Back by popular demand (Volatility 1/3)

Over the next three posts, I will try to address these questions:

1. What are the basic properties of volatility and implied volatility?
   
2. How do quants measure volatility? Why should there be different implied volatilities for options of different strikes? Why should the put volatility be the same as the call volatility on the same strike?
3. What is dynamic replication? Can I arbitrage volatility?

“Implied volatilities are the focus of interest both in volatility trading and in risk management. As common practice traders directly trade the so called "vega", i.e. the sensitivity of their portfolios with respect to volatility changes. In order to establish vega trades market professionals use delta-gamma neutral hedging strategies which are insensitive to changes in the underlying and to time decay, Taleb (1997). To accomplish this, traders depend on reliable estimates of implied volatilities and, most importantly, their dynamics.” -- Applied Quantitative Finance

Volatility and implied volatility are different

Volatility is the standard measure for how 'active' a stock is. Volatility:

• is linear
• measures the stock's spread of distribution
• is numerically debatable because the past isn’t supposed to fully reflect the future
• has an associated time frame. Vol = V per year. Assuming the volatility of the stock stays constant over more years, Vol per t years : V * √(t)
• has market components. Even though the company itself has not announced any news, the volatility can grow or shrink due to the market becoming more or less volatile. This is why Principal Components Analysis sounds like a good idea (covered in next post)
  

Since it seems unsettling to draw conclusions from the past to expect market behavior, the focus shifted to implied volatilities, Dumas, Fleming and Whaley (1998). Implied volatility is inferred from option prices given some sort of formula for option prices like Black-Scholes. Implied Volatility shares the above properties but has a couple of its own properties:

• is different for different strikes
• is the same for puts and calls of the same strike
• sometimes doesn’t make sense for exotic options
• incorporates a distribution adjustment due to stocks not being totally lognormal. This is because Black-Scholes formula assumes lognormality
• incorporates volatility premiums due to supply and demand. This means that the stock isn’t expected to have this volatility but the price has been bid up and the implied volatility is higher to match the price. This is somewhat controversial and mostly original with me, so don’t think that it is right. This is due to the unavailability of pure arbitrage and further difficulty of balancing in the short term. (What I’m saying is the dynamic recreation doesn’t work completely. The problem with the last idea of course is knowing when the 4^th reason is driving premiums and when the 5^th reason is driving up premiums.)