Archive for September, 2010

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Stochastic Volatility Modelling

Understanding volatility is critical to many players in finance. Risk managers need to know the likelihood that their financial instruments are going to decline in value in the future, option traders need to know the volatility that can be expected over the life of the contract since it affects the chance of it being exercised, and marker makers need to understand the future volatility so the bid/ask spread can be set accordingly.

Known Behaviour
There have been many studies to examine volatility in financial markets and several common factors usually result:

  1. Volatility is mean reverting. In other words, regardless of the current volatility there is evidence to suggest that it will tend towards some long-term average value. If you think about it this makes sense. If the price/barrel of WTI fell to US$7 due to overproduction you would expect the price to eventually rise as producers decrease supply. These expectations are intuitive in nature and are supported by many empirical studies.

  2. Exogenous variables may affect volatility. For example, in FX the GBPUSD cross rate price and volatility will likely be affected by the release of macroeconomic data from either the UK or the States.

  3. Market events may have asymmetric impact on volatility. Specifically negative "shocks" such as major market corrections tend to have a greater impact on volatility than do optimistic announcements and market rallies.

Modelling
Modelling volatility can be done in many ways. The most simplistic is the assumption of constant volatility. Here you use historical volatility observed from data to provide your estimate of future volatility. Whilst simplistic and easy, this approach is not applicable to instruments that are heteroskedastic (have differing variance/volatility over time).

If you don't think historical volatility is a good predictor of future volatility in, say, an option or an instrument with embedded optionality like an interest rate cap used to hedge interest rate risk, you can look at the implied volatility of the option contract/instrument. That is the volatility that needs to be plugged into the Black-Scholes formula, or whatever pricing model you are using, such that the model yields a theoretical value for the option equal to the current market value.

As an aside...for portfolios of options that also contain their underlying assets it's possible to construct delta-neutral portfolios meaning that the portfolio as a whole is relatively insensitive to changes in the price of the underlying asset. Delta, of course, is the option value change relative to the change in the underlying asset. By executing a delta neutral position, one can profit from a change in volatility with no directional risk when the underlying stock moves insignificantly. This option trading strategy is extremely useful when implied volatility is expected to change drastically soon. But you got to have a forecast of volatility to make that call right? So let's get back on track regarding volatility models that you might consider...

Geometric Brownian Motion
Many financial models employ "random walks" (Geometric Brownian Motion) but clearly the Markov memoryless property of such are at odds with our mean reversion property. To address this, a GBM model with Mean Reversion can be used. The pure GBM-based models will obviously produce a much greater spread of values since each deviation is independent as the following 2 graphs show (source = Financial Engineering Associates):



Whilst the mean-reverting GBM model is a better fit to reality the downside is you now have to estimate a few more parameters: the long-run mean price and the rate at which prices revert to this value.

In fact, the famous Black-Scholes option pricing model assumes the underlying asset has constant volatility over the life of the contract. Clearly this assumption is egregious in many cases. The volatility smiles prevalent in implied volatility surfaces in FX are testament to this.

GARCH
An alternative to GBM is the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. From what I've read, it seems the GARCH model continues to substantially outperform the Black-Scholes model even when the Black-Scholes model is updated every period while the parameters of the GARCH model are held constant. The improvement is due largely to the ability of the GARCH model to describe the correlation of volatility with spot returns. This allows the GARCH model to capture strike price biases in the Black-Scholes model that give rise to the skew in implied volatilities in the index options market.

As well, GARCH option pricing models have the inherent advantage that volatility is observable from discrete asset price data and only a few parameters need to be estimated even in a long time series of options records.

The details around GARCH models are varied and complex, so I'll defer that to a future post...

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A Stat Arb Opportunity or Not?

A while ago I was familiarising myself with jet fuel hedging strategies and came across "crack spreads". If you must know, a crack spread effectively represents the spread made by the refinery to "crack" crude oil products and produce more refined products, but that's not what really caught my eye. I also stumbled across this graph with provoked some thought about possible statistical arbitrage opportunities...

It appears that between 1998-2009 Heating Oil and Natural Gas seemed to be highly correlated and good candidates for a pair trade since they are, in theory, genuine substitutes. However since the start of 2009 the co-integration argument doesn't seem to hold. Rather, they have diverged.

I've read articles detailing technological breakthroughs in natural gas extraction (hydraulic fracturing of shale) to explain the price drop/ supply increase in natural gas but because they are substitutes I would have thought that Heating Oil would have to become cheaper relative to natural gas due to consumers switching to natural gas. Natural gas is also cleaner than heating oil so you'd think it's price would increase relative to heating oil but it hasn't.

Anyone got a plausible explanation for this? I guess it goes to show you need the statistics to validate a theory rather than blindly following the stats, but them again, I'm no StatArb guru.