Bermudan Swaptions Research Articles

A Semi-Static Replication Method for Bermudan Swaptions under an Affine Multi-Factor Model

We present a semi-static replication algorithm for Bermudan swaptions under an affine, multi-factor term structure model. In contrast to dynamic replication, which needs to be continuously updated as the market moves, a semi-static replication needs to be rebalanced on just a finite number of instances. We show that the exotic derivative can be decomposed into a portfolio of vanilla discount bond options, which mirrors its value as the market moves and can be priced in closed form. This paves the way toward the efficient numerical simulation of xVA, market, and credit risk metrics for which forward valuation is the key ingredient. The static portfolio composition is obtained by regressing the target option’s value using an interpretable, artificial neural network. Leveraging the universal approximation power of neural networks, we prove that the replication error can be arbitrarily small for a sufficiently large portfolio. A direct, a lower bound, and an upper bound estimator for the Bermudan swaption price are inferred from the replication algorithm. Additionally, closed-form error margins to the price statistics are determined. We practically study the accuracy and convergence of the method through several numerical experiments. The results indicate that the semi-static replication approaches the LSM benchmark with basis point accuracy and provides tight, efficient error bounds. For in-model simulations, the semi-static replication outperforms a traditional dynamic hedge.

Laplace transforms of stochastic integrals and the pricing of Bermudan swaptions

In this paper, we derive the Laplace transforms, i.e., expected values of the exponentials, of time integrals of a class of Extended Cox–Ingersoll–Ross (CIR) processes which are given as the product of a standard CIR process and a positive deterministic function. Assuming such a process for the short-rate gives rise to an affine term structure model for which we can now provide the bond price formulas that had so far not been accessible by attempting to solve the usual Riccati equation. By conditioning the exponentiated time integral on the potential states of the integrated process at its endpoint, we obtain the building blocks to evaluate certain interest rate derivatives, namely Bermudan swaptions. This is achieved by an algorithm that determines the exercise decision at each state by comparing the available payoff with an explicit lower bound built from such conditional Laplace transforms, without resorting to estimation techniques such as American Monte Carlo.

On the Deterministic-Shift Extended CIR Model in a Negative Interest Rate Framework

In this paper, we propose a new exogenous model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model with a perfect fit to the observed term-structure. We use the difference between two independent CIR processes and apply the deterministic-shift extension technique. To allow for a fast calibration to the market swaption surface, we apply the Gram–Charlier expansion to calculate the swaption prices in our model. We run several numerical tests to demonstrate the strengths of this model by using Monte-Carlo techniques. In particular, the model produces close Bermudan swaption prices compared to Bloomberg’s Hull–White one-factor model. Moreover, it finds constant maturity swap (CMS) rates very close to Bloomberg’s CMS rates.

Between Scylla and Charybdis: The Bermudan Swaptions Pricing Odyssey

Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate products, there is an inherent “tug of war” between the pursuit of calibration and pricing precision, tractability, and implementation efficiency. After reviewing the main ideas and implementation techniques underlying both single- and multi-factor models, we offer our own approach based on dimension reduction via Markovian projection. Specifically, on the theoretical side, we provide a reinterpretation and extension of the classic result due to Gyöngy which covers non-probabilistic, discounted, distributions relevant in option pricing. Thus, we show that for purposes of swaption pricing, a potentially complex and multidimensional process for the underlying swap rate can be collapsed to a one-dimensional one. The empirical contribution of the paper consists in demonstrating that even though we only match the marginal distributions of the two processes, Bermudan swaptions prices calculated using such an approach appear well-behaved and closely aligned to counterparts from more sophisticated models.

Learning Bermudans

American and Bermudan-type financial instruments are often priced with specific Monte Carlo techniques whose efficiency critically depends on the effective dimensionality of the problem and the available computational power. In our work we focus on Bermudan Swaptions, well-known interest rate derivatives embedded in callable debt instruments or traded in the OTC market for hedging or speculation purposes, and we adopt an original pricing approach based on Supervised Learning (SL) algorithms. In particular, we link the price of a Bermudan Swaption to its natural hedges, i.e. the underlying European Swaptions, and other sound financial quantities through SL non-parametric regressions. We test different algorithms, from linear models to decision tree-based models and Artificial Neural Networks (ANN), analyzing their predictive performances. All the SL algorithms result to be reliable and fast, allowing to overcome the computational bottleneck of standard Monte Carlo simulations; the best performing algorithms for our problem result to be Ridge, ANN and Gradient Boosted Regression Tree. Moreover, using feature importance techniques, we are able to rank the most important driving factors of a Bermudan Swaption price, confirming that the value of the maximum underlying European Swaption is the prevailing feature.

Deep MVA: Deep Learning for Margin Valuation Adjustment of Callable Products

Regulatory initial margin (IM) is being implemented across the financial industry in accordance with BCBS-IOSCO requirements. The regulations target uncleared over-the-counter derivative trading and, among other issues, aim to mitigate counter party credit risk by defining a comprehensive set of rules for initial margin between trading parties. Computing the funding costs of IM, or Margin Valuation Adjustment (MVA), is a major challenge for xVA systems as it requires the future projection of dynamic IM positions. This is particularly challenging for callable products, such as Bermudan swaptions which are complex to price and require path wise exercise tracking in exposure simulations. Brute force simulation of future IM is not feasible due to the excessive computational demands of model calibration and numerical pricing methods. Approximate MVA methods, such as regression techniques, are difficult to design due to the high-dimensionality of the problem. In this paper, we propose a method based on Deep Neural Networks to approximate the Bermudan swaption pricing function and sensitivities. We exploit neural network's high-dimensionality and universal approximation properties to train networks based on prices and sensitivities generated from existing numerical pricing models. The trained neural networks are then used for extremely fast IM simulation where computationally intense numerical methods are replaced by optimized and hardware accelerated neural network inference. We demonstrate that the neural network models deliver exceptional performance, capable of pricing Bermudan Swaption MVAs over 100,000 times faster than traditional approaches while maintaining a high degree of accuracy.

The Impact of Initial Margin on Derivatives Pricing with an Application of Machine Learning

In this paper we consider the impact of bilateral initial margin on derivatives pricing. We first introduce the background of bilateral initial margin. Then, we focus on how initial margin effects counterparty credit exposures, capital requirements and funding costs. Nowadays, besides risk-neutral valuation principles, these components are included when pricing derivatives through so-called valuation adjustments (xVAs). Based on this, we present an approach which incorporates initial margin into existing xVA frameworks. Then, we give an overview of the most common methods for calculating future initial margin. Moreover, we present a regression model which is based on machine learning methods. Finally, we show some numerical results for a cross currency swap, an fx option and a Bermudan swaption including a scenario analysis. Here, it turns out that it is much more important to consider margin agreement, regulatory or non-modellable parameters in detail than implementing a most accurate dynamic initial margin model.

Deep Learning-Based BSDE Solver for Libor Market Model with Applications to Bermudan Swaption Pricing and Hedging

The Libor market model, also known as the BGM Model, is a term structure model of interest rates. It is widely used for pricing interest rate derivatives, especially Bermudan swaptions, and other exotic Libor callable derivatives. For numerical implementation the pricing of derivatives with Libor market models is mainly carried out with Monte Carlo simulation. The PDE grid approach is not particularly feasible due to the “Curse of Dimensionality”. The standard Monte Carlo method for American swaption pricing more or less uses regression to estimate the expected value as a linear combination of basis functions as demonstrated in the classical paper by Longstaff and Schwartz [Longstaff01]. However, [Longstaff01] only provides the lower bound for American option price. Another complexity arises from applying Monte Carlo simulation is the computation of the sensitivities of the option, the so-called “Greeks”, which are fundamental for a trader's hedging activity. Recently, an alternative numerical method based on deep learning and backward stochastic differential equations (BSDEs) appeared in quite a few researches [Weinan17, Jiequn17]. For European style options the feedforward deep neural networks (DNN) show not only feasibility but also efficiency to obtain both prices and numerical Greeks. The standard LMM implementation requires dimension five or higher in factor space even after PCA, which cannot be solved by traditional PDE solvers, such as finite differences or finite elements methods. In this paper, a new backward DNN solver is proposed for Bermudan swaptions. Our approach is representing financial pricing problems in the form of high dimensional stochastic optimal control problems, FBSDEs, or equivalent PDEs. We demonstrate that using backward DNN the high-dimension Bermudan swaption pricing and hedging can be solved effectively and efficiently. A comparison between Monte Carlo simulation and the new method for pricing vanilla interest rate options manifests the superior performance of the new method. We then use this method to calculate prices and Greeks of Bermudan swaptions as a prelude for other Libor callable derivatives.

Efficient XVA Computation under Local LLvy Models

Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FBSDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local Levy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.

On the American swaption in the linear-rational framework

We study American swaptions in the linear-rational (LR) term structure model introduced in Filipović et al. [J. Finance., 2017, 72, 655–704]. The American swaption pricing problem boils down to an optimal stopping problem that is analytically tractable. It reduces to a free-boundary problem that we tackle by the local time-space calculus of Peskir [J. Theoret. Probab., 2005a, 18, 499–535]. We characterize the optimal stopping boundary as the unique solution to a non-linear integral equation that can be readily solved numerically. We obtain the arbitrage-free price of the American swaption and the optimal exercise strategies in terms of swap rates for both fixed-rate payer and receiver swaps. Finally, we show that Bermudan swaptions can be efficiently priced as well.

Initial Margin Forecast - Bermudan Swaption Methodology and Case Study

The introduction of mandatory margining for non-cleared portfolios has major implications for the pricing and risk measurement of OTC derivatives. In particular, it requires a model for estimating future initial margin to determine pricing adjustments and counterparty credit exposures. Recent literature on the topic suggests that regression techniques, as described in, are capable of estimating future initial margin for vanilla products, but are challenged by option products, in particular products with multiple exercise rights, and diversified portfolios. In this paper we investigate how these limitations can be overcome, using the the example of Bermudan Swaptions. Building on recent research we show how the calculation of accurate “forward sensitivities” can be effectively embedded into the Longstaff Schwartz or American Monte Carlo (AMC) pricing algorithm using Algorithmic Differentiation (AD). We benchmark our AMC/AD results against lattice and semi-analytical calculations, and investigate the impact of differentiating the “indicator functions” involved in making exercise decisions. We then show that the AMC/AD approach is efficient from both a computational speed and a memory usage point of view and is hence a promising candidate for application across portfolios of multi-callable exotics.

The Co-Terminal Swap Market Model with Bergomi Stochastic Volatility

In this article, we apply the forward variance modeling approach by L.Bergomi to the co-terminal swap market model. We build an interest rate model for which all the market price changes of hedging instruments, interest rate swaps and European swaptions, are interpreted as the state variable variations, and no diffusion parameter calibration procedure is required. The model provides quite simple profit and loss (PnL) formula, with which we can easily understand where a material PnL trend comes from when it appears, and consider how we should modify the model parameters. The model has high flexibility to control the model dynamics because parameter calibration is unnecessary and the model parameters can be used solely for the purpose of the model dynamics control. With the model, the position management of the exotic interest rate products, e.g. Bermudan swaptions, can be carried out in a more sophisticated and systematic manner. A numerical experiment is performed to show the effectiveness of the approach for a Canary swaption, which is a special form of a Bermudan swaption.

Bermudan swaption model risk analysis: a local volatility approach

Prices of callable interest rate derivatives, such as Bermudan swaptions, can be strongly affected by our choice of interest rate model as well as its parameterization and calibration strategy. This paper develops a simple way to handle the inherent model uncertainty. The approach is based on analyzing the exercise deferral premium of Bermudan swaptions with exactly two exercise dates in a “minimal” nonparametric local volatility interest rate model. It is shown that the value of the right to defer exercise once, at a future call date, depends on the inputs obtained from a calibrated nonparametric local volatility model and can, moreover, be decomposed into a portfolio of standard European swaptions. Such a “fair” price can then be compared with one obtained using a calibrated Bermudan Monte Carlo pricer, yielding a practical way of handling the model risk involved in the chosen Bermudan pricer. Numerical examples from the US dollar swaption market are provided.

Neural Network for CVA: Learning Future Values

A new challenge to quantitative finance after the recent financial crisis is the study of credit valuation adjustment (CVA), which requires modeling of the future values of a portfolio. In this paper, following recent work in [Weinan E(2017), Han(2017)], we apply deep learning to attack this problem. The future values are parameterized by neural networks, and the parameters are then determined through optimization. Two concrete products are studied: Bermudan swaption and Mark-to-Market cross-currency swap. We obtain their expected positive/negative exposures, and further study the resulting functional form of future values. Such an approach represents a new framework for modeling XVA, and it also sheds new lights on other methods like American Monte Carlo.

Sub-Simulation-Free Upper Bounds for Bermudan Derivatives

In this paper, we propose a new regression-based sub-simulation-free upper bound. The method is based on approximating the martingale component of the lower bound process. The convergence and tightness of this upper bound are analyzed. In addition, the implementation details and enhancement techniques are discussed as well. Overall, our methods make a good trade-off between the time-efficiency of algorithms and the tightness of upper bounds. For a 20 rates Bermudan swaption, the distance between upper bounds with power enhancements and lower bounds is smaller than 10 basis points.

FFT network for interest rate derivatives with Lévy processes

This paper extends the fast Fourier transform (FFT) network to interest derivative valuation under the Hull–White model driven by a Levy process. The classical trinomial tree for the Hull–White model is a widely adopted approach in practice, but fails to accommodate the change in the driving stochastic process. Recent finance research supports the use of a Levy process to replace Brownian motion in stochastic modeling. The FFT network overcomes the drawback of the trinomial approach but maintains its advantages in super-calibration to the term structure of interest rate and efficient computation to various kinds of interest rate derivatives under Levy processes. The FFT network only requires knowledge of the characteristic function of the Levy process driving the interest rate process, but not of the interest rate process itself. The numerical comparison between the closed-form solutions of interest rate caps and swaptions and those from FFT network confirms that the proposed network is accurate and efficient. We also demonstrate its use in pricing Bermudan swaptions and other American-style options. Finally, the FFT network is expanded to accommodate path-dependent variables, and is applied to interest rate target redemption notes and a range of accrual notes.

PV and XVA Greeks for Callable Exotics by Algorithmic Differentiation

We generalize the algorithmic differentiation method proposed by Antonov (2016) from price Greeks to XVA Greeks. This method, named Backward Differentiation (BD), was developed in the context of computing price or PV Greeks for individual callable exotic trades.We start by treating cases where cashflow derivatives are sufficient for computing PV/XVA Greeks, i.e., where the differentiation of conditional expectations (or regression functions) is not necessary. For example, PV Greeks for Bermudan swaptions can be computed without having to perform the complicated step of regression function differentiation. We modify the Backward Differentiation algorithm to calculate Greeks for such instruments: the method is applied during the backward pricing procedure and has almost no overhead with respect to a pure backward pricing (without the Greeks).A general XVA calculation cannot be done using only the cashflow derivatives - some exceptions are listed in this article - instead, the differentiation of future instrument values that are results of the regression may be required. We leverage the algorithmic calculation of future values (Algorithmic Exposures) and describe the Adjoint Differentiation (AD) and the new BD for XVA Greeks. The latter algorithm is much simpler than the former, in particular, it does require the use of the instrument tape, i.e., it does not require the storage of certain payoff derivatives during the pricing procedure as is the case for AD. At the same time, both AD and BD enjoy a similar level of performance.

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 shows that the proposed approximation results in much smaller errors than the standard approach of using the regression function values.

Algorithmic Differentiation for Callable Exotics

In this article, we study the algorithmic calculation of present values greeks for callable exotic instruments. The speed of greeks evaluations becomes important with recent initial margin rules, including the ISDA standard model SIMM, requiring sensitivity calculations for non-cleared deals (e.g. callable exotic/structured products) for a large set of risk factors and on a daily basis.In general, present values of callable trades are obtained by the backward repetitive application of conditional expectation via least-squares Monte Carlo (regression) and arithmetic operations encoded in so called pricing (payoff) script. By nature, this pricing procedure is known to be computationally intensive leading to very slow greeks, if computed by a simple bump-and-reprice. The fast alternative is the adjoint differentiation (AD) method. However, a direct application of the AD is not straightforward because regressions introduce path interdependency, whereas traditional AD is applied on a path-by-path basis. We describe how to extend the traditional AD method to include the regressions. Usually, the procedure must include the recording of information in a data structure, often referred to as the instrument tape, during the backward pricing. This is followed by a subsequent playback of the tape in which the recorded information is used to compute the greeks. The tape adds a significant complexity to the AD implementation and can introduce memory issues (due to the significant storage demands), coding and debugging difficulties, etc.Next, we propose a new tapeless differentiation method, called backward differentiation (BD), for exotic deals. It is applied during the backward pricing procedure following the payoff script. Importantly, it completely avoids the tape and all of its related complications.We demonstrate the efficiency of the BD on the example of a Bermudan swaption calculated with a one-factor Hull-White model. For 50-100 greeks, the BD calculation time is only 2-6 times slower than a single pricing which leads to up 20 times acceleration with respect to the bump-and-reprice. We also give a numerical convergence analysis demonstrating the high quality of the BD greeks.

Model Independent Bounds for Accreting and Amortising Bermudan Swaptions

We derive model independent bounds for amortising and accreting Bermudan swaptions in terms of portfolios of standard Bermudan swaptions. In addition to the well-known upper bounds, we derive new lower bounds for both amortisers and accreters. Numerical results show the bounds to be tight in many situations. These two sided bounds provide useful checks against model arbitrage and super-replication of both long and short positions in the non-standard Bermudan.