Encyclopedia of Quantitative Finance, IV Volume Set

Uncertain Volatility Model

Black–Scholes and Realized Volatility

What happens when a trader uses the Black–Scholes ((BS) in the sequel) formula to dynamically hedge a call option at a given constant volatility while the realized volatility is not constant?

It is not difficult to show that the answer is the following: if the realized volatility is lower than the managing volatility, the corresponding profit and loss (P&L) will be nonnegative. Indeed, a simple, yet, clever application of Itô’s formula shows us that the instantaneous P&L of being short a delta-hedged option reads

where Γ is the gamma of the option (the second derivative with respect to the underlying, which is positive for a call option), and σ_t the spot volatility, for example, the volatility at which the option was sold and represents the realized variance over the period [t, t + dt]. Note that this holds without any assumption on the realized volatility, which will certainly turn out to be nonconstant. This result is fundamental in practice: it allows traders to work with neither exact knowledge of the behavior of the volatility nor a more complex toolbox than the plain BS formula; an upper bound of the realized ...