Solving The Inverse Pricing Model Function
In general, a pricing model function, f, does not have a closed-form solution for its inverse, g. Instead, a root finding technique is used to solve the equation:
While there are many techniques for finding roots, two of the most commonly used are Newton's method and Brent's method. Because options prices can move very quickly, it is often important to use the most efficient method when calculating implied volatilities.
Newton's method provides rapid convergence; however, it requires the first partial derivative of the option's theoretical value with respect to volatility; i.e., which is also known as vega (see The Greeks). If the pricing model function yields a closed-form solution for vega, which is the case for Black–Scholes model, then Newton's method can be more efficient. However, for most practical pricing models, such as a binomial model, this is not the case and vega must be derived numerically. When forced to solve for vega numerically, it usually turns out that Brent's method is more efficient as a root-finding technique.
Read more about this topic: Implied Volatility
Famous quotes containing the words solving the, solving, inverse, model and/or function:
“You are right to demand that an artist engage his work consciously, but you confuse two different things: solving the problem and correctly posing the question.”
—Anton Pavlovich Chekhov (1860–1904)
“Cultural expectations shade and color the images that parents- to-be form. The baby product ads, showing a woman serenely holding her child, looking blissfully and mysteriously contented, or the television parents, wisely and humorously solving problems, influence parents-to-be.”
—Ellen Galinsky (20th century)
“Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.”
—Ralph Waldo Emerson (1803–1882)
“One of the most important things we adults can do for young children is to model the kind of person we would like them to be.”
—Carol B. Hillman (20th century)
“Think of the tools in a tool-box: there is a hammer, pliers, a saw, a screwdriver, a rule, a glue-pot, nails and screws.—The function of words are as diverse as the functions of these objects.”
—Ludwig Wittgenstein (1889–1951)