### Author Topic: Delta-Gamma-Theta Approximation  (Read 3316 times)

0 Members and 1 Guest are viewing this topic.

#### CRS245

• *Senior Staff
• MVP
• Join Date: Jan 2009
• Posts: 30324
• Bonus inPoints: 27
• Fantasy Sport:
##### 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 inPoints and the Invitationals.

### Quick Profile

Did you miss your activation email?

### Chat Room

• indiansnation: Looks Eric im good right know
Yesterday at 10:07:39 PM
• indiansnation: Oooops
Yesterday at 10:07:52 PM
• Eric: Really nobody? I don't believe that lol
Yesterday at 10:32:13 PM
• indiansnation: I actually like my team
Yesterday at 11:11:59 PM
• Eric: Anybody here, cept Brian lol
Yesterday at 11:56:43 PM
• indiansnation: Really use to like u eric
Today at 12:17:56 AM
• indiansnation: I have to have time evaluate my teams has even been a full day yet
Today at 12:19:13 AM
• Eric: hahah im giving you a hard time
Today at 12:21:23 AM
• Corey: Free NFL keeper league..... 5 spots left    [link]
Today at 12:22:11 AM
• Eric: Whats up Corey
Today at 12:27:05 AM
• Corey: Just got home from the mini vaca
Today at 12:28:20 AM
• Eric: Nice, i workedall weekend
Today at 12:29:48 AM
• Corey: Not cool
Today at 12:30:58 AM
• Eric: Nope, it was friggen hot too. 102 today
Today at 12:31:58 AM
• Corey: Dang. tad hot lol
Today at 12:32:57 AM
• Eric: yeah, went outside as little as possible
Today at 12:33:09 AM
• Corey: Laying on the floor next to the crib
Today at 12:46:12 AM
• indiansnation: Pansy eric
Today at 12:46:26 AM
• Corey: To old to be sleeping on the floor
Today at 12:46:40 AM
• indiansnation: 91 in ohio
Today at 12:46:53 AM
• BHows: Corey- PM
Today at 12:47:26 AM
• Eric: been there done that! How is your little one Corey?
Today at 12:47:57 AM
• indiansnation: Corey how old is your child
Today at 12:48:02 AM
• redbeard82: Hot as BooYah! here in Dallas.  Sat in the sun for the Giants - Rangers game. Thank god it was fast (though crappy outcome)
Today at 12:57:53 AM
• Corey: 9.5 months
Today at 01:02:05 AM
• Corey: She has a nasty cold and well im protective lol
Today at 01:02:48 AM
• Corey: Thank you Rick
Today at 01:03:12 AM
• Eric: Doesnt that suck Corey, seems like there isnt much you can do
Today at 01:05:23 AM
• redbeard82: hope she gets better soon corey. that'll be me in about a year
Today at 01:06:17 AM
• Corey: her 1st one
Today at 01:06:23 AM
• Eric: Teething SUCKS
Today at 01:06:40 AM
• Corey: never even had a runny nose yet
Today at 01:06:52 AM
• Corey: We're in the middle of teething now.
Today at 01:07:27 AM
• Eric: Yeah it sucks lol. He is getting his molars now
Today at 01:08:01 AM
• Corey: Red your gonna love it tho
Today at 01:08:57 AM
• Eric: It is great, nothing better in life tbh
Today at 01:09:14 AM
• Corey: Boy or girl?
Today at 01:09:26 AM
• Corey: im out for the night,  see you guys in 7 hrs
Today at 01:12:40 AM
• Eric: later Corey
Today at 01:13:00 AM
• indiansnation: Night corey
Today at 01:20:26 AM
• JMAC: anyone having an error on fantrax for 108? says cannot access.
Today at 01:35:05 AM
• Jwalk100: yes, i get the same error
Today at 01:36:52 AM
• JMAC: says not a member or league not valid. wtf. i need to set my lineup and it is a money league
Today at 01:37:52 AM
• JMAC: someone with admin rights needs to get fixed. lol
Today at 01:38:19 AM
• redbeard82: looks like a fantrax issue to me JMAC
Today at 01:39:48 AM
• redbeard82: don't have admin rights, but i'd assume if you posted your moves on the board they''d be able to be input retroactively
Today at 01:40:24 AM
• JMAC: ill hope its fixed before games tomorrow. odd as all my other leagues are accessible but not 108
Today at 01:41:36 AM
• redbeard82: yeah. it does that sometimes. i have no idea why but have encountered it with the one i commish
Today at 01:49:09 AM
• themarksman13: Man...the Maitan hype is unreal. Do I take JJ Schwarz, Ryan Boldt and Starlin for Maitan
Today at 01:51:43 AM
• Eric: Martin, sent you a pm
Today at 02:10:41 AM