2010/01/15 by Mark H. Davis, Mark H. A. Davis, Davis, Mark H. A. +5 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · #Economic theories and models #FOS: Economics and business #Pricing of Securities (q-fin.PR) #Risk and Portfolio Optimization #Stochastic processes and financial applications #q-fin.PR
paper · pdf · doi:10.48550/arxiv.1001.2678
25 pages, 4 figures
openalex publication_date 2010/01/15 · arxiv created 2012/09/18 · arxiv updated 2012/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Föllmer'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.