714 research outputs found
Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach
We propose an efficient method to evaluate callable and putable bonds under a
wide class of interest rate models, including the popular short rate diffusion
models, as well as their time changed versions with jumps. The method is based
on the eigenfunction expansion of the pricing operator. Given the set of call
and put dates, the callable and putable bond pricing function is the value
function of a stochastic game with stopping times. Under some technical
conditions, it is shown to have an eigenfunction expansion in eigenfunctions of
the pricing operator with the expansion coefficients determined through a
backward recursion. For popular short rate diffusion models, such as CIR,
Vasicek, 3/2, the method is orders of magnitude faster than the alternative
approaches in the literature. In contrast to the alternative approaches in the
literature that have so far been limited to diffusions, the method is equally
applicable to short rate jump-diffusion and pure jump models constructed from
diffusion models by Bochner's subordination with a L\'{e}vy subordinator
Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes
Fast pricing of American-style options has been a difficult problem since it
was first introduced to financial markets in 1970s, especially when the
underlying stocks' prices follow some jump-diffusion processes. In this paper,
we propose a new algorithm to generate tight upper bounds on the Bermudan
option price without nested simulation, under the jump-diffusion setting. By
exploiting the martingale representation theorem for jump processes on the dual
martingale, we are able to explore the unique structure of the optimal dual
martingale and construct an approximation that preserves the martingale
property. The resulting upper bound estimator avoids the nested Monte Carlo
simulation suffered by the original primal-dual algorithm, therefore
significantly improves the computational efficiency. Theoretical analysis is
provided to guarantee the quality of the martingale approximation. Numerical
experiments are conducted to verify the efficiency of our proposed algorithm
Regularity of the Optimal Stopping Problem for Jump Diffusions
The value function of an optimal stopping problem for jump diffusions is
known to be a generalized solution of a variational inequality. Assuming that
the diffusion component of the process is nondegenerate and a mild assumption
on the singularity of the L\'{e}vy measure, this paper shows that the value
function of this optimal stopping problem on an unbounded domain with
finite/infinite variation jumps is in with . As a consequence, the smooth-fit property holds.Comment: To Appear in the SIAM Journal on Control and Optimizatio
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