A variance swap pays realised variance minus a strike. Carr-Madan: that strike is replicable today by a *static* portfolio of OTM puts and calls weighted $2 dK / K^2$.
A variance swap is a forward contract on realised variance. The buyer gets paid the difference between the realised annualised variance of the underlying and a strike $K_{\text{var}}$ agreed at inception. The Carr-Madan (1998) miracle is that this payoff can be replicated *statically* today using only a portfolio of vanilla European options: hold $2\,dK / K^2$ of each out-of-the-money call (above the forward) and put (below). No daily rebalancing, no model assumption. The resulting fair strike is exactly the model-free expectation of integrated variance under the risk-neutral measure. For a constant-vol Black-Scholes world, this collapses to $K_{\text{var}} = \sigma^2$; for a market with a smile, it overshoots ATM IVΒ² by exactly the smile's curvature contribution. The VIX is a 30-day variance-swap strike with a finite truncation.
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
A non-dividend stock has $S_0 = 100$, BS volatility $\sigma = 20\%$ for all strikes (constant-vol world), $r = 5\%$, $T = 1$ year. (a) Show analytically that the fair 1-year var-swap strike is $K_{\text{var}} = \sigma^2$. (b) Verify by discretely summing the Carr-Madan replication portfolio over OTM puts (strikes $5$ to $\lfloor F \rfloor$) and calls (strikes $\lceil F \rceil$ to $200$) at $\Delta K = 0.5$.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
The Carr-Madan replication is model-free in the sense that no diffusion is assumed when deriving it — it is just ItΓ΄'s lemma on $\ln S$. So why does $K_{\text{var}} > \sigma_{\text{ATM}}^2$ in markets with smile? What is the smile actually adding to the variance forecast?