Author Topic: Delta-Gamma-Theta Approximation  (Read 4769 times)

0 Members and 1 Guest are viewing this topic.

Offline Colby

  • MLFB Founder
  • MVP
  • ***
  • Join Date: Jan 2009
  • Posts: 30420
  • Bonus inPoints: 27
  • Fantasy Prestige 57
    • View Profile
    • Buy me a beer
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 :Commish: 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:
Type the letters shown in the picture
Listen to the letters / Request another image
Type the letters shown in the picture:
Last name of MLB all-time HR leader:
New England Patriots QB last name:

Forum Search


Quick Profile

 
 
Welcome, Guest. Please login or register.
Did you miss your activation email?

* Chat Room

Refresh History
  • indiansnation: Fnhl wild looking to move jeff carter for specs and picks
    Yesterday at 10:49:49 AM
  • AntMan: PM for JC Indians
    Yesterday at 11:05:40 AM
  • indiansnation: Antman looking at it right know
    Yesterday at 11:10:04 AM
  • indiansnation: Antmanpm
    Yesterday at 11:20:26 AM
  • AntMan: Sounds acceptable
    Yesterday at 11:26:08 AM
  • AntMan: Posted Indians
    Yesterday at 11:55:45 AM
  • blkhwkfn: Good Afternoon FNHLheads
    Yesterday at 03:42:40 PM
  • hockeygoon33: Whats up blkhwk!
    Yesterday at 04:11:05 PM
  • blkhwkfn: Not a heck of a lot just having a 007 marathon how about you
    Yesterday at 05:33:47 PM
  • hockeygoon33: Just turned on Goon2 going over rosters.
    Yesterday at 06:16:40 PM
  • blkhwkfn: Nice, break out that feather duster lol
    Yesterday at 06:19:02 PM
  • hockeygoon33: Lol. Yeah figured its going to be a while!
    Yesterday at 06:21:03 PM
  • mcrow: Looking for some new owners for a 7-player keeper league. DM me if interested.
    Yesterday at 06:23:05 PM
  • blkhwkfn: Daddy would kill in that league crow
    Yesterday at 06:25:47 PM
  • blkhwkfn: do new owners have the option to carry on the previous owners roster to pick keepers
    Yesterday at 06:26:31 PM
  • blkhwkfn: not both one or the other I would imagine.
    Yesterday at 06:26:58 PM
  • mcrow: Lots of good keeper options available during new owner draft ... Burns, Ovechkin, Tavares,  Lundqvist,  Pacioretty, Getzlaf, Oshie ... just to name a few
    Yesterday at 06:28:44 PM
  • blkhwkfn: thats a strong list!
    Yesterday at 06:29:14 PM
  • mcrow: Plan is to have a new owner keeper draft from available players
    Yesterday at 06:29:30 PM
  • blkhwkfn: My issue with the league the last couple years was the inability of CBS to work propperly on my mobile
    Yesterday at 06:30:15 PM
  • blkhwkfn: I am back on line with a computer because my job calls for it thank god lol
    Yesterday at 06:30:49 PM
  • mcrow: Yeah, it is on CBS
    Yesterday at 06:31:00 PM
  • blkhwkfn: this is the first computer that I bought all the others were the companies that I worked for
    Yesterday at 06:31:38 PM
  • mcrow: Only $35 for the year. Hit me up if interested.
    Yesterday at 06:32:00 PM
  • blkhwkfn: I think it will fill up quick
    Yesterday at 06:32:38 PM
  • blkhwkfn: I am already in silly
    Yesterday at 06:32:49 PM
  • blkhwkfn: blkice
    Yesterday at 06:32:52 PM
  • blkhwkfn: are you going to have paypal set up?
    Yesterday at 06:33:44 PM
  • mcrow: I know :)
    Yesterday at 06:34:08 PM
  • mcrow: That was for others
    Yesterday at 06:34:13 PM
  • mcrow: Yeah, I will collect via paypal
    Yesterday at 06:34:26 PM
  • blkhwkfn: LMAO oops
    Yesterday at 06:34:36 PM
  • blkhwkfn: sweet thats the best way
    Yesterday at 06:35:00 PM
  • indiansnation: Evening all
    Yesterday at 06:47:36 PM
  • blkhwkfn: howdy
    Yesterday at 06:48:52 PM
  • indiansnation: Man the blackhawks look good this yr
    Yesterday at 06:49:05 PM
  • blkhwkfn: they could be stanley cup winners of the preseason
    Yesterday at 06:51:19 PM
  • blkhwkfn: they are coached well and Donato on the PK special teams is a great addition
    Yesterday at 06:51:56 PM
  • indiansnation: Watching them know 3-0 over redwings.
    Yesterday at 06:53:27 PM
  • blkhwkfn: Granato
    Yesterday at 06:53:28 PM
  • blkhwkfn: and actually Ulf Samualsson is leading the PK coaching but the crew behind the bench is impressive
    Yesterday at 06:54:42 PM
  • indiansnation: Ok 4-0
    Yesterday at 06:54:46 PM
  • blkhwkfn: it keeps going  lol
    Yesterday at 07:30:08 PM
  • blkhwkfn: gotta run
    Yesterday at 07:30:22 PM
  • blkhwkfn: Pretty quiet around here
    Today at 01:33:20 AM
  • mcrow: Calm before the reg season
    Today at 10:10:26 AM
  • blkhwkfn: Hurricane FNHL
    Today at 10:20:37 AM
  • indiansnation: Johnathan interested in your. leo at work send me a pm what u want
    Today at 06:29:17 PM
  • indiansnation: Will reply tonight around 1130pm
    Today at 06:29:47 PM
  • cho34: hey brian need your keepers for the farm so we can start draft
    Today at 06:51:45 PM