### Author Topic: Delta-Gamma-Theta Approximation  (Read 3713 times)

0 Members and 1 Guest are viewing this topic.

#### CRS245

• *Senior Staff
• MVP
• Join Date: Jan 2009
• Posts: 30364
• Bonus inPoints: 27
• Fantasy Sport:
##### Delta-Gamma-Theta Approximation
« on: March 23, 2010, 08:50:52 PM »

## Delta-Gamma-Theta Approximation

The definition of the Taylor Series is:

$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2+\ldots$

Let's do some one-to-one substitutions to make the Taylor Series fit our subject, option pricing.  Let  $x=S_t$ and $E=S_t-S_0$  then if the "output" is the Option Price ($f(x)=C(s_t)$), the 1st derivative with respect to stock price will be Delta and the 2nd derivative with respect to stock price will be Gamma.  The infinite amount of terms following the 3rd term would, in most cases, be relatively small.  Therefore, in this case, those terms can be replaced by one simple error term.

Rewriting the Taylor Series equation gives us:

$C(S_t)=S_0+\Delta E + \frac{1}{2}\Gamma E^2 + \text{error term}$

Example
Using the same parameters from the 2nd example, estimate the change in call's value if the stock price increases to 210.

Recall that $S_0=200, r=0.05,\sigma=0.2$  and $\Delta=0.8554$ , so the approximate value of the call is:

$C(S_t)=C(S_0)+\Delta E + \frac{1}{2}\Gamma E^2$
$C(S_t)=27.95+0.8554(10)+0.5\Gamma(100)=36.504+50\Gamma$

The value of the call is highly dependent upon Gamma, the 2nd derivative of option price with respect to stock price.  Concavity and convexity will only forecast if the stock will level off or change even more in value, so those aspects of Calculus are important for forecasting stock prices.  What can we expect Gamma to be in this example?

There really isn't enough information given to calculate or predict the Gamma.  All that we know is that Gamma will be the same whether it is a call or a put.  Assuming Gamma to be zero makes the option follow a more linear pattern which is not a good estimation of the option itself, so an arbitrarily small value for Gamma will suffice.

Let's assume that:  $\Gamma=0.02$

$36.504+50\Gamma=37.504$

The call value is expected to increase by 9.554, which is less than 10, the change in stock price.  This follows the laws of arbitrage and the increase in the call can be expected with the increase in the stock price.

The Error Term in Hedging
The approximation used in the last example is actually Delta-Gamma approximation.  To apply Delta-Gamma-Theta approximations to option values, the parameter of time must be introduced to the equation.  Specifically, the Greek of Theta must be used.  The additional error term, albeit small, will most often reduce the approximate option value because Theta is usually negative.  Why is Theta usually negative?  Theta measures the increase in option price with respect to the decrease in time to maturity, and options increase in value as maturity time increases due to extra room for volatility.

Less maturity time => Lower variance => Decreased expected value

The error term is measured in days, and time in the Delta-Gamma-Theta approximation is measured in years, so in order to keep all time variables equal, it must be converted like so:

$C(S_t)=S_0+\Delta E+\frac{1}{2}\Gamma E^2 + \frac{t\theta}{365}$

The number of days in a year is a matter of convention.  The banker may use 360 days whereas the actuary would use 365.25 days.  The investor may use actual number of days, 365 or 366, depending whether the current year is a leap year or not.

### References

W. McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006

### Contributors

Colby
« Last Edit: November 05, 2010, 06:35:16 PM by Colby »
Learn about inPoints and the Invitationals.

With Quick-Reply you can write a post when viewing a topic without loading a new page. You can still use bulletin board code and smileys as you would in a normal post.

Warning: this topic has not been posted in for at least 120 days.
Unless you're sure you want to reply, please consider starting a new topic.

Name: Email:
Verification:
Listen to the letters / Request another image
Type the letters shown in the picture:
Last name of MLB all-time HR leader:
Who won the 2015 mens NCAA basketball championship (school name only):

### Quick Profile

Did you miss your activation email?

### Chat Room

• redbeard82: sounds good brian. can you post? gotta get an mb2 one up
Today at 12:42:13 AM
• indiansnation: Just redo it red
Today at 12:42:26 AM
• redbeard82: can do ours after if necessary though
Today at 12:42:35 AM
• redbeard82: aight
Today at 12:42:46 AM
• indiansnation: I'm on my phone red.pain in ass to do it
Today at 12:42:58 AM
• redbeard82: i'll get it brian after i finish my MB2 one.
Today at 12:44:10 AM
• patjossom: Jake pm
Today at 12:45:37 AM
• redbeard82: posted sky!
Today at 12:47:20 AM
• indiansnation: Looking to move Phillips in ml
Today at 12:53:56 AM
• redbeard82: posted brian
Today at 12:54:23 AM
• Sky: Indians, which phillips?
Today at 12:54:39 AM
Today at 12:54:48 AM
• redbeard82: cheers!
Today at 12:54:57 AM
• redbeard82: andyscott OTC in 108
Today at 12:55:32 AM
• redbeard82: sky think you're on deck if you want to leave a proxy for the AM
Today at 12:57:21 AM
• Orange Country: aight dave enjoy doing that draft
Today at 12:58:14 AM
• Orange Country: you did BLB, 108, and MB II, I did BTL/SD
Today at 12:58:27 AM
• Orange Country: still gotta get BLB fantrax squared away
Today at 12:58:41 AM
• redbeard82: it mostly is
Today at 12:59:17 AM
• Orange Country: what we do for this site
Today at 01:00:00 AM
Today at 01:01:30 AM
• redbeard82: got it. thanks sky!
Today at 01:03:54 AM
• patjossom: Rob be polite and pick in FLG. Lol
Today at 01:07:11 AM
• Orange Country: like I said, crazy pat
Today at 01:07:27 AM
• patjossom: Cmon that's funny.
Today at 01:10:09 AM
• patjossom: You see what I did there.....right.....lol
Today at 01:10:43 AM
• patjossom: Giantjake hit me back. Let's work something out.
Today at 01:12:21 AM
• Giantjake: Gettin back to you now
Today at 01:13:46 AM
• Orange Country: eh you belong in the looney bin
Today at 01:14:41 AM
• redbeard82: Jake - please take a look at the Hoover thread when you get a chance. need a new contract in 108
Today at 01:16:22 AM
• Giantjake: ok ill take a look now. Thanks
Today at 01:17:24 AM
• Money ball: i just did a new segment on the 108 media check it out
Today at 01:44:00 AM
• ajm5551: Love the post Money ball
Today at 01:51:36 AM
• redbeard82: nice money
Today at 01:57:51 AM
• redbeard82: reid's built a pretty impressive system in too
Today at 01:58:03 AM
• redbeard82: speaking of 108...could use some bullpen arms there if anybody has to deal
Today at 02:01:55 AM
• Smurrman: i got hunter
Today at 02:20:56 AM
• redbeard82: not quite what i'm looking for
Today at 02:27:54 AM
• Smurrman: aw man, how about joe nathan? he is good
Today at 02:29:40 AM
• patjossom: Red is there a way to look at 2015 fantrax for 108?
Today at 02:35:03 AM
• patjossom: Or anyone 108 gms. Is it as easy as just sending a link?
Today at 02:36:47 AM
• Orange Country: workin on it pat
Today at 02:38:21 AM
• patjossom: Thanks
Today at 02:39:18 AM
• redbeard82: haha. if only we could turn back the clock smurr
Today at 02:39:27 AM
• redbeard82: pat - what are you wanting to know?
Today at 02:39:36 AM
• Orange Country: good to go pat
Today at 02:39:51 AM
• Orange Country: go to my leagues tab and look for 2015 108
Today at 02:40:15 AM
• patjossom: Just to see where teams finished last yr.
Today at 02:40:58 AM
• patjossom: Thanks Reid.
Today at 02:41:07 AM
• patjossom: See where the completion in the NL is
Today at 02:43:44 AM