## Friday, July 16, 2010

## Thursday, July 9, 2009

### Stochastic Volatility Models

**From:** Collector's Blog | July 08, 2009

Every stochastic volatility model assumes yes stochastic volatility. All the stochastic volatility models I have looked into however assume constant volatility of volatility. Empirical research (mostly unpublished) shows the volatilRead more at Collector's Blog »

## Comments (2)

## Sunday, July 5, 2009

### Using Volatility to Predict Market Direction

**Decomposing Asset Returns**

We can decompose the returns process R_{t} as follows:

Now consider the implications of dependence and hence forecastability in the sign of asset returns, or, equivalently, the direction-of-change. It may be possible to develop profitable trading strategies if one can successfully time the market, regardless of whether or not one is able to forecast the returns themselves.

There is substantial evidence that sign forecasting can often be done successfully. Relevant research on this topic includes Breen, Glosten and Jaganathan (1989), Leitch and Tanner (1991), Wagner, Shellans and Paul (1992), Pesaran and Timmerman (1995), Kuan and Liu (1995), Larsen and Wozniak (10050, Womack (1996), Gencay (1998), Leung Daouk and Chen (1999), Elliott and Ito (1999) White (2000), Pesaran and Timmerman (2000), and Cheung, Chinn and Pascual (2003).

There is also a huge body of empirical research pointing to the conditional dependence and forecastability of asset volatility. Bollerslev, Chou and Kramer (1992) review evidence in the GARCH framework, Ghysels, Harvey and Renault (1996) survey results from stochastic volatility modeling, while Andersen, Bollerslev and Diebold (2003) survey results from realized volatility modeling.

#### Sign Dynamics Driven By Volatility Dynamics

Let the returns process R_{t} be Normally distributed with mean m and conditional volatility s_{t}.

The probability of a positive return Pr[R_{t+1} >0] is given by the Normal CDF F=1-Prob[0,f]

For a given mean return, m, the probability of a positive return is a function of conditional volatility s_{t}. As the conditional volatility increases, the probability of a positive return falls, as illustrated in Figure 1 below with m = 10% and s_{t} = 5% and 15%.

In the former case, the probability of a positive return is greater because more of the probability mass lies to the right of the origin. Despite having the same, constant expected return of 10%, the process has a greater chance of generating a positive return in the first case than in the second. Thus volatility dynamics drive sign dynamics.

Figure 1

**Email me at jkinlay@investment-analytics.com.com for a copy of the complete article.**

### EuroPlace Finance Forum Paris

## Monday, May 18, 2009

### Quant Congress USA NY July 14-16, 2009

The subject of my presentation will be Forecasting and Trading Volatility.

Details here: http://web.incisive-events.com/rma/2009/07/quant-congress-usa/index.html

### Volatility Forecasting and Trading Strategies Summit May 20, 2009 New York

## Thursday, May 7, 2009

### Volatility Metrics

Part of the reason was that the claimed efficiency improvements of the Parkinson, Garman-Klass and other estimators failed to translate into practice when applied to real data. Or, at least, no one could quite be sure whether such estimators really were superior when applied to empirical data since volatility, the second moment of the returns distribution, is inherently unknowable. You can say for sure what the return on a particular stock in a particular month was simply by taking the log of the ratio of the stock price at the month end and beginning. But the same cannot be said of volatility: the standard deviation of daily returns during the month, often naively assumed to represent the asset volatility, is in fact only an

*estimate*of it.

All that began to change around 2000 with the advent of high frequency data and the concept of Realized Volatility developed by Andersen and others (see Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2000), “The Distribution of Exchange Rate Volatility,” Revised version of NBER Working Paper No. 6961). The researchers showed that, in principle, one could arrive at an estimate of volatility arbitrarily close to its true value by summing the squares of asset returns at sufficiently high frequency. From this point onwards, Realized Volatility became the "gold standard" of volatility estimation, leaving other estimators in the dust.

Except that, in practice, there are often reasons why Realized Volatility may not be the way to go: for example, high frequency data may not be available for the series, or only for a portion of it; and bid-ask bounce can have a substantial impact on the robustness of Realized Volatility estimates. So even where high frequency data is available, it may still make sense to compute alternative volatility estimators. Indeed, now that a "gold standard" estimator of true volatility exists, it is possible to get one's arms around the question of the relative performance of other estimators. That was my intent in my research paper on Estimating Historical Volatility, in which I compare the performance characteristics of the Parkinson, Garman-Klass and other estimators relative to the realized volatility estimator. The comparison was made on a number of synthetic GBM processes in which the simulated series incorporated non-zero drift, jumps, and stochastic volatility. A further evaluation was made using an actual data series, comprising 5 minute returns on the S&P 500 in the period from Jan 1988 to Dec 2003.

The findings were generally supportive of the claimed efficiency improvements for all of the estimators, which were superior to the classical standard deviation of returns on every criterion in almost every case. However, the evident superiority of all of the estimators, including the Realized Volatility estimator, began to decline for processes with non-zero drift, jumps and stochastic volatility. There was even evidence of significant bias in some of the estimates produced for some of the series, notably by the standard deviation of returns estimator.

Finally, analysis of the results from the study of the empirical data series suggested that there were additional effects in the empirical data, not seen in the simulated processes, that caused estimator efficiency to fall well below theoretical levels. One conjecture is that long memory effects, a hallmark of most empirical volatility processes, played a significant role in that finding.

The bottom line is that, overall, the log-range volatility estimator performs robustly and with superior efficiency to the standard deviation of returns estimator, regardless of the precise characteristics of the underlying process.

Send me an email at jkinlay@investment-analytics.com if you would like to receive a copy of the paper.

The models are phenomenological and require input of the parameters and so can never be predictive for all scenarios. Sometimes (as in Hull's book) you find that people use a model such as (if I remember correctly) Black's model (rather than Black and Scholes) and forgive me if I got the name wrong, to model for example products that depend on interest rates (which according to Hull have mean reversion and so are not behaving like a Brownian motion) and somehow this model "works out" or "works better" than the B-S because although not intended for this it makes the right outputs. The point to remember is that all of these models are phenomenological and depend on estimates of the inputs such as "historical volatility" and volatility of volatility, so that they are both of academic interest and of engineering interest (see how well they might work in practice). As an analogy consider turbulence modelling which makes some assumptions to close the equations - it works for some geometries but needs a lot of adjustment or fails for others. So I guess the thing to do is construct and study these models and then somehow evaluate them for different scenarios and issue recommendations as to how and when to use them? More than one model (and quite differently motivated models) may give the same outputs. In recent years ways of speaking in averages (fuzzy logic) have been as effective as complex control theories in practical engineering.

Daniel HowardDirector at Howard Science Limitedposted 54 minutes ago

I am going to be presenting a paper on Volatility Modeling and Trading at the upcoming Quant USA conference in New York next week in which I discuss a very effective stochastic volatility of volatility model, the ARFIMA-GARCH model. It models volatility as a long memory process which is disturbed by shocks from the volatility of volatility process, which evolves in GARCH form.

The paper evaluates the performance of the model in trading S&P options.

More on the conference here: http://web.incisive-events.com/rma/2009/07/quant-congress-usa/index.html

More details on my Quantitative Investment and Trading blog to come:http://quantinvestment.blogspot.com/

Jonathan Kinlay, PhD (jkinlay@investment-analytics.com)