Why Implied Volatility Skew Can Flatten at Longer Maturities

Summary

The document collects several explanations for why an option volatility smile or skew may become flatter at longer maturities. One account invokes the central limit theorem: under broad assumptions, sums of log returns tend toward a normal distribution, consistent with the Black–Scholes model’s flat smile. Other answers question whether this alone explains observed flattening and offer different mechanisms, including mean reversion in volatility or volatility of volatility, changing return autocorrelation, and practical limits on long-dated volatility trading.

A mathematical argument uses no-arbitrage bounds on put prices, the Black–Scholes formula, and Mills ratio bounds to show that the strike sensitivity of implied volatility tends to zero as maturity grows under its assumptions. Another explanation notes that steep skew can produce negative digital option prices or other arbitrage issues, with vega increasing roughly with the square root of time. These are model-based explanations, not a single settled account of market behavior; the collection presents competing views and assumptions.

Key ideas

• Under broad assumptions, aggregation of log returns can make their distribution more nearly normal at longer horizons.
• A Black–Scholes-based argument derives declining strike sensitivity of implied volatility from no-arbitrage bounds and Mills ratio estimates.
• Mean reversion in volatility or volatility of volatility is proposed as another cause of flatter long-dated smiles.
• Steep skew can conflict with no-arbitrage constraints, including bounds implied by digital options.
• The answers offer competing explanations, and the document does not establish one universal cause.

Tags

# Bloomberg terminal swap zero curve calculation


# Bloomberg terminal swap zero curve calculation












I would like to ask about swap zero curve calculation algorithm by Bloomberg terminal. This is a plain vanilla CZK interest rate swap, fixing the Prague IBOR. My task is to calculate zero rates from market rates, however I have only managed to get accurate zero rates from 2 years onwards. I tried to bootstrap spot rates from FRAs (CKFR0F1 is FRA 6x12 and CKFR011 is 12x18) with this formula:

$(1+r_{0;t_{0}}\frac{t_{0}}{360})*(1+r_{t_{0};t_{0}+t{u}}\frac{t_{u}}{360})= (1+r_{0;t_{0}+t{u}}\frac{t_{u}+t_{0}}{360})$

Where $r_{0;t_{0}}= 0.0056$, $r_{t_{0};t_{0}+t{u}} = 0.0095$, $t_{0}=182$ and $t_{u}=183$. By solving this equation I get $r_{0;t_{0}+t{u}}=r_{0;1}= 0.007568827$, which is off only by a tiny fraction. I guess the mistake will be in day count conventions, however this is the closest I have come to the correct solution. Can someone explain how the calculation should be done?

I have also attached screenshots of the swap yield curve and cash-flows.

## Answer by Olaf (score 3, accepted)

https://quant.stackexchange.com/a/36442

It looks like you should use a different convention for the zero rates. I tried the following:

$$\left(1+r_{0;t_{0}}\frac{t_{0}}{360}\right) \times \left(1+r_{t_{0};t_{0}+t{u}}\frac{t_{u}}{360}\right) = \left(1+r_{0;t_{0}+t{u}}\right)^{\frac{t_{u}+t_{0}}{360}}$$

Solving with the same input gives $r_{t_{0};t_{0}+t{u}}=0.00756843$, in agreement with Bloomberg.

The right hand side convention kicks in because $(t_u+t_0)/360 > 1~year$. It's a convention used for zero rates, and it looks Bloomberg is using it.

The only source I'm aware of which treats this convention is Brigo and Mercurio:

https://books.google.ie/books?id=C31l_fs-mMkC&lpg=PA57&pg=PA9#v=onepage&q&f=false

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.