No-Arbitrage Conditions

A static arbitrage opportunity that arises when the risk-neutral probability density becomes negative. This condition is violated if the price of a butterfly spread—constructed by buying two outer options and selling two at-the-middle options—is negative.

• Mathematical Constraint: Requires option prices to be strictly convex across strikes.
• Detection: Identified by checking the second derivative of the call price with respect to strike, which must be non-negative.
• Consequence: A violation implies the existence of a trading strategy with a non-positive cost today that guarantees a non-negative, and potentially positive, future payoff.

An arbitrage condition ensuring that the total implied variance is monotonically increasing with time to expiration. It prevents a scenario where a longer-dated option is cheaper than a shorter-dated one on the same underlying and strike.

• Monotonicity Rule: For two maturities T₁ < T₂, the price of a call option must satisfy C(K, T₂) ≥ C(K, T₁).
• Forward Variance: This condition guarantees that forward variance between any two future dates remains non-negative.
• Practical Impact: Violations often occur in stressed markets or due to discrete dividend modeling errors, allowing traders to sell short-term options and buy long-term options for a credit.

The Breeden-Litzenberger formula establishes that the risk-neutral density (RND) is the discounted second derivative of the call price with respect to strike. For a surface to be arbitrage-free, this density must integrate to one and be non-negative everywhere.

• Positivity: The RND must be ≥ 0 for all strikes, directly enforcing the butterfly arbitrage condition.
• Normality: The integral of the density across all strikes must equal the discount factor, ensuring the martingale property of the underlying.
• Tail Behavior: The density must decay sufficiently fast to ensure finite variance, preventing moment explosions in exotic pricing.

A fundamental no-arbitrage relationship linking European call and put options with the same strike and expiration. It states that a portfolio of a long call and a short put replicates a forward contract on the underlying asset.

• Formula: C - P = S₀ - Ke^{-rT}, where C is the call price, P is the put price, S₀ is the spot price, and K is the strike.
• Synthetic Positions: Violations allow traders to create a synthetic long position cheaper than buying the asset directly, or vice versa.
• Surface Consistency: This parity enforces that the implied volatility for a call and put at the same strike and maturity must be identical, ensuring a single volatility surface.

The process of fitting a parametric or non-parametric model to market-quoted option prices to construct a smooth, arbitrage-free implied volatility surface. This is a constrained optimization problem balancing market fit against no-arbitrage conditions.

• Objective Function: Minimizes the squared difference between model and market prices, often weighted by bid-ask spreads or liquidity.
• Regularization: Penalty terms are added to enforce smoothness and prevent overfitting to noisy quotes, ensuring stable Greeks for hedging.
• Global Optimization: Techniques like simulated annealing or differential evolution are used to avoid local minima in highly non-convex parameter spaces.

A critical distinction in derivatives pricing. Static arbitrage can be locked in today with a portfolio that requires no future rebalancing, while dynamic arbitrage relies on continuous trading strategies that may fail under transaction costs or jumps.

• Static Arbitrage: Violations of butterfly or calendar spread conditions. These are considered 'true' arbitrages as they are model-independent.
• Dynamic Arbitrage: Exploits mispricings relative to a specific model assumption, such as delta-hedging a mispriced option. These are model-dependent and carry execution risk.
• Surface Construction: A robust surface must be free of static arbitrage to be considered valid for pricing any exotic derivative.