This paper discusses the semiparametric boundary problem of the upper or lower mean of the European butterfly option under normal condition, unimodal condition and bimodal condition, and draws good conclusions. The research is based on dual method and measure transformation

We utilise functional time series (FTS) techniques to characterise and forecast implied volatility in foreign exchange markets. In particular, we examine the daily implied volatility curves of FX options, namely; EUR-USD, EUR-GBP, and EUR-JPY. Based on existing techniques in the literature, the...