6,423 research outputs found
Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models
We consider a class of stochastic path-dependent volatility models where the
stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is
multiplied by a (leverage) function of the spot price, its running maximum, and
time. We propose a Monte Carlo simulation scheme which combines a log-Euler
scheme for the spot process with the full truncation Euler scheme or the
backward Euler-Maruyama scheme for the squared stochastic volatility component.
Under some mild regularity assumptions and a condition on the Feller ratio, we
establish the strong convergence with order 1/2 (up to a logarithmic factor) of
the approximation process up to a critical time. The model studied in this
paper contains as special cases Heston-type stochastic-local volatility models,
the state-of-the-art in derivative pricing, and a relatively new class of
path-dependent volatility models. The present paper is the first to prove the
convergence of the popular Euler schemes with a positive rate, which is
moreover consistent with that for Lipschitz coefficients and hence optimal.Comment: 34 pages, 5 figure
Option pricing in affine generalized Merton models
In this article we consider affine generalizations of the Merton jump
diffusion model [Merton, J. Fin. Econ., 1976] and the respective pricing of
European options. On the one hand, the Brownian motion part in the Merton model
may be generalized to a log-Heston model, and on the other hand, the jump part
may be generalized to an affine process with possibly state dependent jumps.
While the characteristic function of the log-Heston component is known in
closed form, the characteristic function of the second component may be unknown
explicitly. For the latter component we propose an approximation procedure
based on the method introduced in [Belomestny et al., J. Func. Anal., 2009]. We
conclude with some numerical examples
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