Author Topic: Delta-Gamma-Theta Approximation  (Read 3289 times)

0 Members and 1 Guest are viewing this topic.

Offline CRS245

  • *Senior Staff
  • MVP
  • *****
  • Join Date: Jan 2009
  • Posts: 30324
  • Bonus inPoints: 27
    • :PIT:
    • :PIT-NFL:
    • :Blank:
    • :Blank:
    • :PennState:
    • :UnitedStates:
    • :Blank:
    • View Profile
    • ProFSL
    • Buy me a beer
  • 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.

 

Forum Search


Quick Profile

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

ProFSL Fantasy Sports

* Chat Room

Refresh History
  • Vollmernator: The highest bidder wins players you don't know who wins player Til the deadline is
    Today at 04:12:59 PM
  • Vollmernator: I kinda like them
    Today at 04:13:32 PM
  • Jonathan: I don't like how if you bid way over the second highest bidder, that you still got to overpay. That is very unrealistic.
    Today at 04:14:47 PM
  • ajm5551: Hmmm
    Today at 04:15:18 PM
  • ajm5551: Interesting
    Today at 04:15:22 PM
  • Jonathan: Should only have to pay x more than second highest bidder.
    Today at 04:15:41 PM
  • AdamBombs: How are they done though? Via PM?
    Today at 04:17:52 PM
  • redbeard82: Adam - back
    Today at 04:18:15 PM
  • Jonathan: Looks like Joe Ross is going to re-appear.
    Today at 04:31:41 PM
  • redbeard82: Adam - back
    Today at 04:40:39 PM
  • scottnva: anyone ant to buy my MB2 team?
    Today at 04:47:32 PM
  • OUDAN: Why Scott?
    Today at 04:47:58 PM
  • OUDAN: PM me about it
    Today at 04:48:09 PM
  • redbeard82: Adam - replied
    Today at 04:51:04 PM
  • ajm5551: I'm looking for a LOR style type NBA league to join
    Today at 04:51:06 PM
  • ajm5551: Knicks looking to open up
    Today at 05:11:38 PM
  • ajm5551: *cap in LOR
    Today at 05:11:52 PM
  • Brent: Jonathan, I agree with you 100% on thd blind bidding.
    Today at 05:11:53 PM
  • Brent: Afternoon guys.
    Today at 05:12:06 PM
  • Lindner: Hey Brent.
    Today at 05:14:00 PM
  • Jdwalter21: Brent my question is what do you do if 2nd highest bid is 60m then top bidder bid 120m to ensure he got the player knowing his bid would be dropped down a lot? Sorry if it was a poor explanation
    Today at 05:15:05 PM
  • AdamBombs: If i have say 30m in cap.space. can i only have 30m out in bids?
    Today at 05:17:53 PM
  • Jonathan: Also, it shouldn't be anyone who has a team, determining who is winning any bids. I think with rules in place it can work.
    Today at 05:20:30 PM
  • Brent: Figure out a set amount to be bid over second.  < $10M = $.5M over, < $20M = $1M over, then go up in increments till all possible bid amounts are covered.
    Today at 05:20:57 PM
  • AdamBombs: Why not just a regular auction?
    Today at 05:25:44 PM
  • Jonathan: People hate auctions. LOL  Its like death.
    Today at 05:26:11 PM
  • AdamBombs: I guess the silent is quicker though
    Today at 05:26:25 PM
  • AdamBombs: Are the bids sent via PM?
    Today at 05:28:20 PM
  • Brent: Silent could be the answer to the messes we have now, but it just needs some tweeks.
    Today at 05:28:24 PM
  • Jdwalter21: For BoB, i will get everyone's email address. Don't reply all, but reply to me with your bids or passes for each player. once i get everyone's responses i will post the winners and go on to the next round of players
    Today at 05:43:22 PM
  • AdamBombs: Got it.  Who will we email?  I agree it should be someone who isnt bidding
    Today at 05:49:35 PM
  • AdamBombs: Nvm i see you said wed email you
    Today at 05:49:56 PM
  • Jonathan: JPP about to be a FA. Dumbass
    Today at 05:50:00 PM
  • Jonathan: Plaxico 2.0
    Today at 05:54:24 PM
  • ajm5551: Knicks block just posted on LOR
    Today at 06:44:24 PM
  • Daddy: Doesnt look like your team page is updated ajm
    Today at 07:03:57 PM
  • Daddy: Maybe post the contracts on your block to help fan interests
    Today at 07:04:38 PM
  • ajm5551: Ha I put the updates team on my block. Just didn't put contracts in
    Today at 07:09:17 PM
  • ajm5551: Ya*
    Today at 07:10:07 PM
  • Daddy: D Rose contract is a monster Alex
    Today at 07:11:32 PM
  • Daddy: If your willing to eat some of it, we can talk for sure
    Today at 07:12:21 PM
  • ajm5551: Some? Yea not too much. It already has 3m paid on it this year and 4.5m next year
    Today at 07:14:00 PM
  • ajm5551: Shoot me an offer
    Today at 07:14:10 PM
  • ajm5551: Jdwaklter.... Pm back ur way
    Today at 07:14:28 PM
  • Daddy: I will
    Today at 07:14:48 PM
  • Daddy: Gotta be careful trading in the division :)
    Today at 07:18:54 PM
  • Vollmernator: jdwalter I posted my keepers in bob for browns
    Today at 07:41:51 PM
  • jblum: USA two goals in 5 minutes
    Today at 08:09:17 PM
  • nedsports: 3-0
    Today at 08:14:13 PM
  • nedsports: 4-0
    Today at 08:15:36 PM