823 research outputs found
Convergence of Monte Carlo simulations involving the mean-reverting square root process
The mean-reverting square root process is a stochastic differential equation (SDE) that has found considerable use as a model for volatility, interest rate, and other financial quantities. The equation has no general, explicit solution, although its transition density can be characterized. For valuing path-dependent options under this model, it is typically quicker and simpler to simulate the SDE directly than to compute with the exact transition density. Because the diffusion coefficient does not satisfy a global Lipschitz condition, there is currently a lack of theory to justify such simulations. We begin by showing that a natural Euler-Maruyama discretization provides qualitatively correct approximations to the first and second moments. We then derive explicitly computable bounds on the strong (pathwise) error over finite time intervals. These bounds imply strong convergence in the limit of the timestep tending to zero. The strong convergence result can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. We spell this out for a bond with interest rate given by the mean-reverting square root process, and for an up-and-out barrier option with asset price governed by the mean-reverting square root process. We also prove convergence for European and up-and-out barrier options under Heston's stochastic volatility model - here the mean-reverting square root process feeds into the asset price dynamics as the squared volatility
A Comparison of Biased Simulation Schemes for Stochastic Volatility Models
When using an Euler discretisation to simulate a mean-reverting square root process, one runs into the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the Heston stochastic volatility model, where the variance is modelled as a square root process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the upward bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to a recent quasi-second order scheme of Kahl and Jäckel and the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme by far outperforms all biased schemes in terms of bias, root-mean-squared error, and hence should be the preferred discretisation method for simulation of the Heston model and extensions thereof
A subordinated CIR intensity model with application to Wrong-Way risk CVA
Credit Valuation Adjustment (CVA) pricing models need to be both flexible and
tractable. The survival probability has to be known in closed form (for
calibration purposes), the model should be able to fit any valid Credit Default
Swap (CDS) curve, should lead to large volatilities (in line with CDS options)
and finally should be able to feature significant Wrong-Way Risk (WWR) impact.
The Cox-Ingersoll-Ross model (CIR) combined with independent positive jumps and
deterministic shift (JCIR++) is a very good candidate : the variance (and thus
covariance with exposure, i.e. WWR) can be increased with the jumps, whereas
the calibration constraint is achieved via the shift. In practice however,
there is a strong limit on the model parameters that can be chosen, and thus on
the resulting WWR impact. This is because only non-negative shifts are allowed
for consistency reasons, whereas the upwards jumps of the JCIR++ need to be
compensated by a downward shift. To limit this problem, we consider the
two-side jump model recently introduced by Mendoza-Arriaga \& Linetsky, built
by time-changing CIR intensities. In a multivariate setup like CVA,
time-changing the intensity partly kills the potential correlation with the
exposure process and destroys WWR impact. Moreover, it can introduce a forward
looking effect that can lead to arbitrage opportunities. In this paper, we use
the time-changed CIR process in a way that the above issues are avoided. We
show that the resulting process allows to introduce a large WWR effect compared
to the JCIR++ model. The computation cost of the resulting Monte Carlo
framework is reduced by using an adaptive control variate procedure
Pricing Weather Derivatives
This paper presents a general method for pricing weather derivatives. Specification tests find that a temperature series for Fresno, California follows a mean-reverting Brownian motion process with discrete jumps and ARCH errors. Based on this process, we define an equilibrium pricing model for cooling degree day weather options. Comparing option prices estimated with three methods: a traditional burn-rate approach, a Black-Scholes-Merton approximation, and an equilibrium Monte Carlo simulation reveals significant differences. Equilibrium prices are preferred on theoretical grounds, so are used to demonstrate the usefulness of weather derivatives as risk management tools for California specialty crop growers.derivative, jump-diffusion process, mean-reversion, volatility, weather, Demand and Price Analysis,
First order strong approximations of scalar SDEs with values in a domain
We are interested in strong approximations of one-dimensional SDEs which have
non-Lipschitz coefficients and which take values in a domain. Under a set of
general assumptions we derive an implicit scheme that preserves the domain of
the SDEs and is strongly convergent with rate one. Moreover, we show that this
general result can be applied to many SDEs we encounter in mathematical finance
and bio-mathematics. We will demonstrate flexibility of our approach by
analysing classical examples of SDEs with sublinear coefficients (CIR, CEV
models and Wright-Fisher diffusion) and also with superlinear coefficients
(3/2-volatility, Ait-Sahalia model).
Our goal is to justify an efficient Multi-Level Monte Carlo (MLMC) method for
a rich family of SDEs, which relies on good strong convergence properties
Convergence of numerical methods for stochastic differential equations in mathematical finance
Many stochastic differential equations that occur in financial modelling do
not satisfy the standard assumptions made in convergence proofs of numerical
schemes that are given in textbooks, i.e., their coefficients and the
corresponding derivatives appearing in the proofs are not uniformly bounded and
hence, in particular, not globally Lipschitz. Specific examples are the Heston
and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia
model with rational coefficient functions. Simple examples show that, for
example, the Euler-Maruyama scheme may not converge either in the strong or
weak sense when the standard assumptions do not hold. Nevertheless, new
convergence results have been obtained recently for many such models in
financial mathematics. These are reviewed here. Although weak convergence is of
traditional importance in financial mathematics with its emphasis on
expectations of functionals of the solutions, strong convergence plays a
crucial role in Multi Level Monte Carlo methods, so it and also pathwise
convergence will be considered along with methods which preserve the positivity
of the solutions.Comment: Review Pape
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