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)
“More than a decade after our fellow citizens began bedding down on the sidewalks, their problems continue to seem so intractable that we have begun to do psychologically what government has been incapable of doing programmatically. We bring the numbers down—not by solving the problem, but by deciding it’s their own damn fault.”
—Anna Quindlen (b. 1952)
“The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
—Aldous Huxley (1894–1963)
“The Battle of Waterloo is a work of art with tension and drama with its unceasing change from hope to fear and back again, change which suddenly dissolves into a moment of extreme catastrophe, a model tragedy because the fate of Europe was determined within this individual fate.”
—Stefan Zweig (18811942)
“The function of the actor is to make the audience imagine for the moment that real things are happening to real people.”
—George Bernard Shaw (1856–1950)