# Implied volatility general properties

Even though investors take implied volatility into account when making investment decisions, and this dependence inevitably has some impact on the prices themselves, there is no guarantee that an option's price will follow the predicted pattern. Another important point to note is that implied volatility does not predict the direction in which the price change will go. This talk investigates implied volatility in general classes of stock price models. To begin with, we take a very general view. We find that implied volatility is always, everywhere, and for every expiry well-defined only if the stock price is a non-negative martingale.

We also derive sufficient and close to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from static arbitrage means that the call price surface generated by the implied volatility surface is free from static arbitrage.

We also investigate the small time to expiry behaviour of implied volatility. We used S to denote the current stock price, K to be a option strike price, T denotes time to expiry, and C K, T the price of the K strike option expiring in T time units.

Under these weak assumptions, we obtain exact asymptotic formulae relating the call price surface and the implied volatility surface close to expiry. We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility in a variety of models.

We consider exponential Levy models, obtaining new and somewhat surprising results. We then investigate the behaviour close to expiry of stochastic volatility models in the at-the-money case.

Our results generalise what is already known and by a novel method of proof. In the not at-the-money case, we consider local volatility models using classical results of Varadhan.

In obtaining the asymptotics for local volatility models, we use a representation of the European call as an integral over time to expiry. We may also consider review representations of the European call option.

## Or why your option prices can be less stable than a one-legged duck

He is interested in mathematical finance.Title: Implied Volatility: General Properties and Asymptotics Abstract words maximum: (PLEASE TYPE) This thesis investigates implied volatility in general classes of stock price models. Oftentimes, options traders look for options with high levels of implied volatility to sell premium.

This is a strategy many seasoned traders use because it captures decay. Implied volatility represents the expected volatility of a stock over the life of the option. As expectations change, option premiums react appropriately. Implied volatility is directly influenced by the supply and demand of the underlying options and by the market's expectation of the share price's direction.

In general, it is not possible to give a closed form formula for implied volatility in terms of call price. However, in some cases (large strike, low strike, short expiry, large expiry) it is possible to give an asymptotic expansion of implied volatility in terms of call price. We analyze the properties of the implied volatility, the commonly used volatility estimator by direct option price inversion.

It is found that the implied volatility is subject to a systematic bias in the presence of pricing errors, which makes it inconsistent to the underlying volatility.

He also derived general properties of “rational. Government Properties Income Trust That is because the Sep 21, \$ Put had some of the highest implied volatility of all equity options today.

What is Implied Volatility?

Implied volatility - Wikipedia