Author Topic: Delta-Gamma-Theta Approximation  (Read 2950 times)

0 Members and 1 Guest are viewing this topic.

Offline CRS245

  • *Senior Staff
  • MVP
  • *****
  • Join Date: Jan 2009
  • Posts: 30191
  • Bonus inPoints: 27
    • :PIT:
    • :PIT-NFL:
    • :Blank:
    • :Blank:
    • :PennState:
    • :UnitedStates:
    • :Blank:

    • 0_: :EPL13INVITEE: :NFL13INVITEE: :EPL12INVITEE: :NHL12INVITEE: :NFL12INVITEE: :NBA12INVITEE: :MLB12INVITEE: :NFL11FINALIST: :NHL11INVITEE: :NBA11INVITEE: :MLB11INVITEE: :MLB10INVITEE: :MLB09CHAMP:
    • View Profile
    • ProFSL
  • Fantasy Sport: :MLB:
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.

 

Quick Profile

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

ProFSL Fantasy Sports

* Chat Room

Refresh History
  • Jonathan: 1st overall was LUCK  :P
    Today at 04:53:07 AM
  • Jonathan: talk about a drop off
    Today at 04:53:19 AM
  • Brent: Yeah, holy crap
    Today at 04:57:14 AM
  • Brent: I do have Mariota, Ty Montgomery and Su'A Cravens.
    Today at 04:58:30 AM
  • Jonathan: 1 bad draft sets you back a lot here
    Today at 04:58:41 AM
  • Brent: In the pipeline
    Today at 04:58:43 AM
  • Jonathan: My  last class was Charles Sims lol
    Today at 04:59:05 AM
  • Jonathan: You had like half the draft
    Today at 04:59:39 AM
  • Jonathan: NFLC is gonna be all about draft too, those picks are gold
    Today at 05:01:40 AM
  • Jonathan: only way to get cheap contracts.
    Today at 05:07:24 AM
  • Jonathan: I found the trade icon, 10/22/13 a year ago
    Today at 05:10:00 AM
  • Jonathan: alright, enough for 1 night, down to 31 from 40 pages
    Today at 05:15:06 AM
  • Jdwalter21: Morning everyone
    Today at 09:54:21 AM
  • andyscott23: Good Morning
    Today at 10:15:42 AM
  • Brent: Morning guys
    Today at 10:32:22 AM
  • Jdwalter21: anything exciting happening today?
    Today at 10:48:16 AM
  • Brent: LSU vs Ole Miss
    Today at 10:51:51 AM
  • Tubbs: El Classico
    Today at 10:52:19 AM
  • Jdwalter21: Anyone looking to deal in any league i'm in
    Today at 11:12:41 AM
  • nedsports: Ole Miss all day
    Today at 11:20:00 AM
  • Brent: Based on just rankings sure, but there is more to it than that.
    Today at 11:22:45 AM
  • Vollmernator: Brent did you get my payment for NFLC
    Today at 11:25:07 AM
  • Brent: I did, you are updated.
    Today at 11:25:39 AM
  • Brent: Thank you
    Today at 11:25:50 AM
  • andyscott23: Can the LOR TC vote on Clip/Jazz deal. It has been posted for 5 days. Thanks
    Today at 11:34:21 AM
  • Orange Country: dang it Brent
    Today at 12:24:22 PM
  • Brent: What's up Reid?
    Today at 12:27:44 PM
  • Brent: What did I do?
    Today at 12:29:33 PM
  • Orange Country: took someone I wanted
    Today at 12:31:29 PM
  • Brent: Ah, must have been McKinney or Judge.
    Today at 12:37:00 PM
  • redbeard82: judge was definitely on my short list
    Today at 12:41:34 PM
  • redbeard82: though you're probably saving me from myself...i keep winding up with too many OFs
    Today at 12:41:57 PM
  • Orange Country: yea I was looking at both
    Today at 12:42:27 PM
  • Orange Country: no trading or FA for brent til 2016
    Today at 12:42:52 PM
  • redbeard82: nuts though albies was higher on my list
    Today at 12:46:48 PM
  • Corey: notthat took my next pick
    Today at 12:50:52 PM
  • Brent: That is okay, want to see my guys develop.
    Today at 12:54:33 PM
  • Tubbs: Madrid -Barca already awesome. Ray Hudson already 4 orgasms
    Today at 01:11:30 PM
  • Jdwalter21: Anyone wanna do some dealing?
    Today at 01:20:35 PM
  • Brent: I always want Jarvis Landry.
    Today at 01:30:36 PM
  • Jdwalter21: Yea, won't be moving him.
    Today at 01:32:19 PM
  • Jdwalter21: Delmas is about the only one in my starting lineup that i could move
    Today at 01:35:16 PM
  • OUDAN: :DAL: always looking
    Today at 01:37:11 PM
  • Jdwalter21: I way overpaid for Chris Hogan. I just see him as the second coming of Wes Welker
    Today at 01:39:03 PM
  • Brent: Yeah, I wouldn't trade the next Anquan Boldin either.
    Today at 01:39:59 PM
  • OUDAN: Looking to move a LB for DE or CB
    Today at 01:46:23 PM
  • Jdwalter21: Brent what are u looking for?
    Today at 01:46:59 PM
  • Brent: I non injured RB.
    Today at 01:55:20 PM
  • Brent: A non injured RB.
    Today at 01:55:28 PM
  • icehouse325: Brent just curious why you havent made any free agent bids in NBA invitational
    Today at 01:57:07 PM

Forum Search