Bermudan Swaptions Research Articles

An exact and efficient method for computing cross-Gammas of Bermudan swaptions and cancelable swaps under the Libor market model

An exact and efficient method for computing cross-Gammas of Bermudan swaptions and cancelable swaps under the Libor market model

Unspanned Stochastic Volatility & Conformal Symmetry (Presentation Slides)

We employ conformal symmetries to provide a generic tractable framework for interest rate modelling. The approach combines calibration flexibility of market models with tractability and computational efficiency of shot rate models. The methodology enables robust calibration to the whole variety of caps and swaptions with various expirations, strikes and tenors. In the same time, low dimensionality enables backward induction allowing efficient valuation of bermudan swaptions without resorting to suboptimal American Monte Carlo.

Counterparty Credit Exposures for Interest Rate Derivatives using the Stochastic Grid Bundling Method

ABSTRACTThe regulatory credit value adjustment (CVA) for an outstanding over-the-counter (OTC) derivative portfolio is computed based on the portfolio exposure over its lifetime. Usually, the future portfolio exposure is approximated using the Monte Carlo simulation, as the portfolio value can be driven by several market risk-factors. For derivatives, such as Bermudan swaptions, that do not have an analytical approximation for their Mark-to-Market (MtM) value, the standard market practice is to use the regression functions from the least squares Monte Carlo method to approximate their MtM along simulated scenarios. However, such approximations have significant bias and noise, resulting in inaccurate CVA charge. In this paper, we extend the Stochastic Grid Bundling Method (SGBM) for the one-factor Gaussian short rate model, to efficiently and accurately compute Expected Exposure, Potential Future exposure and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates how different measures, for instance spot and terminal measure, can simultaneously be employed in the SGBM framework, to significantly reduce the variance and bias of the solution.

Fast and accurate exercise policies for Bermudan swaptions in the LIBOR market model

This paper describes an American Monte Carlo approach for obtaining fast and accurate exercise policies for pricing of callable LIBOR Exotics (e.g., Bermudan swaptions) in the LIBOR market model using the Stochastic Grid Bundling Method (SGBM). SGBM is a bundling and regression based Monte Carlo method where the continuation value is projected onto a space where the distribution is known. We also demonstrate an algorithm to obtain accurate and tight lower–upper bound values without the need for nested Monte Carlo simulations.

A Short Note on the Exact Stochastic Simulation Scheme of the Hull-White Model and Its Implementation

In this short note we derive an exact simulation scheme for the joint distribution of (r(t),N(t)), where r denotes the short rate following a Hull-White model and $N$ denotes the numeraire.To sample the correct joint distribution of (r(t),N(t)) our scheme requires a two-factor Brownian driver. In other words: the time-discrete Hull-White model is a two factor model.We investigate the performance of this scheme compared to the classical schemes for correlation dependent products like the LIBOR in arrears and Bermudan swaptions. We show that the exact sampling of the joint distribution is important in some applications, e.g., the valuation of Bermudan options using American Monte-Carlo method. Traditional approximation of the numeraire may generate biases if the model uses larger simulation time-steps.Large time-step simulation of short rate models and the American Monte-Carlo method is of interest in the calculation of portfolio effects (xVAs), where usually many risk factors have to be simulated and computational resources need to be saved.

Adaptive Curve Evolution Model

We employ conformal symmetries to provide a generic tractable framework for interest rate modelling. The approach combines calibration flexibility of market models with tractability and computational efficiency of shot rate models. The methodology enables robust calibration to the whole variety of caps and swaptions with various expirations, strikes and tenors. In the same time, low dimensionality enables backward induction allowing efficient valuation of bermudan swaptions without resorting to suboptimal American Monte Carlo.

Least Squares Monte Carlo Credit Value Adjustment with Small and Unidirectional Bias

Credit value adjustment (CVA) and related charges have emerged as important risk factors following the Global Financial Crisis. These charges depend on uncertain future values of underlying products, and are usually computed by Monte Carlo simulation. For products that cannot be valued analytically at each simulation step, the standard market practice is to use the regression functions from least squares Monte Carlo method to approximate their values. However, these functions do not necessarily provide accurate approximations to product values over all simulated paths and can result in biases that are difficult to control. Motivated by a novel characterization of the CVA as the value of an option with an early exercise opportunity at a stochastic time, we provide an approximation for CVA and other credit charges that rely only on the sign of the regression functions. The values are determined, instead, by pathwise deflated cash flows. A comparison of CVA for Bermudan swaptions and cancellable swaps show that the proposed approximation results in much smaller errors than the standard approach of using the regression function values.

Efficient computation of exposure profiles on real-world and risk-neutral scenarios for Bermudan swaptions

This paper presents a computationally efficient technique for the computation of exposure distributions at any future time under the risk-neutral and some observed real-world probability measures; these are needed for the computation of credit valuation adjustment (CVA) and potential future exposure (PFE). In particular,we present a valuation framework for Bermudan swaptions. The essential idea is to approximate the required value function via a set of risk-neutral scenarios and use this approximated value function on the set of observed real-world scenarios. This technique significantly improves the computational efficiency by avoiding nested Monte Carlo simulation and using only basic methods such as regression. We demonstrate the benefits of this technique by computing exposure distributions for Bermudan swaptions under the Hull-White and G2++ models. (Less)

Efficient Computation of Exposure Profiles on Real-World and Risk-Neutral Scenarios for Bermudan Swaptions

This paper presents a computationally efficient technique for the computation of exposure distributions at any future time under the risk-neutral and some observed real-world probability measures; these are needed for the computation of credit valuation adjustment (CVA) and potential future exposure (PFE). In particular, we present a valuation framework for Bermudan swaptions. The essential idea is to approximate the required value function via a set of risk-neutral scenarios and use this approximated value function on the set of observed real-world scenarios. This technique significantly improves the computational efficiency by avoiding nested Monte Carlo simulation and using only basic methods such as regression. We demonstrate the benefits of this technique by computing exposure distributions for Bermudan swaptions under the Hull-White and G2 models.

Numerical valuation of derivatives in high-dimensional settings via partial differential equation expansions

We propose a new numerical approach to solving high-dimensional partial differential equations (PDEs) that arise in the valuation of exotic derivative securities. The proposed method is extended from the work of Reisinger andWittum and uses principal component analysis of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. The approximation is related to anchored-analysis-of-variance decompositions and is expected to be accurate whenever the covariance matrix has one or few dominating eigenvalues. We give a careful analysis of the numerical accuracy and computational complexity compared with state-of-the-art Monte Carlo methods, using Bermudan swaptions and ratchet floors, which are considered difficult benchmark problems, as examples. We demonstrate that, for problems with medium to high dimensionality and moderate time horizons, the PDE method presented delivers results comparable in accuracy to the Monte Carlo methods considered here in a similar or (often significantly) faster run time.

An Exact and Efficient Method for Computing Cross-Gammas of Bermudan Swaptions and Cancellable Swaps Under the Libor Market Model

We introduce a new simulation algorithm for computing the Hessians of Bermudan swaptions and cancelable swaps. The resulting pathwise estimates are unbiased and accurate. Given the exercise strategy, the pathwise angularities are removed by a sequence of measure changes. The change of measure at each exercise time is chosen to be optimal in terms of minimizing the variance of the likelihood ratio terms. Numerical results for the Hessian of cancelable swaps are presented to demonstrate the speed and efficacy of the method.

Replication & XVA: The Unique Counting Approach

In the large context of valuation adjustments, collectively referred to as “XVA”, we treat replication strategies as a unique way to calculate fair price equations. We consider different replication strategies (Piterbarg, Burgard-Kjaer and our contribution) and identify the valuation adjustments for the different pricing equations. We reduce a (large) list of pricing PDE’s to two main cases and calculate the XVAs for each case. We also consider the funding cost counting issue raised by Hull and White and treat it mathematically. This issue relates to the fact that any funding cost adjustment associated with unsecured funding should, in theory, be offset by a corresponding debit valuation adjustment. The framework is generalized to allow for stochastic hazard rates, as has been treated by others, and we also consider the case where collateralized vanilla derivatives are used as hedging instruments, demonstrating that there is no impact upon pricing outcomes. The special case of a “back-to-back” hedge with another institution is examined, and it is seen that the funding valuation adjustment is driven by differences between the CSA with the trade counterparty and the CSA with the hedge counterparty. We also provide numerical results for an interest rate swap and a related Bermudan swaption.

Robust and accurate Monte Carlo simulation of (cross-) Gammas for Bermudan swaptions in the LIBOR market model

Robust and accurate Monte Carlo simulation of (cross-) Gammas for Bermudan swaptions in the LIBOR market model

Counterparty Credit Exposures for Interest Rate Derivatives Using the Stochastic Grid Bundling Method

The regulatory credit value adjustment (CVA) for an outstanding over-the-counter (OTC) derivative portfolio, is computed based on the portfolio exposure over its lifetime. Usually the future portfolio exposure is approximated using Monte Carlo simulation, as the portfolio value can be driven by several market risk-factors. For derivatives, such as Bermudan swaptions, that do not have an analytical approximation for their Mark-to-Market (MtM) value, the standard market practice is to use the regression functions from least squares Monte Carlo method to approximate their MtM along simulated scenarios. However, such approximations have significant bias and noise, resulting in inaccurate CVA charge. In this paper we extend the Stochastic Grid Bundling Method (SGBM) for the one-factor Gaussian short rate model, to efficiently and accurately compute Expected Exposure, Potential Future exposure and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates, how different measures, for instance spot and terminal measure, can simultaneously be employed in the SGBM framework, to significantly reduce the variance and bias of the solution.

How Does Stochastic Basis Change Mean Reversion? A Study on Bermudan Swaptions

The main goal of this work is to analyse the impact of the stochastic nature of basis spreads in the price of Bermudan swaptions. In this paper we find a relationship between the effect of stochastic basis spreads and an effective mean reversion of the traditional one-factor Hull-White model. We develop an extended Hull-White model which considers two different curves: one used to discount and the other used to estimate cash-flows, with a stochastic basis spread between both curves. We derive formulas for the Libor rate and the stochastic discount factor. We also derive semi-analytical formulas for Euopean swaptions used to calibrate the model. The impact of stochastic basis spreads in Bermudan swaptions is analysed for different volatility and correlation scenarios. We show that the impact may clearly become non-negligible and is captured in a natural way by means of an effective mean reversion in the simplest one-factor Hull-White model. This permits to manage positions in an effective way, incorporating stochastic basis spreads automatically.

IMPLICATIONS FOR HEDGING OF THE CHOICE OF DRIVING PROCESS FOR ONE-FACTOR MARKOV-FUNCTIONAL MODELS

In this paper, we study the implications for hedging Bermudan swaptions of the choice of the instantaneous volatility for the driving Markov process of the one-dimensional swap Markov-functional model. We find that there is a strong evidence in favor of what we term "parametrization by time" as opposed to "parametrization by expiry". We further propose a new parametrization by time for the driving process which takes as inputs into the model the market correlations of relevant swap rates. We show that the new driving process enables a very effective vega-delta hedge with a much more stable gamma profile for the hedging portfolio compared with the existing ones.

Funding Value Adjustment for General Financial Instruments: Theory and Practice

In the first part of this paper (Antonov-Bianchetti, 2013) we developed the theoretical framework for pricing financial instruments under multiple sources of funding, leading to a non-linear pricing PDE and to Funding Value Adjustment (FVA).In this second part we develop the numerical framework for computing FVA for general financial instruments including callable features. Our main technical result is an efficient approximation of the FVA in a fully universal way. Usage of non-linear effective discounting rates permits an exact handling of all solvable special cases (the collateral as linear function of the value) by a single formula. Furthermore, the formula delivers a very accurate approximation for general instruments (barriers, Bermudans, etc.). The proof is based on our second technical result: exact calculation of prices of automatically exercisable instruments (e.g. European-style or Barrier) having different stochastic discount rates before and after exercise.We also address the implementation workflow of the FVA calculation, allowing parallel deal-by-deal computation, and we provide a concrete example of FVA calculation for a Bermudan Swaption with partial collateralization, proving that the quality of the numerical approximation is excellent.

Fast and Accurate Exercise Policies for Bermudan Swaptions in Libor Market Model

This paper describes an American Monte Carlo approach for obtaining fast and accurate exercise policies for pricing of callable LIBOR Exotics (e.g., Bermudan swaptions) in the LIBOR market model using the Stochastic Grid Bundling Method (SGBM). SGBM is a bundling and regression based Monte Carlo method where the continuation value is projected onto a space where the distribution is known. We also demonstrate an algorithm to obtain accurate and tight lower-upper bound values without the need for nested Monte Carlo simulations.

Efficient Monte Carlo Simulation of the Delta Vector of a Bermudian Swaption in the LIBOR Market Model

We consider various methods for an efficient numerical computation of the Delta vector of a Bermudan swaption in a LIBOR market model setting. All methods are based on the least-squares Montel Carlo method of Longstaff & Schwartz (2001). Among them, we present three new approaches: a new version of the adjoint method introduced by Leclerc et al. (2009), a path-wise method based on the use of the forward drift, and a likelihood ratio approach. The new version of the adjoint method shows superior performance compared with the other methods.

Pricing Interest Rate Derivatives in a Multifactor HJM Model with Time

We investigate the partial differential equation (PDE) for pricing interest derivatives in the multi-factor Cheyette Model, which involves time-dependent volatility functions with a special structure. The high dimensional parabolic PDE that results is solved numerically via a modified sparse grid approach, that turns out to be accurate and efficient. In addition we study the corresponding Monte Carlo simulation, which is fast since the distribution of the state variables can be calculated explicitly. The results obtained from both methodologies are compared to the known analytical solutions for bonds and caplets. When there is no analytical solution, both European and Bermudan swaptions have been evaluated using the sparse grid PDE approach that is shown to outperform the Monte Carlo simulation.