1,113 research outputs found
Randomisation and recursion methods for mixed-exponential Levy models, with financial applications
We develop a new Monte Carlo variance reduction method to estimate the
expectation of two commonly encountered path-dependent functionals:
first-passage times and occupation times of sets. The method is based on a
recursive approximation of the first-passage time probability and expected
occupation time of sets of a Levy bridge process that relies in part on a
randomisation of the time parameter. We establish this recursion for general
Levy processes and derive its explicit form for mixed-exponential
jump-diffusions, a dense subclass (in the sense of weak approximation) of Levy
processes, which includes Brownian motion with drift, Kou's double-exponential
model and hyper-exponential jump-diffusion models. We present a highly accurate
numerical realisation and derive error estimates. By way of illustration the
method is applied to the valuation of range accruals and barrier options under
exponential Levy models and Bates-type stochastic volatility models with
exponential jumps. Compared with standard Monte Carlo methods, we find that the
method is significantly more efficient
Explicit solution of an inverse first-passage time problem for L\'{e}vy processes and counterparty credit risk
For a given Markov process and survival function on
, the inverse first-passage time problem (IFPT) is to find a
barrier function such that the survival
function of the first-passage time is given
by . In this paper, we consider a version of the IFPT problem
where the barrier is fixed at zero and the problem is to find an initial
distribution and a time-change such that for the time-changed process
the IFPT problem is solved by a constant barrier at the level zero.
For any L\'{e}vy process satisfying an exponential moment condition, we
derive the solution of this problem in terms of -invariant
distributions of the process killed at the epoch of first entrance into the
negative half-axis. We provide an explicit characterization of such
distributions, which is a result of independent interest. For a given
multi-variate survival function of generalized frailty type, we
construct subsequently an explicit solution to the corresponding IFPT with the
barrier level fixed at zero. We apply these results to the valuation of
financial contracts that are subject to counterparty credit risk.Comment: Published at http://dx.doi.org/10.1214/14-AAP1051 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Some remarks on first passage of Levy processes, the American put and pasting principles
The purpose of this article is to provide, with the help of a fluctuation
identity, a generic link between a number of known identities for the first
passage time and overshoot above/below a fixed level of a Levy process and the
solution of Gerber and Shiu [Astin Bull. 24 (1994) 195-220], Boyarchenko and
Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript
(1999), SIAM J. Control Optim. 40 (2002) 1663-1696], Chan [Original unpublished
manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100
(2002) 75-107], Mordecki [Finance Stoch. 6 (2002) 473-493], Asmussen, Avram and
Pistorius [Stochastic Process. Appl. 109 (2004) 79-111] and Chesney and
Jeanblanc [Appl. Math. Fin. 11 (2004) 207-225] to the American perpetual put
optimal stopping problem. Furthermore, we make folklore precise and give
necessary and sufficient conditions for smooth pasting to occur in the
considered problem.Comment: Published at http://dx.doi.org/10.1214/105051605000000377 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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A Novel Fourier Transform B-spline Method for Option Pricing
We present a new efficient and robust framework for European option pricing under continuous-time asset models from the family of exponential semimartingale processes. We introduce B-spline interpolation theory to derivative pricing to provide an accurate closed-form representation of the option price under an inverse Fourier transform.
We compare our method with some state-of-the-art option pricing methods, and demonstrate that it is extremely fast and accurate. This suggests a wide range of applications, including the use of more realistic asset models in high frequency trading. Examples considered in the paper include option pricing under asset models, including stochastic volatility and jumps, computation of the Greeks, and the inverse problem of cross-sectional calibration
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