Calculating Vega and other Greeks in Black-Scholes

A bunch of people have been Google searching for "How is Vega Calculated", and finding my post on Black-Scholes. I left out the calculation of vega, so I'll give the details.

Recall that a call price is a function

CP(S, X, t, v, r), where

S is the current stock price
X is the strike price
t is the time to expiration
v is the volatility assumption
r is the interest rate assumption and assumption for the expected growth rate of the underlying

Vega is defined to be the partial derivative d(CP)/dv.

Typically, the function CP is calculated via a binomial tree expansion. For American options this is necessary, because of the early exercise right. For European options, there is an explicit formula you can differentiate.

To calculate vega, you do a numerical approximation of the derivative. Suppose epsilon = 0.001. In that case, vega is (CP(v + 0.001) - CP(v) / 0.001). You can't make epsilon too small, because floating point round-off will introduce error if epsilon is really small. A value of 0.001 is good enough.

To calculate vega, you construct two binomial trees, numerically approximating the derivative.

That's how you calculate vega. It's very simple. It's just numerically approximating the derivative.

Rho is calculated similarly. Rho is d(CP)/dr. You calculate two separate trees and numerically approximate the derivative.

Delta, gamma, and theta can be calculated directly from the binomial tree. Delta is d(CP)/dS. Delta is your correct hedge, which you calculated when you priced the option. Gamma is d(delta)/dS = d(CP^2)/(dS)^2. Gamma can be calculated by looking at delta at the root and delta at the first level of the tree. Theta is d(CP)/dt. Theta can also be read directly off the tree, by comparing the option price at the root with the prices at the first level of the tree. Alternatively, you could calculate a second binomial tree; some options traders think the latter method is better.

Based on my experience, these "sophisticated financial models" are usually a direct application of the compound interest formula and numerically approximating a derivative or integral. Really complicated financial models usually are complete nonsense.