Monday, June 15, 2009

Fixed Delta and Risk Graphs

I've received a few questions as to how to manage or adjust a delta neutral, positive theta position (see my last post on Clash of the Greeks). I've already introduced the idiosyncrasies of delta in past posts, but want to start a new series of posts covering delta hedging. I'll be using risk graphs quite frequently so let's kick off the new series by highlighting the intricacies of delta on a risk graph.

In my original post on delta I stated,

"Rate of Change: This is the characteristic of Delta, most people are familiar with. Delta measures an option’s sensitivity to movements in the stock price. More specifically it measures the change in an options value given a $1 increase in the underlying. Bullish positions have positive delta, bearish positions have negative delta. Remember the following table:
Positive Delta = Long stock, Long Calls, Short Puts
Negative Delta = Short stock, Long Puts, Short Calls "

There are 2 types of Delta: Fixed and Variable
Stock has fixed delta, Options have variable delta.

Fixed Delta: If I were to buy 100 shares of a stock, my position delta would be +100. This means if stock were to increase $1, my stock position would profit $100. Whether the stock increases or decreases in price, my delta position is fixed at +100. On the other hand were I to short 100 shares of stock, my position delta would be -100. If the stock were to increase $1, my position would lose $100. Whether the stock increases or decreases in price, my delta position is fixed at -100. Simply put, for stock positions the delta never changes.

What's it look like on a risk graph?

Long 100 shares of SPY
What can we glean from the risk graph?
1.As you can see, the curvature of the line never changes. Why dat? Because the underlying position has a fixed delta.
2. Positive delta positions increase from bottom left to top right

Short 100 shares of SPY

1. Once again, the curvature of the line never changes as the position has a fixed delta.
2. Negative delta positions decrease from top left to bottom right.

A Delta Neutral Position

1. The curvature of line never changes as in this example the position has a fixed delta.
2. Delta neutral positions are shown as a horizontal line. Whether the stock increases or decreases, the profit/loss remains the same.

Before I delve into variable delta it's important to re-iterate what happens to an option at expiration. Remember an options premium is comprised of both intrinsic and extrinsic value (Premium = IV + EV).
Intrinsic Value is the amount the option is In-The-Money (IV = stock price - strike price).
Extrinsic Value is the amount you pay primarily for time and implied volatility (EV = Premium - IV).
At expiration there is no Extrinsic value left in option premiums. Out-of-the-money options expire worthless and In-the-money options will trade at their intrinsic value (sometimes called trading at parity). At expiration, delta for an option will either be 0 or 100.

If SPY was trading at $92 at options expiration, then all call options with a strike <92>92 will have a delta of 0. Take a look at the expiration risk graph of a long call:
When ITM the risk graph looks just like a long 100 shares risk graph (delta = 100). When OTM the risk graph looks like a delta neutral position (delta = 0)

In our next installment we'll explore variable delta and how it plays out on a risk graph.