Arbitrage Bounds for Prices of Weighted Variance Swaps

TL;DR

This paper derives model-independent no-arbitrage bounds for weighted variance swaps using a pathwise stochastic calculus approach, providing explicit bounds and hedging strategies based solely on observed option prices.

Contribution

It introduces a novel, probability-free framework for pricing weighted variance swaps and derives explicit no-arbitrage bounds using semi-infinite linear programming techniques.

Findings

Market quotes are close to the model-free lower bounds.

Explicit upper bounds are derived.

Hedging strategies are constructed within a model-independent setup.

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.