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A proof of the smoothness of the finite time horizon american put option for jump diffusions
, 2008
"... We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a free boundary equation. We also show that the value function is C¹ across the optimal stopp ..."
Abstract

Cited by 3 (3 self)
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We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a free boundary equation. We also show that the value function is C¹ across the optimal stopping boundary. Our proof, which only uses the classical theory of parabolic partial differential equations of [7, 8], is an alternative to the proof that uses the theory of vicosity solutions (see [14]). This new proof relies on constructing a monotonous sequence of functions, each of which is a value function of an optimal stopping problem for a geometric Brownian motion, converging to the value function of the American put option for the jump diffusion uniformly and exponentially fast. This sequence is constructed by iterating a functional operator that maps a certain class of convex functions to classical solutions of corresponding free boundary equations. On the other hand, since the approximating sequence converges to the value function exponentially fast, it naturally leads to a good numerical scheme. We also show that the assumption that [14] makes on the parameters of the problem, in order to guarantee that the value function is the unique classical solution of the corresponding free boundary equation, can be dropped.
An efficient method for pricing american options for jump diffusions, http://arxiv.org/abs/0706.2331
, 2007
"... We approximate the price of the American put for jump diffusions by a sequence of functions, which are computed iteratively. This sequence converges to the price function uniformly and exponentially fast. Each element of the approximating sequence solves an optimal stopping problem for geometric Bro ..."
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We approximate the price of the American put for jump diffusions by a sequence of functions, which are computed iteratively. This sequence converges to the price function uniformly and exponentially fast. Each element of the approximating sequence solves an optimal stopping problem for geometric Brownian motion, and can be numerically constructed using the classical finite difference methods. We present examples to illustrate our algorithm’s numerical performance.