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For real-valued functions of a real variable, y = f(x), its ordinary derivative dy/dx is geometrically the gradient of the tangent line to the curve y = f(x) at all points in the domain. Partial derivatives extend this idea to tangent hyperplanes to a curve. The second order partial derivatives can be calculated for every pair of variables:
For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the fundamental theorem of calculus. This states that differentiation is the reverse process to integration.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
This "real" probability distribution of the market prices is typically denoted by the blackboard font letter "", as opposed to the "risk-neutral" probability "" used in derivatives pricing. Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss ...
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) {\textstyle \arctan(y,x)} .
The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R n. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold
Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to
The first application to option pricing was by Phelim Boyle in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how to price Asian options by Monte Carlo. An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options with early exercise features.