We already mentioned that there is a major distinction between the Delta and the Theta measure. Delta measures sensitivity of the call option value with respect to the random movements of the stock price. We shall later see that we may need to take account of a more precise approximation to the call value. The Delta approximation, or hedging, amounts to utilizing the slope of the graph of the call option value at a certain point to produce a linear approximation to the true value of the call option. It is, in fact, the Taylor series approximation to the function. However, if the function is very "nonlinear" around the point in question, the approximation will not be very accurate and may not be adequate for the purpose intended.

To increase the quality of the approximation one can take account of the second derivative of the call option value with respect to the stock price. The second derivative of the call price with respect to the stock allows us to include some information about the curvature of the graph around that point. The Gamma measure is the second derivative of the call price with respect to the stock price. Correspondingly, Gamma is the first derivative of Delta with respect to the stock price. Gamma is represented by the Greek letter and can be stated as

= .

(7.10)

The appendix to this Chapter shows that the equation for Gamma is

N'( ).

(7.11)

We use MAPLE to define Gamma as the second derivative of the call option value with respect to the stock price as below:

> Gamma:=unapply((diff(Bs(K,t,Pr,sigma,rf),Pr$2)),K,t,Pr,sigma,rf):

> Delta:=unapply((diff(Bs(K,t,Pr,sigma,rf),Pr)),K,t,Pr,sigma,rf):

.

(7.12)

Equation (7.12) and Figure 7.12 illustrates this second-order approximation. In this figure the true value of the call option (the green curve with positive y-intercept) is graphed in the same plane along with the second-order approximation to the value of the call option which is calculated (the red curve with negative y-intercept) using Gamma. Note that this time, unlike the approximation which used Delta, the approximating function is quadratic.

> plot([Bs(70,.5,S,0.18,0.12),[[80,0],[80,Bs(70,.5,80,0.18,0.12)]],Bs(70,.5,80,0.18,0.12)+(S-80)*Delta(70,.5,80,0.18,0.12)+((S-80)^2)*Gamma(70,.5,80,0.18,0.12)/2],S=60..100,color=[green,blue,red],labels=[`Stock Price`,`Call Value`],thickness=2,title=`Figure 7.12: Quadratic Approximation Using Gamma and Delta`,titlefont=[TIMES,BOLD,10]);

The reader can also execute the command below to plot the linear approximation with Delta only (the lower brown curve with the largest x-intercept) the quadratic approximation with Gamma (the middle red curve with the smaller positive x-intercept) in the same plane as the value of the call option (the upper green curve with positive y-intercept). This is demonstrated in Figure 7.13. Note that the approximation with both Gamma and Delta is better than the one with Delta only.

> plot([Bs(70,.5,S,0.25,0.08),[[80,0],[80,Bs(70,.5,80,0.25,0.08)]],Bs(70,.5,80,0.25,0.08)+(S-80)*Delta(70,.5,80,0.25,0.08)+((S-80)^2)*Gamma(70,.5,80,0.25,0.08)/2,Bs(70,.5,80,0.25,0.08)+(S-80)*Delta(70,.5,80,0.25,0.08)],S=60..100,color=[green,blue,red,brown],labels=[`Stock Price`,`Call Value`],view=[70..90,0..27],thickness=2,title=`Figure 7.13: Quadratic Approximation vs. Linear Approximation`,titlefont=[TIMES,BOLD,10]);

We can also plot, Figure 7.14, the graph of Gamma for an option which is in-the-money (the lowest red graph), at-the-money (the upper blue graph hat does not intersect the axes), and out-of-the-money (the middle green graph that intersects the at-the-money graph).

> plot([Gamma(70,t,80,0.18,0.12),Gamma(80,t,80,0.18,0.12),Gamma(90,t,80,0.18,0.12)],t=0..1,labels=[`Time to Maturity`,Gamma],color=[red, blue,green],thickness=2,title=`Figure 7.14: Gamma of an In-the-Money, At-the-Money, and Out-of-the-Money Option`,titlefont=[TIMES,BOLD,10]);

We now move on to the investigation of other measures of sensitivity. Given the detailed explanation provided for these first three sensitivity measures, the exposition and description for the others can be shortened. The measures to be defined henceforth are all based on the first-order Taylor series expansion. In other words, these measures use only the slope (not any measure of curvature, as was the case with Gamma) of the graph of the option value with respect to the parameter in question. Hence, these measures induce a linear approximation to the true option value.

The sensitivity measures to be discussed in the rest of this Chapter are with respect to parameters, which are assumed to remain constant in the Black-Scholes derivation. They even differ from the measure Theta. Theta measures the sensitivity with respect to time. The time indeed proceeds to maturity, albeit in a very deterministic way. The Delta and Gamma measures were with respect to random movements of the stock price.

The remaining sensitivity measures are defined with respect to interest rate and volatility. They differ from Delta, Gamma, and Theta since the parameters are assumed fixed. These measures will be informative regarding how to approximate errors in the calculation of the option price if, for example, there were an error in estimation of either the volatility or the risk free rate of interest. These measures can also be used as an ad hoc technique for calculating the value of the option if the interest rate is not fixed or if the volatility is assumed to be nonstationary, but rather varies with time.