When is option value inversely related to expected volatility?

Question

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?

Answer 1

You are on the right track IMO, except that there are no exceptions to "the greater the expected value, the higher the option value" rule, since an option premium's is precisely the discounted expectation of its payout by absence of arbitrage - at least under the risk-neutral measure.

As far as the influence of volatility on the option value is concerned:

• The example of a corridor option (or a double knock-out barrier) is even more illuminating I would say. Clearly, as volatility increases, the chances of the option getting knocked out increase (meaning that the chances of the option paying out nothing increase), hence the expected payout decreases, hence the option price decreases.

• 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.

For a knock-out barrier option, you'll indeed observe that Vega becomes negative close to the barrier.