## Tuesday, March 23, 2010

### The Straddle, Volatility Plays, and the Greeks – Part 1

Long overdue for this blog is a discussion of the Greeks. While I highly recommend a more detailed study for someone who is considering or has just begun trading options, this should provide a decent overview. For this example we will use the purest of volatility plays: the straddle, which involves either buying or selling the same strike and month call and put. Most option traders are familiar with this position and understand that there is considerable risk associated with it. Usually the choice is for the at-the-money options since this maximizes premium collection on the short side and provides the least expensive way to bet on significant movement in either direction on the long side. Break even calculations are fairly straight forward; strike price +/- total premium collected or paid.

Since both options ultimately decay to zero or parity, the holder of the short position wants the stock, or in this example the SPYs, to go nowhere over the holding period, whereas the holder of a long straddle wants the stock to move as far from the strike as possible in either direction. Simple enough, right? Wrong. As a side note, I chose the SPYs because the bid/ask spreads are narrow and in this case there was only a $.03 difference between buying and selling the straddle, however for most individual equities the spreads are much wider.

The probability of a stock or ETF (or whatever the underlying is) staying exactly in one place or moving strongly in only one direction is small, so one must consider hedging the position at some point. Before you can do this you must understand the Greeks. We talked about vega (v) in the discussion on implied volatility, now we need to consider delta (∆), gamma (Γ), and theta (θ). In the case of the long straddle, a position of 10 contracts of both calls and puts, the Greeks are 1) long 110 ∆'s (when I priced this example the SPYs were around 116.50 and implied volatility was around 14.5%, so the calls had 11 more ∆'s than the puts), 2) long 180 Γ, which means that for every $1 move higher the position will get longer by approximately 180 ∆'s and for every $1 move lower the position will get shorter by approximately 180 ∆'s, 3) short -$75 in θ, meaning that each day, as a result of time decay, the value of the straddle will decline by approximately $75, and 4) long 240 v, so that for a change of 1 point in implied volatility the position will gain approximately $240 in value if the move is higher or lose the same amount if lower. Of course all the Greeks are reversed for the short straddle and the reason I keep saying "approximately" is that as prices change and time passes these values change incrementally.

For the long straddle one must be aware of the vega since changes in implied vol also impact the other Greeks, but as a standalone position there is not much you can do about it as far as hedging. So the focus is on the other three. You can of course do nothing during your holding period and hope that the SPY finishes more than $3.57 (the total premium paid) away form 116, but many people opt to ∆ hedge. The most you can ever be long or short with this position is 1000 ∆'s (10 contracts * 100 shares per contract), but unless the SPYs gap largely in one direction or another, the straddle is likely to have no more than 700 ∆'s if there is any meaningful time left to expiration and that would still require a fairly significant move.

One approach to ∆ hedging is on a percentage basis. The straddle cost approximately 3% of the value of the index, so if you are actively hedging you might choose to flatten out on a 1.5% move as an initial threshold. Let's say that that move happens after holding the position for 3 days and the SPY is trading at 118.25. The Greeks are now: ∆ = +440, Γ = +150, and θ = -$72. The straddle is now worth approximately $3.85 (we are assuming constant implied vol for simplicity, although, as always, this may not be the most accurate assumption). Since we are ∆ hedging we decide to sell 440 SPY at 118.25. As you may know, if you are familiar with ∆, or may have guessed by reading this, one definition of ∆ is the hedge ratio and it tells us how much of the underlying we need to buy or sell to remain neutral to price direction.

After two more days the SPY dips to 116 and net of our short stock position we are short approximately 440 ∆'s, which we buy back and pocket $990, and the straddle is now worth approximately $310. We are again price neutral but our Γ has increased to +200 and the options are decaying at a net rate of $85 per day. These are two important points. As expiration nears and the stock stays near the straddle strike gamma increases along with the rate of decay (θ). Intuitively this makes sense because the closer we are to expiration the greater the probability is that the put or the call will finish with 100 ∆'s as they approach their terminal value of either zero or parity.

Obviously this is a discussion that is going to take more than one posting so we will continue this next time.

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