The aim of this study is to examine the volatility smile based on European options Shanghai stock exchange 50 ETF. data gives evidence existence a well-known U-shaped implied for SSE ETF market in China. For those near-month options, smirk also observed. And remains high short maturity and decreases as increases. patterns indicate that in-the-money out-of-the-money are more expensive relative a...

We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European options which are written in terms of the implied volatility are given. In particular, we prove elegant formulas for the fair strikes of the variance swap and the...

Despite its success, the Black-Scholes formula has become increasingly unreliable over time in the very markets where one would expect it to be most accurate. In addition, attempts by financial economists to extract probabilistic information from option prices have been puny in comparison to what is clearly possible. This paper develops a new method for inferring risk-neutral probabilities (or ...

Finding a constraint network that will be efficiently solved by a constraint solver requires a strong expertise in Constraint Programming. Hence, there is an increasing interest in automatic reformulation. This paper presents a general framework for learning implied global constraints in a constraint network assumed to be provided by a non-expert user. The learned global constraints can then be...

This paper studies implied volatility smirk quantitatively. We first propose a new concept of smirkness, which is defined as a triplet of at-the-money implied volatility, skewness (slope at the money) and smileness (curvature at the money) of implied volatility – moneyness curve. The moneyness is the logarithm of the strike price over the forward price, normalized by the standard deviation of e...

The illiquidity of long-maturity options has made it difficult to study the term structures of option spanning portfolios. This paper proposes a new estimation and inference framework for these option-implied term structures that addresses long-maturity illiquidity. By building a sieve estimator around the risk-neutral valuation equation, the framework theoretically justifies (fattailed) extrap...