Calculate the Area under a Curve

Question

I would like to calculate the area under a curve to do integration without defining a function such as in integrate().

My data looks as this:

Date          Strike     Volatility 2003-01-01    20         0.2 2003-01-01    30         0.3 2003-01-01    40         0.4 etc. 

I plotted plot(strike, volatility) to look at the volatility smile. Is there a way to integrate this plotted "curve"?

Answer 1

The AUC is approximated pretty easily by looking at a lot of trapezium figures, each time bound between x_i, x_{i+1}, y{i+1} and y_i. Using the rollmean of the zoo package, you can do:

library(zoo)  x <- 1:10 y <- 3*x+25 id <- order(x)  AUC <- sum(diff(x[id])*rollmean(y[id],2)) 

Make sure you order the x values, or your outcome won't make sense. If you have negative values somewhere along the y axis, you'd have to figure out how exactly you want to define the area under the curve, and adjust accordingly (e.g. using abs() )

Regarding your follow-up : if you don't have a formal function, how would you plot it? So if you only have values, the only thing you can approximate is a definite integral. Even if you have the function in R, you can only calculate definite integrals using integrate(). Plotting the formal function is only possible if you can also define it.

Answer 2

Just add the following to your program and you will get the area under the curve:

require(pracma) AUC = trapz(strike,volatility) 

From ?trapz:

This approach matches exactly the approximation for integrating the function using the trapezoidal rule with basepoints x.