Author Topic: Delta-Gamma-Theta Approximation  (Read 2895 times)

0 Members and 1 Guest are viewing this topic.

Offline CRS245

  • *Senior Staff
  • MVP
  • *****
  • Join Date: Jan 2009
  • Posts: 29943
  • 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

 

Quick Profile

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

ProFSL Fantasy Sports

Community Chat Room

Refresh History
  • JohnnyWeissflog: Hey JD, afternoon to you too. Or evening in my case
    Yesterday at 03:06:09 PM
  • Orange Country: I am
    Yesterday at 03:32:55 PM
  • Orange Country: be home soon to sort bids
    Yesterday at 03:33:09 PM
  • kylerap: was just gonna mention fantrax says draft hasn't started so can't add guys that just started being won in premium fb
    Yesterday at 03:35:41 PM
  • Ace: commish needs to sort out transactions settings: allow claims and drops before draft
    Yesterday at 03:41:15 PM
  • Orange Country: OK will fix soon
    Yesterday at 03:42:05 PM
  • Gilly: friggin late bids
    Yesterday at 03:49:08 PM
  • Daddy: Happy early Birthday Sooner!!!
    Yesterday at 04:05:30 PM
  • Daddy: 30 aint chit! Life gets better and easier my friend.
    Yesterday at 04:06:18 PM
  • Tubbs: Gill!!
    Yesterday at 04:26:08 PM
  • Gilly: yes tubby
    Yesterday at 04:28:24 PM
  • ripper: Gilly give me your trext.....these pms are too hard
    Yesterday at 04:47:20 PM
  • Gilly: lol ok
    Yesterday at 04:47:51 PM
  • e40: Gilly when you get a chance sent you another pm
    Yesterday at 04:48:59 PM
  • Gilly: Im addressing all that right now e40....give me 10 mins
    Yesterday at 04:49:35 PM
  • e40: np take your time no rush
    Yesterday at 04:50:13 PM
  • Gilly: U can go ahead with what u asked tho...the entire draft itself will be ending shortly
    Yesterday at 04:51:27 PM
  • e40: Ok thanks
    Yesterday at 04:51:56 PM
  • Tubbs: Sorry gill u still there? Is Hogh Stick a one year league?
    Yesterday at 04:58:37 PM
  • Gilly: yes
    Yesterday at 05:00:18 PM
  • Gilly: one and done
    Yesterday at 05:00:21 PM
  • Gilly: No DL spots, No minors...if u grab guys on the DL or that is a prospect they take up a roster spot
    Yesterday at 05:02:49 PM
  • Gilly: only 8 bench spots too, very challenging :)
    Yesterday at 05:03:16 PM
  • Tubbs: Cool. Thx.
    Yesterday at 05:22:10 PM
  • norrya66: Gilly, you need to re-select in Puck Galore
    Yesterday at 05:41:44 PM
  • norrya66: guru, you here?
    Yesterday at 05:42:00 PM
  • Ace: anyone want a team in 2nd year nfl dynasty, free 10 team league starting 2 QB, rookie/FA draft starts tomorrow, team has RG3, Aaron Rodgers, MJD, Lesean McCoy, Vereen, Roddy White, Reggie Wayne, Robert Woods, Colby Fleener, Jordan Reed, Blackmon if he comes back, plus 6th pick in draft
    Yesterday at 05:49:55 PM
  • Anelson: that look like a decent team there
    Yesterday at 05:51:35 PM
  • kylerap: yeah not bad
    Yesterday at 05:54:21 PM
  • Ace: it is, owner has left site though (richowl)
    Yesterday at 05:55:40 PM
  • Gilly: i will
    Yesterday at 06:07:27 PM
  • Brent: Afternoon guys
    Yesterday at 06:10:09 PM
  • norrya66: Ace, what league is it?
    Yesterday at 06:23:43 PM
  • Ace: united states football league norry, using old usfl franchises, scoring is quite good to if i don't say so myself
    Yesterday at 06:27:48 PM
  • Ace: same as this league, formatting not very friendly though
    Yesterday at 06:35:44 PM
  • Ace: [link]
    Yesterday at 06:35:51 PM
  • Gilly: dont want me eh ace lol
    Yesterday at 06:48:23 PM
  • norrya66: i think i'll pass ace...thanks for the info though
    Yesterday at 06:50:17 PM
  • Ace: gills, when you said i will i thought u were answering norry
    Yesterday at 06:51:24 PM
  • Ace: who said something about a re-pick
    Yesterday at 06:51:36 PM
  • Gilly: oh hahahaha
    Yesterday at 06:52:10 PM
  • Ace: i'd be honoured to have you if you want in
    Yesterday at 06:53:49 PM
  • Gilly: oh muffin , I like this honoured stuff lol :)
    Yesterday at 06:58:40 PM
  • Ace: u in then?
    Yesterday at 07:04:40 PM
  • Ace: i assume yes so just added you on profsl and fantrax, check the scoring, check last year's player scoring as a better guideline than the projections which don't track all of our stats, read the rules and pm me if you have any questions
    Yesterday at 07:51:02 PM
  • Ace: sunday night feature is up for anyone in mlb casino [link]
    Yesterday at 07:55:20 PM
  • Corey: oudan on the clock
    Yesterday at 09:14:15 PM
  • Ace: corey 5 posts away from 40k
    Today at 12:48:17 AM
  • Brent: Anyone online?
    Today at 02:23:27 AM
  • Ace: yeah
    Today at 02:29:09 AM

Forum Search