elle
(Elle)
#1
The different parameters have the following meaning:
St
Price/level of the underlying at time t
Constant volatility (i.e., standard deviation of returns) of the underlying
K
Strike price of the option
T
Maturity date of the option
r
Constant riskless short rate
Consider now that an option quote for a European call option C* is given. The implied
volatility
imp
is the quantity that solves the implicit Equation 3-2.
Equation 3-2. Implied volatility given market quote for option
There is no closed-form solution to this equation, such that one has to use a numerical
solution procedure like the Newton scheme to estimate the correct solution. This scheme
iterates, using the first derivative of the relevant function, until a certain number of
iterations or a certain degree of precision is reached. Formally, we have Equation 3-3 for
some starting value and for 0 < n < ∞.
Equation 3-3. Newton scheme for numerically solving equations
The partial derivative of the option pricing formula with respect to the volatility is called
Vega and is given in closed form by Equation 3-4.
Equation 3-4. Vega of a European option in BSM model
