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

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#### CRS245

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##### 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

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