\sigma)d E$$ then compute the `vega` of portfolio: $\textrm{Vega}_P= \dfrac{\partial P}{\partial\sigma},$ then let $$\dfrac{\partial \textrm{Vega}_P }{\partial S}=0$$ obtain $f(E) = \dfrac{k}{E^2}.$
His conclusion is `variance swaps can be hedged with vanilla option, using the 'one over strike squared' rule.` I can not understand:
1.What's the meaning of the representation of $P$(why take the integral) i.e how do we implement this portfolio by vanilla option, hold $F(E)$ share?
2.Why we need constant vega against $S$ i.e how to hedge?
## Answer by Quantuple (score 2)
https://quant.stackexchange.com/a/32869
- An integral is used because the portfolio contains an infinite number of instruments. More specifically, the idea is to hold a continuous strip of $f(E)$ units of European vanilla options struck at $E$ for each $E \in [0,\infty[$. Of course this remains a theoretical concept: to build a similar portfolio in practice, one must consider a partition of the truncated strike domain (hence a finite number of options): $$ P = \sum_{i=1}^N f(E_i) V(S,E_i;\sigma) $$
- You need a constant, non-zero, Vega against $S$ because you would like a product which is sensitive to volatility and independent of the path the asset will take, i.e. a pure volatility bet.
Have a look at this well-known deck by JP Morgan here. It may help you better understand the practical implications.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.