Correlation Trading

Correlation trading is something that hasn't yet been discussed on this blog. In principle, one can bet on the correlation(s) among n assets of one's choice. However most commonly correlation trading is done on an index. In other words, you are betting on the correlation of the constituents of the index. You can approach a bank and ask them to quote the strike of a correlation swap for you. Then you can sit back and let the bank worry about all the dirty hedging works.

There are sound reasons not to go down this route. First, what if you want to speculate on a basket that is not a common index? That may not be readily quotable at the bank. Secondly, the strike of the correlation swap quoted by the bank is likely to have a large spread buffer to compensate for the risk they take.

An alternative is to proxy the correlation swap. The technique is generally known as dispersion trading. It's a fancy name of long-shorting an index versus its components, or vice versa.

Straddles
The most primitive way of dispersion trading is to long the index straddle and short the constituents straddles (this would be betting on the correlation among the constituents to go up). In this case, each of the straddles proxy the specific variances (of the index or of the constituents), and the long-short portfolio of straddles in turn proxies the correlation. Such a simplistic scheme would of course leave a lot to be desired (e.g. the need to rehedge, see here).

Variance Swaps
Can we do better? Absolutely. It's almost a no-brainer: replace the straddles with variance swaps! So instead of longing the index straddle and shorting the constituents straddles, you long the index variance swap and short the constituents variance swaps to bet the correlation going up. Thanks to the characteristics of variance swaps, you don't have to worry about rehedging anymore. Sounds good, doesn't it? But there is still room for improvement. Although hedging is not necessary, the weights of the variance swaps have to be dynamically maintained. As an example, suppose our (value weighted) index I has only two components A and B. Initially the the index is made up of 50% each of A and B. You try to trade correlation by longing $100 notional of variance swap on I and shorting $50 notional of variance swaps on A and B each. Over time, the underlyings move and the weights of A and B become, say, 43% and 57% respectively. You'd have to dynamically re-balance your variance swap portfolio to keep in line with the changing weights.

Note: The above discussion assumes a value weighted index.

As a side note, the strike of the long index variance swap, short constituents variance swap portfolio would in general differ from a correlation swap quote. The reason is non-zero vol-of-vol. This paper gives an in-depth investigation into the matter.

Gamma Swaps
Knowing the shortcoming of dispersion trading using variance swaps (i.e. the need for reallocation), what alternative do we have? We can use a kind of swap that 'scales' itself according to the underlying level, a feature that is provided by gamma swaps. You long the index gamma swap and short the constituents gamma swaps to bet the correlation going up. Due to the structure of a gamma swap, your exposure to each index component would in fact automatically match the desired weightings as the underlyings change.

Common Option Strategies: The Greeks

Naked, single option trading is not only risky, but also it is difficult to single out specific exposures. Hence option combo strategies are more common. But what strikes to choose? Hopefully this post would give you some hints.

Assumptions:
- Volatility smile = $0.15 - 0.1\times (K-K_{ATM}) + 0.1 \times (K-K_{ATM})^2 $
- Sticky strike
- Flat volatility term structure

Since $\Theta$ and Vega are similar to $\Gamma$, we skip presenting those. We are showing:
- Price
- Delta
- Gamma
- Vanna
- Volga
- Charm
- $ \frac {\partial}{\partial t} Vega$

Note:
$Vanna  = \frac{\partial}{\partial S}  \frac{\partial P}{\partial \sigma}$
$Volga  =  \frac{\partial^2 P}{\partial \sigma^2}$
$Charm = \frac{\partial}{\partial S}  \frac{\partial P}{\partial T}$

Strangle

Short Bear Spread

Back Spread

Calendar Spread

Diagonal Spread

Butterfly

Broken Wing Butterfly

Iron Condor

Short Note: VIX Derivatives Pricing

- Lest it has not been emphasized enough in the previous post, note that there is no straightforward no-arbitrage relationship to get VIX future price from sopt VIX.

- Perhaps one day, the market would be so well-developed and liquidly-traded that we could have a "Volatility Market Model." Until then, we will have to rely on more modest approaches.

- Why do people say that VIX options are priced off VIX future instead of spot? Well, for options on, say, FX or stock, there is no need to distinguish between pricing options off futures or spots because spot and future are strictly connected by no-arbitrage; for VIX, however, such connection is absent. If you start of with VIX spot, you first have to price the future contract before you can price the option. So it is better to directly use VIX future price as input when pricing option on VIX, so as to avoid "an extra layer of error."

-  Currently the different approaches of pricing VIX derivatives are:
1. Modeling the instantaneous spot variance as a stochastic process
This is similar to modeling the short rate in interest rate derivatives pricing. Instantaneous spot variance, much like short rate, is neither observed nor tradable. Yet, they are rather well-studied and people are quite familiar with them, at least in the context of vanilla and exotic stock options. The idea is to pick your favorite model that describes the evolution of the instantaneous variance (say, Heston or 3/2 model). Calibrate it to the current volatility surface (considering VIX is a 30-day measure, you may want to calibrate only to the front and forthcoming month). Then use the calibrated parameters to price the VIX derivatives with numerical integration.

2. Modeling the instantaneous forward variance as a stochastic process
This is similar to the HJM framework in interest rate derivatives pricing. One example is the model proposed by Bergomi (see, for example, Bergomi's Smile Dynamics series).

3. Modeling VIX itself as a stochastic process
This differs from the previous two cases in the sense that VIX itself, which is observed, is modeled as a stochastic process.

Bottom line:
In addition to calibration efficiency, one important desirable feature is for the model to fit to both the SPX option prices and the VIX option prices.

Reference
Mencía and Sentana, 2012, Valuation of VIX Derivatives
http://www.bde.es/f/webbde/SES/Secciones/Publicaciones/PublicacionesSeriadas/DocumentosTrabajo/12/Fich/dt1232e.pdf
Wang and Daigler, 2008, The Performance of VIX Options Pricing Models: Empirical Evidence Beyond Simulation
http://www2.fiu.edu/~zwang001/Research/papers/VIX_Option_Pricing_FMA.pdf

VIX Futures Pricing

Recently I came across a blog called macroption, which seems to carry some interesting materials. 

The blogger specifically explains the difference between the term structure and futures curve of VIX, which can sometimes be confusing. To eliminate the ambiguity, we can use the full notation, with three arguments, to denote forward volatility: $ V(t,S,T) $ (which is similar to the full notation of LIBOR). $ V(t,S,T) $ is, of course, the forward $T-S$ day volatility starting at a future time $S$ as observed at time $t$. With this in mind, the term structure of VIX would be a graph of $V(0,0,T)$ against $T$; while the futures curve of VIX would be a graph of $V(0,S,S+\delta)$ against $S$ ($\delta$ in this case would be 30 days).

Now that we have clarified the notation, let's talk about VIX futures pricing and how convexity plays a role. $VIX^2$ is the fair strike of a variance swap, which can be replicated using vanillas in a model-independent manner (recap here). To express it in terms of expectation, we can write

$$ VIX_0^2 = E^Q_0[V(0,0,30)] $$

However, the VIX index and hence its related derivatives (futures and options) are quoted not in $VIX^2$ terms, but simply in $VIX$ itself (in other words, as we see that VIX is at 14.5, the fair strike of a variance swap is 0.145 * 0.145 = 0.021). If we consider a future contract, in general if the underlying is commodities, FX or stocks we can simply apply the no-arbitrage rule and arrive at the future price using the appropriate funding/dividend/storage/convenience yields. However, these simply don't exist for VIX, which is not a tradable entity. So, we have to rely on

$$ F(0,T) = E^Q_0[\sqrt{VIX_T^2}] \neq \sqrt{E^Q_0[VIX_T^2]}$$

This is where the convexity adjustment comes in (in fact it should be a 'concavity adjustment', as the square root function $\sqrt{x}$ is concave in $x$). We can approximate the future price as

$$ F(0,T) \approx \sqrt{E^Q_0[VIX_T^2]} - \frac{var^Q_0[VIX_T^2]}{8[E^Q_0[VIX_T^2]]^{2/3}}$$

(see, for example, Zhu and Lian,  An Analytical Formula for VIX Futures and its Applications, 2010) The negative sign of the higher order correction term clearly shows concavity.