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A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
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A note on contracts on quadratic variation.

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This study shows how option portfolios can price futures under stochastic volatility models, generalizing variance swap pricing. It also demonstrates designing new variance contracts using local volatility models and Dupire

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Area of Science:

  • Quantitative Finance
  • Financial Derivatives
  • Stochastic Modeling

Background:

  • The classical Black stochastic volatility model is foundational in derivatives pricing.
  • Variance swaps are key financial instruments for hedging and speculating on future volatility.
  • Existing models often lack flexibility in pricing complex volatility-dependent derivatives.

Purpose of the Study:

  • To generalize the representation of derivative prices using option portfolios.
  • To extend the classical variance swap representation to a broader class of payoffs.
  • To explore the design of novel variance-related financial contracts.

Main Methods:

  • Utilizing a Black stochastic volatility model for a future F.
  • Applying a general function g to represent the derivative price.
  • Leveraging Dupire's formula within a local volatility framework.

Main Results:

  • The price of a specific future payoff can be represented by portfolios of put and call options.
  • This representation generalizes the established result for variance swaps.
  • An example demonstrates the design of variance contracts with specific characteristics.

Conclusions:

  • The findings provide a powerful tool for pricing and constructing exotic derivatives sensitive to volatility.
  • The methodology offers flexibility in creating customized variance-related financial products.
  • This research contributes to the advancement of derivatives pricing and risk management strategies.