Necessary and Sufficient

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...

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.

What is an IR Swap?
A plain-vanilla interest rate (IR) swap is a bilateral OTC contractual arrangement where the counterparties agree to swap streams of future interest payments based on a specified "notional" principal for a pre-defined period of time. In the "plain vanilla" case this is done using a common currency on a fixed-for-floating basis whereby one party exchanges a stream of fixed interest payments for a stream of floating interest payments that are based on the future movement of a specified floating reference rate like BBSW or LIBOR. Of course, the other party does the reciprocal operation contemporaneously. The interest payments of the swap are based on a notional principal meaning there is no upfront exchange of principal - it wouldn't make any sense since the net exchange would be zero. The start and end dates, the coupon frequency, the notional principal, the day count convention to be used, along with a number of other input parameters are all negotiated up-front.

What Exactly is a Reference Rate?
The reference rate is the published rate that determines the future floating interest payments. Obviously, it is not published until it is known so the floating interest amount is not known until the time it is to be paid. It's the same situation if you have a variable rate home loan - you don't know what that rate will be in 9 months or 2 years time.

One very popular reference rate base is LIBOR, the London Interbank Offered Rate. LIBOR is the interest rate that banks with high credit ratings from ratings agencies (rating of AA- or above) charge each other for short-term financing. LIBOR is set daily by the British Banking Association (BBA) and is considered the benchmark for floating short-term interest rates. Here's an extract of published LIBOR rates taken from the Financial Times on Aug 4.

Notice that LIBOR is quoted in numerous currencies and numerous "tenors" (maturities). In general, the tenor of the reference rate used in an IR swap will be the same as the coupon payment frequency of the fixed leg of the swap.

In Australia, the 90-day Bank Bill Swap rate (BBSW) and the Australian Bank Bill Swap Bid Rate (BBSY) are heavily used reference rates. BBSW is compiled by the Australian Financial Markets Association (AFMA) at 10am every business day based on the average MID-price of market rates supplied by a select number of Australian bank bill market makers. BBSY is calculated similarly except that the average BID-price is used instead of the average MID-price. You can read more about their methodology here. Note that BBSW and BBSY are NOT to be confused with the RBA "cash rate".

The key takeaway is this: no single bank sets these reference rates, they are compiled by independent authorities using a transparent algorithm based on information from numerous financial institutions.

Also note that often the floating rate used by IR swaps is the reference rate plus (or minus) a margin. e.g. LIBOR + 0.5% (50 basis points).

Why Use IR Swaps?
There are many reasons, but put simply... swaps allow an entity to rapidly modify the nature of it's IR exposure from fixed to floating or vice versa. In the context of risk management an IR swap can be used to get greater certainty about future interest payments and lower your effective cost of funding. Having greater certainty about cash flows allows companies to ensure capital adequacy for future obligations.

Note that counterparty risk is still present because the other party may default, and interest rate risk still exists because you could have locked in a higher rate and miss out of favourable floating rate moves.

Working Through An Example
Assume two entities, AAA Corp and BBB Corp, enter into a 5 year fixed-for-floating USD vanilla swap with AAA Corp paying fixed and receiving floating, and BBB Corp doing the opposite.

Typically we say the entity that is paying fixed is "long" the swap, whilst the other entity is short the swap. So here AAA Corp is long and BBB Corp is short the swap.

Furthermore, assume the contract specifies that the fixed rate is 6% semi-annual, the floating interest payments will be pegged to 3-month LIBOR plus 50 basis points, the notional principal is 100 million USD and the floating payment frequency is quarterly.

This implies that AAA Corp will pay 6.0% of $100 million USD on a semi-annual basis (so 300k USD twice a year) to BBB Corp for the next 5 years, whilst BBB Corp will pay, approximately, 1/4 * (3-month LIBOR + 0.5% of $100 million) four times per year. So if the 3-month LIBOR rate is 2.3% on the day the payment amount is calculated (the "reset" date), then BBB Corp is obligated to pay 1/4 * (2.3% + 0.5%) = 0.7% * $100 million = 700k USD to AAA Corp for that period.

In practice, the above fractions are likely to differ depending on the day count convention specified in the contract.

In the next post we'll talk about how to value an IR Swap.