Payoff terms of the form g(x) = 1 - x² are found in a typical contract called a variance swap.

Judge against the standard definition of a variance swap. The payoff at maturity is notional × (realised variance − strike variance), and realised variance is a sum of squared returns. TRUE: the payoff of one side of a typical variance swap is of the form constant − (squared variable), so that g(x)=1−x² matches it up to normalisation (unit strike, scaling). For example, the short-variance side receives strike minus realised variance. MOSTLY TRUE: the form matches only after a modest reinterpretation of x, scaling or sign that the context implies. MOSTLY FALSE: variance swaps pay on squared returns, but not in a form that reasonably reduces to 1−x². FALSE: variance swap payoffs are not quadratic in the underlying variable.

Set before the research began.

1. E1 Quantitative Strategies Goldman, Sachs & Co. (Quantitative Strategies Research Notes; authors Demeterfi, Derman, Kamal, Zou) · March 1999

   “the life of the contract, Kvar is the delivery price for variance, and N is the notional amount of the swap in dollars per annualized volatility”

2. E2 Quantitative Strategies Goldman, Sachs & Co. (Quantitative Strategies Research Notes; authors Demeterfi, Derman, Kamal, Zou) · March 1999

   “This is a good approximation to the variance of daily returns used in the contract terms of most variance swaps.”

3. E3 Pricing and Hedging Volatility Derivatives Mark Broadie (Columbia University) and Ashish Jain (Lehman Brothers), working paper · January 2008

   “The holder of a variance swap at expiration receives N dollars for every unit by which the stock’s realized variance Vd(0, n, T ) exceeds the variance strike K.”

4. E4 Variance swap - Wikipedia Wikipedia · 2 March 2005

   “Thus the net payoff to the counterparties will be the difference between these two and will be settled in cash at the expiration of the deal”

E3 gives the variance swap payoff as notional times the amount by which realised variance exceeds the variance strike, and E4 describes the net payoff as a difference settled in cash at expiry. E2 notes that the contract terms use the variance of daily returns, meaning a sum of squared returns. So the short-variance side receives the strike minus realised variance, a constant minus a squared quantity. With unit strike and scaling this matches g(x)=1−x², the TRUE case in the criteria.