# Volatility

Volatility is annualized standard deviation of returns.

## Definitions

• Actual volatility is a measure of the amount of randomness in a financial quantity at any point in time. It's difficult to measure, and even harder to forecast but it's one of the main inputs into option pricing models.
• Realized/historical volatilities are associated with a period of time. We might say that the daily volatility over the last sixty days has been 27%. This means that we take the last sixty days' worth of daily asset prices and calculate the volatility.
• Implied volatility is the number you have to put into the Black-Scholes option-pricing equation to get the theoretical price to match the market price. Often said to be the market's estimate of volatility. Implied volatility levels the playing field so you can compare and contrast option prices across strikes and expirations.
• There is also forward volatility. The adjective forward is added to anything financial to mean values in the future. So forward volatility would usually mean volatility, either actual or implied, over some time period in the future.

Since volatility is so difficult to pin down it is a natural quantity for some interesting modelling. Here are some of the approaches used to model or forecast volatility.

## Realized/historical volatilities

When we speak about realized volatility, it is clear that we can only build an estimator of the underlying volatility.

### Daily volatility

The more common estimator which is used is the close to close volatility, which is the squre root of the empirical quadratic variations of the price sampled at each market close.

This estimator as a large variance, and some attempts have been made to build other estimators that have less variance. A good example of such estimators is the Garman & Klass one, which is based on open, high, low and close prices of the instrument $\sigma^2 = (H-L)^2 /2 - (2\ln(2)-1) (C-O)^2$

### Intra day volatility

The estimation of intra day realized volatility (using high frequency datasets) is more complicated. One has to take into account :

• the effect of discretization of the price
• the effect of the jumps in the price

Those three effects have been theoretically studied by Jacod and Delattre for the first one ; Barndorff-Nielsen and Shephard for the second one ; and Zhang, Mykland, and Aït-Sahalia for the last one.

### Econometric estimations

These models use various forms of time series analysis to estimate current and future expected actual volatility. They are typically based on some regression of volatility against past returns and they may involve autoregressive or moving-average components. In this category are the GARCH type of models.

## Models

### Deterministic models

The simple Black-Scholes formulae assume that volatility is constant or time dependent. But market data suggests that implied volatility varies with strike price. Such market behaviour cannot be consistent with a volatility that is a deterministic function of time. One way in which the Black-Scholes world can be modified to accommodate strike-dependent implied volatility is to assume that actual volatility is a function of both time and the price of the underlying. This is the deterministic volatilit model.

### Stochastic volatility

Since volatility is difficult to measure, and seems to be forever changing, it is natural to model it as stochastic. The most popular model of this type is due to Heston. Such models often have several parameters which can either be chosen to fit historical data or, more commonly, chosen so that theoretical prices calibrate to the market.

### Poisson processes

There are times of low volatility and times of high volatility. This can be modelled by volatility that jumps according to a Poisson process.

### Uncertain volatility

An elegant solution to the problem of modelling the unseen volatility is to treat it as uncertain, meaning that it is allowed to lie in a specified range but whereabouts in that range it actually is, or indeed the probability of being at any value, are left unspecified. With this type of model we no longer get a single option price, but a range of prices, representing worst-case scenario and best-case scenario.

## References

• Avellaneda, M, Levy, A & Paras, A 1995 Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance 2 73-88
• Barndorff-Nielsen, O. and Shephard, N. (2004). Multipower variation and stochastic volatility. Economics Papers 2004-W30, Economics Group, Nuffield College,University of Oxford.
• Barndorff-Nielsen, O.~E. and Shephard, N. (2003). Econometrics of testing for jumps in financial economics using bipower variation.Technical report, The Centre for Mathematical Physics and Stochastics (MaPhySto), University of Aarhus and Nuffield College, University of Oxford.
• Delattre, S. and Jacod, J. (1997). A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors. ; Bernoulli Journal of Mathematical Statistics and Probability, 3:1--28(28).
• Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7 (2) 32-39 (February)
• Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18-20 (January)
• Garman, M. B. and Klass, M. J. (1980) On the estimation of security price volatility from historical data ; Journal of Business, 53(1):67--78.
• Gatheral, J. (2006), The Volatility Surface: A Practioner's Guide, Wiley
• Heston, S 1993 A closed-form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6 327-343
• Jacod, J. (1996). Hommage a P. A. Meyer et J. Neveu, volume 236, chapter La variation quadratique moyenne du brownien en présence d'erreurs d'arrondi. Asterisque.
• Jacod, J. (1998). Rates of convergence to the local time of a diffusion. Annales de l'institut Henri Poincaré (B) Probabilités et Statistiques, 34(4):505--544.
• Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons
• Lewis, A 2000 Option valuation under Stochastic Volatility. Finance Press
• Lyons, TJ 1995 Uncertain Volatility and the risk-free synthesis of derivatives. Applied Mathematical Finance 2 117-133
• Rubinstein, M 1994 Implied binomial trees. Journal of Finance 69 771-818
• Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second edition. John Wiley & Sons
• Zhang, L., Mykland, P.~A., and Aït-Sahalia, Y. (2005) A tale of two time scales: Determining integrated volatility with noisy high-frequency data ; Journal of the American Statistical Association, 100(472).