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We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.
Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [Adv. Math. Econ. 6, 69–83, 2004] and Lyons–Victoir [Proc. R. Soc. Lond. Ser. A 460, 169–198, 2004], involve the solution to numerous auxiliary ordinary differential equations. With focus on the Ninomiya-Victoir algorithm [Appl. Math. Fin. 15, 107–121, 2008], which corresponds to a concrete level 5 cubature method, we study some parametric diffusion models motivated from financial applications, and exhibit structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies, by introducing a (model-dependent) variation of the Ninomiya-Victoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time.
Density expansions for hypoelliptic diffusions (X1^,...,X^d) are revisited. In particular, we are interested in density expansions of the projection (X^1_T,...,X^l_T) at time $T>0$, with $l \le d$. Global conditions are found which replace the well-known ”not-in-cutlocus” condition known from heat-kernel asymptotics; cf. G. Ben Arous (88). Our small noise expansion allows for a ”second order” exponential factor. Applications include tail and implied volatility asymptotics in some correlated stochastic volatility models; in particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).
A robust implementation of a Dupire type local volatility model is an important issue for every option trading floor. In the present note we provide new analytic insights into the asymptotic behavior of local volatility in the wings. We present a general approximation formula and specialize it to the Heston model, showing that local variance is linear in the wings. This further justifies the choice of certain local volatility parametrizations.
We solve Skorokhod's embedding problem for Brownian motion with linear drift $(W_t+ \kappa t)_{t\geq 0}$ by means of techniques of stochastic control theory. The search for a stopping time
$T$ such that the law of $W_T + \kappa T$ coincides with a prescribed law $\mu$ possessing the first
moment is based on solutions of backward stochastic differential equations of quadratic
type. This new approach generalizes an approach by Bass [Bas] of the classical version of
Skorokhod's embedding problem using martingale representation techniques.
We consider a dynamical system described by the differential equation $\dot{Y}_t = -U^'(Y_t)$
with a unique stable point at the origin. We perturb the system by L\'evy noise of
intensity $\varepsilon$, to obtain the stochastic differential equation $dX^\varepsilon_t = -U^'(X^\varepsilon_{t-})dt + \varepsilon dL_t}.
The process $L$ is a symmetric L\'evy process whose jump measure $\nu$ has exponentially
light tails, $\nu([u, \infty))\sim exp(-u^\alpha), \alpha > 0, u \to\infty$. We study the first exit problem for
the trajectories of the solutions of the stochastic differential equation from the interval
$[-1, 1]$. In the small noise limit $\varepsilon\to 0$ we determine the law and the mean value of the
first exit time, to discover an intriguing phase transition at the critical index $\alpha = 1$.
In this article we propose a novel approach to reduce the computational complexity
of the dual method for pricing American options. We consider a sequence of
martingales that converges to a given target martingale and decompose the original
dual representation into a sum of representations that correspond to dierent levels
of approximation to the target martingale. By next replacing in each representation
true conditional expectations with their Monte Carlo estimates, we arrive at what
one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm
reveals that the computational complexity of getting the corresponding target upper
bound, due to the target martingale, can be signicantly reduced. In particular, it
turns out that using our new approach, we may construct a multilevel version of the
well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is,
regarding complexity, virtually equivalent to a non-nested algorithm. The performance
of this multilevel algorithm is illustrated by a numerical example.