Volatility is Rough Sample Clauses

Volatility is Rough. ‌ The origins of modern mathematical finance as it is currently known can be traced back to the introduction of the Black-Scholes model which even today is considered the canonical model for financial assets in continuous time: dSt/St = µdt + σdWt (1.1) where W is a Brownian Motion. This model was revolutionary in that one could price all European options (and some exotics) with corresponding unique replication strategies i.e. perfect hedging strategies. Such a model was ubiquitous in industry and contributed to the boom in the derivatives market during the end of the twentieth century. Whilst simple this model has some substantial drawbacks. Given market prices for vanilla options we can infer the value of σ required to give the observed market price, the so- called implied volatility. When the implied volatility is computed from market observed equity options prices across strikes we typically see what has come to be known as a volatility smile i.e. a non-constant implied volatility which typically increases as we move sufficiently far away from the "at-the-money" (ATM) strike. This is in contrast to the "flat" implied volatility characteristic of the Black-Scholes model. The gradient of the smile at-the-money (where the strike is the same as the spot value) is known as the skew and it has been empirically observed to diverge as the maturity tends to zero. Faced with the task of building more realistic models for asset prices we can divide the various proposed approaches into two broad types. The first, which we shall not address in this thesis, consists of models that incorporate jumps into the sample paths of the asset prices thus allowing for the framework of (potentially infinite-activity) Lévy processes amongst other things.