Quantitative Derivatives & Model-Free Valuation

Carr-Madan Option Replication Lab

Explore Peter Carr and Dilip Madan's (2001) static spanning theorem. Replicate arbitrary European payoffs, synthesize model-free VIX variance contracts, analyze finite-strike discretization errors, and evaluate synthetic Greeks.

Replicated Option Value
$0.42
Total Synthetic: $0.42
RMSE Discretization Error
0.0028
Max Abs Err: 0.0084
Synthetic Portfolio Gamma
0.0195
Delta: -0.021
Model-Free Vol (VIX)
20.08%
Fair Vol Strike: 20.00% BS
Model Parameters
Replication Principle: Spans any nonlinear payoff by integrating European options weighted by the second derivative $F''(K)\Delta K$.

Mathematical Foundations: The Carr-Madan Spanning Theorem

1. Exact Analytical Decomposition via Taylor Expansion

Peter Carr and Dilip Madan (2001) proved that any twice continuously differentiable payoff $F(S_T)$ can be represented exactly via calculus:

$$F(S_T) = F(S_*) + F'(S_*)(S_T - S_*) + \int_0^{S_*} F''(K)(K - S_T)^+ dK + \int_{S_*}^\infty F''(K)(S_T - K)^+ dK$$

The formula decomposes the terminal payoff into three distinct components: (1) a static bond paying $F(S_*)$, (2) a forward contract with position size $F'(S_*)$, and (3) a continuum of OTM puts ($K < S_*$) and OTM calls ($K \ge S_*$) weighted by the curvature $F''(K)$.

2. Model-Free Variance & The CBOE VIX Index

Applying the theorem to the log contract $F(S) = -2\ln(S / S_*)$ yields the fundamental relation for variance swaps and the CBOE VIX:

$$F''(K) = \frac{2}{K^2} \implies \sigma_{\text{MF}}^2 = \frac{2}{T} \left[ \int_0^{S_*} \frac{e^{rT} P(K)}{K^2} dK + \int_{S_*}^\infty \frac{e^{rT} C(K)}{K^2} dK \right]$$

Because $F''(K) = 2/K^2$, out-of-the-money puts receive rapidly growing weights as strike $K \to 0$, explaining why downside tail risk dominates the variance premium and index skew.

RMSE Discretization Error Sensitivity

Strike Count ($N$) vs. Implied Volatility ($\sigma$)

$N$ \ $\sigma$ 10% 15% 20% 30% 40%
Replication Portfolio Cost Sensitivity ($)

Strike Domain Width vs. Time to Expiry ($T$)

Domain \ $T$ 1 Mo 3 Mo 6 Mo 1 Yr 2 Yr

Interactive Mastery Assessment

1. In Carr-Madan replication of the log contract $F(S) = -2\ln(S/S^*)$, how do option weights scale across strikes?

2. Why does the Carr-Madan theorem specifically utilize out-of-the-money (OTM) calls and puts rather than in-the-money options?

3. What is the primary source of truncation error when calculating model-free implied volatility from market quotes?

4. How does the Carr-Madan framework relate to the official CBOE VIX Index methodology?

Frequently Asked Questions

The Carr-Madan (2001) static option replication theorem states that any twice continuously differentiable terminal payoff F(S_T) can be exactly spanned without dynamic delta hedging by a static portfolio consisting of a risk-free bond, an underlying forward position, and a continuous continuum of out-of-the-money (OTM) European calls and puts weighted by the second derivative F''(K).

The CBOE VIX Index calculates 30-day model-free implied volatility by replicating a short log contract F(S) = -2 * ln(S/S*). Because the second derivative of the log contract is F''(K) = 2/K^2, the fair variance swap rate is spanned by integrating across all OTM call and put option prices with weights inversely proportional to strike squared: (2/T) * int [P(K)/K^2 dK + C(K)/K^2 dK].

Using OTM options (puts for strikes below the reference price S* and calls for strikes above S*) maximizes market liquidity, minimizes bid-ask transaction friction, and avoids early-exercise complications. Furthermore, out-of-the-money options correspond directly to the Taylor expansion remainder terms in the Carr-Madan integral.

Discretization error arises because market exchanges quote options at discrete strike intervals rather than a continuous spectrum, requiring numerical approximations such as the trapezoidal rule. Truncation error occurs because option chains only cover a finite strike domain [K_min, K_max], omitting extreme tail realizations outside the trading band.

The standard Carr-Madan spanning formula directly replicates European-style path-independent payoffs that depend solely on terminal price S_T. However, through extensions like Derman-Ergener-Kani boundary replication and semi-static hedging, related principles can be applied to barrier options, corridor variance swaps, and forward-starting payoffs.
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