Abstract
A variance swap is a derivative with a path-dependent payoff which allows investors to take positions on the future variability of an asset. In the idealised setting of a continuously monitored variance swap written on an asset with continuous paths, it is well known that the variance swap payoff can be replicated exactly using a portfolio of puts and calls and a dynamic position in the asset. This fact forms the basis of the VIX contract.
But what if we are in the more realistic setting where the contract is based on discrete monitoring, and the underlying asset may have jumps? We show that it is possible to derive model-independent, no-arbitrage bounds on the price of the variance swap, and corresponding sub- and super-replicating strategies. Further, we characterise the optimal bounds. The form of the hedges depends crucially on the kernel used to define the variance swap.
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Notes
Throughout this paper, all option contracts have maturity T unless stated otherwise.
This means that we do not need to introduce a notation for the put price, which is convenient since P is already in use for the partition. Put-call parity for the forward says that the price of a put with strike x is the price of a call with the same strike minus f(0)−x.
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Hobson, D., Klimmek, M. Model-independent hedging strategies for variance swaps. Finance Stoch 16, 611–649 (2012). https://doi.org/10.1007/s00780-012-0190-3
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DOI: https://doi.org/10.1007/s00780-012-0190-3