Linearization of Options

In a sense, trading is easy, all you need to remember is "buy low, sell high, repeat."  Obviously such a simple "strategy" misses 2 cases that would make it impossible to guarantee profits: sometimes you buy low and it keeps going lower, and sell high and it keeps going higher.  But this is not what we are going to discuss here.  We will talk about something option traders usually do -- linearize so that they can apply the traders credo of "buy low, sell high". 

Options are intrinsically nonlinear;  such property gives rise to the convexity in an option portfolio.  When managed properly, the convexity makes it possible to have returns that are higher than market average with less correlation with the market. However, directly dealing with the nonlinearity makes the trading decision making process cumbersome.  Linearizing the options makes it possible to talk about them in a similar way as talking about stocks. Volatility as well as the first order Greeks (Delta, Vega, Theta, Rho, etc.) are all part of results from the linearization process.  

Linearizing a convex asset like options is an ingenious idea.  Traders now can talk about volatility and Vega like talking about stocks with prices and shares. We can build a position to certain Vega size, long or short, then have a pretty good idea of the profit and loss when the volatility goes higher or lower.  We can even consider Vega as “inventory”, trade it like commodities.  Volatility trading firms like us can work as a "warehouse" of Vega, just like equity funds are warehouse of stocks.

While we use such linearized attributes in trading, it is crucial to remember that options are still nonlinear, especially in the following two areas:  Gamma, and path-dependency.  First, Gamma, a second derivative term, gives the convexity of options.  Unlike typical goods, where larger quantity usually lower the per-unit prices (a concave behavior), Gamma causes the delta to get longer as the base price increases, or get shorter when decreases.  When option contracts get closer to the expiration, Gamma for the options with strikes close to the spot grows, while options that are farther away from spot, Gamma drops off – both changes rapidly as expiration get near.  Actually, options offer an interesting chance for the mathematical minded to observe the Dirac Delta function in action at 4pm on every option expiration Friday.   While other linearized quantities change also as the base price changes, none is as dynamic as Gamma as time goes.  Such characterization makes it dangerous to apply the “buy low/sell high” credo to volatility, especially when the options are near the expiry and the Gamma change is accelerating.  Buy an option at 20 vol and sell it later at 25 vol doesn’t necessarily mean profit. 

Path dependency is the second monkey wrench that makes it dangerous to think of volatility in term of low/high.  Volatility doesn’t exist in the market per se, instead, it is realized through the hedging process.  When the options move toward expiration, the stock movement pushes an option through areas where its Gamma has big variations.  The profit/loss of the option depends on how the spot moves: does it trending up or trending down? Or chopping? Etc.  The profit/loss can no longer be estimated by looking at the beginning and end point of the volatility.

In short, when talking about Vega, or buy low/sell high in vol, it generally only works for the farther term options.  The closer the options get to expiry, the less meaningful it is to talk about Vega and vol.

In recent years, there are 2 general trends in the US equity option market places:  first, weekly options have become much more common (currently over 200 names have weekly options, with daily volume of nearly 1/5 of the total equity option volume).  Secondly, strike gaps are decreasing.  In the SPY for example, the strike gap in 2006 was $1 to $2.5, but that has narrowed down to 0.5.  Imagine that on an expiration day, a 2% move in S&P 500 can cause SPY to zip through 4 or 5 strikes in the expiring SPY options!  That kind of exciting bumpy ride used to only exist after a company announcing earning (remember GOOG in April 2008 after the first quarter earnings, stock jumped through 8 strikes).  Both of these trends make for more option Gamma plays. Nonlinearity is exciting for option traders, but it is important to remember that the common concept we talk about in trading, like the volatility, Delta and Vega, etc., all entail linear approximation, and we have to keep an alert eye on the nonlinearity and convexity that we both love and hate.