Arbitrage bounds for prices of weighted variance swaps

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Title: Arbitrage bounds for prices of weighted variance swaps
Author(s): Davis, MHA
Obloj, J
Raval, V
Item Type: Journal Article
Abstract: We develop robust pricing and hedging of a weighted variance swap when market prices for a finite number of co--maturing put options are given. We assume the given prices do not admit arbitrage and deduce no-arbitrage bounds on the weighted variance swap along with super- and sub- replicating strategies which enforce them. We find that market quotes for variance swaps are surprisingly close to the model-free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi-infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model-independent and probability-free setup. In particular we use and extend F\"ollmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the 'log contract' and similar connections for weighted variance swaps. Our results take form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk-neutral expectations of discounted payoffs.
Publication Date: 7-Feb-2013
Date of Acceptance: 1-Sep-2012
URI: http://hdl.handle.net/10044/1/42520
DOI: https://dx.doi.org/10.1111/mafi.12021
ISSN: 0960-1627
Publisher: Wiley
Start Page: 821
End Page: 854
Journal / Book Title: Mathematical Finance
Volume: 24
Issue: 4
Copyright Statement: © 2013 Wiley Periodicals, Inc. This is the accepted version of the following article: Davis, M., Obłój, J. and Raval, V. (2014), ARBITRAGE BOUNDS FOR PRICES OF WEIGHTED VARIANCE SWAPS. Mathematical Finance, 24: 821–854., which has been published in final form at https://dx.doi.org/10.1111/mafi.12021
Keywords: Social Sciences
Science & Technology
Physical Sciences
Business, Finance
Economics
Mathematics, Interdisciplinary Applications
Social Sciences, Mathematical Methods
Business & Economics
Mathematics
Mathematical Methods In Social Sciences
weighted variance swap
weak arbitrage
arbitrage conditions
model-independent bounds
pathwise Ito calculus
semi-infinite linear programming
fundamental theorem of asset pricing
model error
PROBABILITIES
OPTIONS
Finance
0102 Applied Mathematics
1502 Banking, Finance And Investment
Publication Status: Published
Appears in Collections:Financial Mathematics
Mathematics
Faculty of Natural Sciences



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