Replicating Variance and Volatility Swaps with Options
Summary
The document contrasts replication approaches for variance swaps and volatility swaps. A variance swap can be replicated theoretically with options across strikes at a single maturity, using holdings weighted inversely to the square of strike, alongside a dynamic position in the underlying. A volatility swap is more difficult: its square-root payoff requires dynamic trading in options, which can make the hedge costly to maintain.
The discussion connects variance replication to the log-contract literature and notes that a related method underlies the CBOE’s VIX calculation. It also cautions that a continuum of options is not feasible in practice, and mentions an approximate dynamic variance-swap hedge using three delta-hedged options. That alternative is presented as a separate contribution, not as a general replacement. The material is a concise literature overview; it does not derive the replication formulas or quantify transaction costs and hedge performance.
Key ideas
- A variance swap can be replicated theoretically with a static strip of options and dynamic trading in the underlying.
- The option holdings in the variance replication are weighted inversely to the square of strike.
- The square-root payoff of a volatility swap makes replication require dynamic option trading.
- The theoretical continuum of options can be impractical because of transaction costs and market constraints.
- The document mentions an approximate three-option dynamic hedge for variance swaps, without detailing its performance.
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# Static and Dynamic Hedging of Vol/Var Swaps # Static and Dynamic Hedging of Vol/Var Swaps Why can a variance swap be perfectly statically hedged whereas a volatility swap requires dynamic hedging? Possible reference request to the corresponding literature. ## Answer by Alex C (score 4, accepted) https://quant.stackexchange.com/a/25470 There has been a lot of work in recent years on the pricing and hedging of volatility derivatives, leading to some non-obvious, even startling results. It is summarized in Mark Joshi's book More Mathematical Finance among other places. It all started with the work of Anthony Neuberger on the Log Contract in 1994, which seemed to be a theoretical result about a non-existent contract. It led to a solution for Var swaps, the famous Derman paper More Than You Ever Wanted To Know About Volatility Swaps 1999 1; see also Bossu's Just What you Need to Know about Variance Swaps 2 for a simpler treatment. Then Peter Carr and Roger Lee wrote Robust Replication of Volatility Derivatives 3in 2009 which addressed Vol Swaps. To summarize: A Variance swap can be replicated by a static position in options plus a dynamic position in the underlying. This is a beautiful and very practical result. You have to hold options of all strikes at the given maturity, with holdings inversely proportional to the square of the strike. A Volatility swap can be replicated with a dynamic position in options. This is not very practical as the transaction costs for continuously buying and selling a large number of options will eat you alive. As a result Vol Swaps are not much traded and Var Swaps are preferred. Finally, the new method for pricing variance swaps was adopted by the CBOE in 2004 as a way of calculating VIX values. Essentially they compute VIX^2 by this method and then publish the square root of this at frequent intraday intervals. So whenever people look at the VIX they are implicitly relying in this method. ## Answer by Frido (score 2) https://quant.stackexchange.com/a/76451 Not sure if still relevant for the OP, but as I lack modesty I'd like to say that it is possible, to a good approximation, to hedge varswaps dynamically using 3 delta-hedged options only. This is something I found out relatively recently, and explained in the following note: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4542475 The static portfolio of a continuum of options (and a dynamic position in the underlying) for varswaps replication is beautiful theoretically, but not feasible in practice. Volswaps is a different matter, the square root really messes things up, and I am not sure there can be an 'easy' hedge for the volswap. As the volswap can be regarded as derivative on the terminal value of a varswap, it can be written as a static strip of options on realized variance (a la Carr-Madan), but not as a static strip of options on the underlying asset.
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