At the beginning of his well known book, Gatheral writes the following

[...] Moreover, unlike alternative models that can fit the smile (such as local volatility models, for example), SV models assume realistic dynamics for the underlying. Although SV price processes are sometimes accused of being ad hoc, on the contrary, they can be viewed as arising from Brownian motion subordinated to a random clock. This clock time, often referred to as trading time, may be identified with the volume of trades or the frequency of trading (Clark 1973); the idea is that as trading activity fluctuates, so does volatility.

I always have the feeling that I'm not fully grasping what he means here. What does it mean by a Brownian motion subordinated to a random clock? Why is the SV model signaled as ad hoc?


  • 5
    $\begingroup$ a "subordinated stochastic process" is a process where the "clock" (i.e. the passage of time) is not constant, time appears to move more quickly or more slowly depending on a factor which is itself random. This mathematical construct is very natural in Finance because we do observe that on some days prices fluctuate more than on other days, it is as if "time is speeded up" on those days. Several days worth of activity appears to take place on such a day. The stock market could be a subordinated GBM, one with a variable "clock" rather than the GBM in Black Scholes. $\endgroup$
    – nbbo2
    Jan 24 at 12:59


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