Author Topic: Delta-Gamma-Theta Approximation  (Read 2994 times)

0 Members and 1 Guest are viewing this topic.

Offline CRS245

  • *Senior Staff
  • MVP
  • *****
  • Join Date: Jan 2009
  • Posts: 30236
  • 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
  • Corey: These players are gone [link]
    Today at 11:19:37 AM
  • Corey: Tons of talent available
    Today at 11:19:43 AM
  • OUDAN: Hard to tell what's left that way is there like an updated fantrax?
    Today at 11:20:48 AM
  • Corey: Added you to my team in fantrax
    Today at 11:23:12 AM
  • OUDAN: You the man
    Today at 11:23:16 AM
  • Corey: Sooner you should join the NBA version
    Today at 11:25:01 AM
  • OUDAN: I thought about it, payouts arent much enough for my taste lol
    Today at 11:26:05 AM
  • Corey: There more after yr 1. IF we start this year then the winner just gets a little bit and the rest of the entry carries to year 2
    Today at 11:27:16 AM
  • Corey: New Owner in Scouting Department [link]
    Today at 11:28:58 AM
  • OUDAN: Ill do the Basketball one hell might as well whats the name of it?
    Today at 11:30:00 AM
  • OUDAN: 3 in the key?
    Today at 11:30:44 AM
  • Corey: yep 3 in the key
    Today at 11:32:46 AM
  • OUDAN: Still looking over BT in fantrax
    Today at 11:33:10 AM
  • OUDAN: I thought Rendon was won?
    Today at 11:33:17 AM
  • Corey: The players stickied on the ProFSL board have not been added to fantrax yet
    Today at 11:33:43 AM
  • Corey: Rendon was won
    Today at 11:33:49 AM
  • OUDAN: Gotcha
    Today at 11:33:52 AM
  • Corey: There is a ton of talent left
    Today at 11:34:03 AM
  • OUDAN: Whats the entry fee?
    Today at 11:34:12 AM
  • OUDAN: Only concern is getting bids in by the 28th
    Today at 11:35:01 AM
  • Corey: $50
    Today at 11:37:23 AM
  • OUDAN: Im leaving town tomorrow for a few days so bids by the 28th I guess id just have to do them today lol
    Today at 11:39:21 AM
  • Corey: I sent mine in yesterday lol
    Today at 11:39:58 AM
  • OUDAN: lol
    Today at 11:40:06 AM
  • OUDAN: Ok I'm in
    Today at 11:41:31 AM
  • OUDAN: Want me to post in the sign up?
    Today at 11:43:12 AM
  • utmbrad: bro hugs all around next group
    Today at 11:43:23 AM
  • OUDAN: this fool over paid Brown lol
    Today at 11:46:55 AM
  • Ace: welcome sooner
    Today at 11:48:07 AM
  • utmbrad: he wanted him for 5 over 5 yrs
    Today at 11:48:08 AM
  • utmbrad: welcome sooner
    Today at 11:48:42 AM
  • OUDAN: Yeah at that I guess wouldnt be bad
    Today at 11:48:47 AM
  • OUDAN: thank you sir
    Today at 11:48:53 AM
  • utmbrad: Corey, Aoki and Tanaka are listed twice
    Today at 12:11:37 PM
  • Corey: Fixing that and the MILR player. Tried to post VIA mobile and it didnt go well
    Today at 12:12:28 PM
  • utmbrad: understand completely
    Today at 12:13:18 PM
  • Corey: There fixed and Renfroe is fixed
    Today at 12:13:50 PM
  • utmbrad: awesome. thanks
    Today at 12:18:26 PM
  • OUDAN: Corey PMed u back
    Today at 12:21:00 PM
  • scottnva: mike05 your up in elite baseball
    Today at 12:22:12 PM
  • Corey: Moneyball :ATL: BLACK FRIDAY DEALS
    Today at 12:46:56 PM
  • Corey: [link]
    Today at 12:47:14 PM
  • Dan Wood: I'm not sitting in a tent for those
    Today at 12:53:37 PM
  • Dan Wood: Scutaro's back resmebles Batman's after Bane broke it
    Today at 12:54:08 PM
  • Dan Wood: I can't write this effing paper anymore
    Today at 12:54:38 PM
  • Vollmernator: :Kentucky:
    Today at 12:54:54 PM
  • Corey: Free for a pick
    Today at 12:56:04 PM
  • Dan Wood: Mark Sanchez or Drew Brees this week?
    Today at 12:59:53 PM
  • Dan Wood: I'm torn... Brees seems hurt and the Cowboys defense is trbl
    Today at 01:00:18 PM
  • Corey: Shaunn Hill
    Today at 01:00:28 PM

Forum Search