Robust option replication for a Black-Scholes model extended with nondeterministic trends

Statistical analysis on various stocks reveals long range dependence behavior of the stock prices that is not consistent with the classical Black and Scholes model. This memory or nondeterministic trend behavior is often seen as a reflection of market sentiments and causes that the historical volatility estimator becomes unreliable in practice. We propose an extension of the Black and Scholes model by adding a term to the original Wiener term involving a smoother process which accounts for these effects. The problem of arbitrage will be discussed. Using a generalized stochastic integration theory [8], we show that it is possible to construct a self financing replicating portfolio for a European option without any further knowledge of the extension and that, as a consequence, the classical concept of volatility needs to be re-interpreted. AMS subject classifications: 60H05, 60H10, 90A09

The real multiple dual

In this paper we present a dual representation for the multiple stopping problem, hence multiple exercise options. As such it is a natural generalization of the method in Rogers (2002) and Haugh and Kogan (2004) for the standard stopping problem for American options. We consider this representation as the real dual as it is solely expressed in terms of an infimum over martingales rather than an infimum over martingales and stopping times as in Meinshausen and Hambly (2004). For the multiple dual representation we present three Monte Carlo simulation algorithms which require only one degree of nesting

Calibration of LIBOR models to caps and swaptions: A way around intrinsic instabilities via parsimonious structures and a collateral market criterion

We expose an intrinsic stability problem in joint calibration of a LIBOR market model to caps and swaptions by direct least squares calibration. This problem typically encounters if one tries to identify jointly the volatility norm behaviour and the correlation structure of the forward LIBORs. As a remedy we propose collateral incorporation of a 'Market Swaption Formula', a rule-of-thumb formula which practitioners tend to use, in the calibration routine. It is shown by experiments with practical data that with this new calibration procedure and suitably parametrized volatility structures LIBOR model calibration to caps and swaptions is stable. The involved calibration routine is based on standard swaption approximation or its refinements by Hull & White, Jäckel & Rebonato. We deal with the issue of differently settled caps and swaptions by accordingly adapting the swap rate formula and give a respective modification of Jäckel and Rebonato's refined swaption approximation formula

Regression on particle systems connected to mean-field SDEs with applications

In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered

Multilevel dual approach for pricing American style derivatives

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 different 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 significantly 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

Projected particle methods for solving McKean--Vlaslov equations

We study a novel projection-based particle method to the solution of the corresponding McKean-Vlasov equation. Our approach is based on the projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method can profit from additional smoothness of the underlying density and leads in many situation to a significant reduction of numerical complexity compared to kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The case of linearly growing coefficients of the McKean-Vlasov equation turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean-Vlasov equations with affine drift

An iterative algorithm for multiple stopping: Convergence and stability

We present a new iterative procedure for solving the discrete multiple stopping problem and discuss the stability of the algorithm. The algorithm produces monotonically increasing approximations of the Snell envelope, which coincide with the Snell envelope after finitely many steps. Contrary to backward dynamic programming, the algorithm allows to calculate approximative solutions with only a few nestings of conditionals expectations and is, therefore, tailor-made for a plain Monte-Carlo implementation

LIBOR rate models, related derivatives and model calibration

Based on Jamshidians framework a general strategy for the quasi-analytical valuation of large classes of LIBOR derivatives will be developed. As a special case we will address the quasi-analytical approximation formula for swaptions of Brace Gatarek and Musiela and show that a similar formula can be derived with Jamshidian's methods as well. As further applications we will study the callable reverse floater and the trigger swap. Then, we will study the thorny issues around simultaneous calibration of (low factor) LIBOR models to cap(let) and swaption prices in the markets. We will argue that a low factor market model cannot be calibrated to these prices in a stable way and propose an, in fact, many factor model with only the same number of loading parameters as a two factor model, but, with much better stability properties

An efficient dual Monte Carlo upper bound for Bermudan style derivatives

Based on a duality approach for Monte Carlo construction of upper bounds for American/Bermudan derivatives (Rogers, Haugh & Kogan), we present a new algorithm for computing dual upper bounds in an efficient way. The method is applied to Bermudan swaptions in the context of a LIBOR market model, where the dual upper bound is constructed from the maximum of still alive swaptions. We give a numerical comparison with Andersen's lower bound method and its dual considered by Andersen & Broadie

Uniform approximation of the CIR process via exact simulation at random times

In this paper we uniformly approximate the trajectories of the Cox-Ingersoll-Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact simulation of the CIR dynamics at some deterministic time grid