# When is option value inversely related to expected volatility?

It is common knowledge that the greater the expected value, the higher the option value. However, there are surely exceptions, as written by Paul Wilmott's FAQs in Quantitative Finance

Q: If you increase volatility what happens to the value of an option?

A:It depends on the option!

So when will an increase in expected value lead to a decrease in option value? One of my guesses is an up-and-out call option where higher volatility will make the underlying price to cross the barrier, hence zero payoff. Is there a case when this property also holds for vanilla options?

• Yes, for European vanilla options the price is a monotonically increasing function of volatility. Just think in terms of Vega. First, note that the call/put parity does not involve volatility, so the Vega of a call is indeed that of a put and it computes as (using the standard notations) $$\nu = P(0,t) F(0,t) \phi(d_1) \sqrt{t}$$ or equivalently $$\nu = P(0,t) K \phi(d_2) \sqrt{t}$$ where in both cases, all terms : discount factor, forward price, strike, probability density function and (square root of ) time... are all positive, hence the Vega is always positive for European call/put options.