Pricing Of Interest Rate Derivatives Research Articles

Discretely Distributed Scheduled Jumps and Interest Rate Derivatives: Pricing in the Context of Central Bank Actions

Interest rate dynamics are influenced by various economic factors, and central bank meetings play a crucial role concerning this subject matter. This study introduces a novel approach to modeling interest rates, focusing on the impact of central banks’ scheduled interventions and their implications for pricing bonds and path-dependent derivatives. We utilize a modified Skellam probability distribution to address the discrete nature of scheduled interest rate jumps and combine them with affine jump-diffusions (AJDs) in order to realistically represent interest rates. We name this class the AJD–Skellam models. Within this class, we provide closed-form formulas for the characteristic functions of a still broad class of interest rate models. The AJD–Skellam models are well-suited for using the interest rate version of the Fourier-cosine series (COS) method for fast and efficient interest rate derivative pricing. Our methodology incorporates this method. The results obtained in the paper demonstrate enhanced accuracy in capturing market behaviors and in pricing interest rate products compared to traditional diffusion models with random jumps. Furthermore, we highlight the applicability of the model to risk management and its potential for broader financial analysis.

Interest rate derivatives in lending to construction companies in the Russian market

Subject. This article examines interest rate derivatives as a way to hedge the risks of construction companies in a period of high uncertainty in financial markets. Objectives. The article aims to build an approach to estimating the price of interest rate derivatives, taking into account the specifics of lending to construction companies in the Russian Federation. Methods. For the study, we used the Black model and a regression analysis. Results. The article presents the first version of the author-developed model for estimating cap options on the key interest rate of the Central Bank of the Russian Federation when lending to construction companies. Conclusions. It is necessary to make an amendment to the traditional model for estimating premiums for the cap option on the key interest rate of the Central Bank of the Russian Federation when lending to construction companies, taking into account the specifics of pricing such loans.

Pricing interest rate derivatives under volatility uncertainty

In this paper, we study the pricing of contracts in fixed income markets under volatility uncertainty in the sense of Knightian uncertainty or model uncertainty. The starting point is an arbitrage-free bond market under volatility uncertainty. The uncertainty about the volatility is modeled by a G-Brownian motion, which drives the forward rate dynamics. The absence of arbitrage is ensured by a drift condition. Such a setting leads to a sublinear pricing measure for additional contracts, which yields either a single price or a range of prices and provides a connection to hedging prices. Similar to the forward measure approach, we define the forward sublinear expectation to simplify the pricing of cashflows. Under the forward sublinear expectation, we obtain a robust version of the expectations hypothesis, and we show how to price options on forward prices. In addition, we develop pricing methods for contracts consisting of a stream of cashflows, since the nonlinearity of the pricing measure implies that we cannot price a stream of cashflows by pricing each cashflow separately. With these tools, we derive robust pricing formulas for all major interest rate derivatives. The pricing formulas provide a link to the pricing formulas of traditional models without volatility uncertainty and show that volatility uncertainty naturally leads to unspanned stochastic volatility.

Effective Markovian projection: application to CMS spread options and mid-curve swaptions

Pricing of interest rate derivatives, such as CMS spread or mid-curve options, depends on the modelling of the underlying single rates. For flexibility and realism, these rates are often described in the framework of stochastic volatility models. In this paper we allow rates to be modelled within a class of general stochastic volatility models, which includes the SABR, ZABR, free SABR and Heston models. We provide a versatile technique called Effective Markovian Projection, which allows a tractable model to be found that mimics the distribution of the more complex models used to price multi-rate derivatives. Three different numerical approaches are outlined and applied to relevant examples from practice. Finally, a new method that involves moment-matching of Johnson distributions is applied to facilitate closed-form pricing formulas.

AMFR-W Numerical Methods for Solving High-Dimensional SABR/LIBOR PDE Models

In this work, we mainly develop a new numerical methodology to solve a PDE model recently proposed in the literature for pricing interest rate derivatives. More precisely, we use high-order-in-time AMFR-W-methods, which belong to a class of W-methods based on approximate matrix factorization (AMF) and are especially suitable in the presence of mixed spatial derivatives. High-order convergence in time allows larger time steps, which, combined with the splitting of the involved operators, highly reduces the computational time for a given accuracy. Moreover, the consideration of a large number of underlying forward rates makes the PDE problem high dimensional in space, so the use of AMFR-W-methods with a sparse grid combination technique represents another innovative aspect, making AMFR-W more efficient than with full grids and opening the possibility of parallelization. Also, the consideration of new homogeneous Neumann boundary conditions provides another original feature to avoid the difficulties associated to the presence of boundary layers when using Dirichlet ones, especially in advection-dominated regimes. These Neumann boundary conditions motivate the introduction of a modified combination technique to overcome a decrease in the accuracy of the standard combination technique.

Finite Difference Method for the Hull–White Partial Differential Equations

This paper reviews the finite difference method (FDM) for pricing interest rate derivatives (IRDs) under the Hull–White Extended Vasicek model (HW model) and provides the MATLAB codes for it. Among the financial derivatives on various underlying assets, IRDs have the largest trading volume and the HW model is widely used for pricing them. We introduce general backgrounds of the HW model, its associated partial differential equations (PDEs), and FDM formulation for one- and two-asset problems. The two-asset problem is solved by the basic operator splitting method. For numerical tests, one- and two-asset bond options are considered. The computational results show close values to analytic solutions. We conclude with a brief comment on the research topics for the PDE approach to IRD pricing.

Yield curve data choice and potential moral hazard: An empirical exercise on pricing callable bonds

AbstractThere are alternative providers of zero‐coupon yield curve datasets. This essential input for most financial purposes can also be estimated from cross‐sectional market price information. Although all these datasets are representations of the same reality, each dataset provides estimations from different baskets of assets and different fitting techniques. As the properties of the yield curves and their time series dynamics are different, the results of pricing interest rate derivatives may be sensitive to the choice among these datasets. We compare our own spot rates estimation from GovPX bond data and three popular available interest rate datasets. From an empirical exercise on pricing callable bonds, we observe relevant differences in economic terms when volatilities from different inputs are used. Therefore, financial institutions may have an incentive for choosing unduly one or another yield curve dataset in their self‐interest, raising a moral hazard problem.

Robust Calibration For SVI Model Arbitrage Free

The purpose of this paper is to study the Stochastic Volatility Inspired model (SVI) as implied volatility model: we study the analytic part of the SVI with the arbitrage conditions, we establish the initial guess and the parameter’s boundaries. Until recently it was not possible to find sufficient conditions that would guarantee an SVI model calibration arbitrage-free. The main contribution in this paper is that we provide two methods to resolve the arbitrage problem (butterfly and calendar spread): the first one is numerical using the Sequential Least-Squares Quadratic Programming (SLSQP) algorithm, and the second one is analytical by using sufficient conditions that guarantee an SVI arbitrage-free. Our method guarantee to get SVI calibration with butterfly and calendar spread arbitrage-free, We provide many numerical examples with arbitrage such as Vogt Axel example and we show how to fix them. The calibration method is tested on 23 equity indexes with 14 maturities each and we get 322 slices fits using the same initial guess and the SVI parameters boundaries for all indexes. This new calibration method is very important and it meets practical need: resolving this arbitrage problem will pave the way to the surface calibration and the transition from implied volatility to local volatility using Dupire’s formula, therefore, it allows price different kind of path-dependent options such as barrier options, and American options. The SVI model could also be applied to price interest rate derivatives such as swaptions, interest rate caps, and floors.

Theory of Cryptocurrency Interest Rates

A term structure model in which the short rate is zero is developed as a candidate for a theory of cryptocurrency interest rates. The price processes of crypto discount bonds are worked out, along with expressions for the instantaneous forward rates and the prices of interest-rate derivatives. The model admits functional degrees of freedom that can be calibrated to the initial yield curve and other market data. Our analysis suggests that strict local martingales can be used for modeling the pricing kernels associated with virtual currencies based on distributed ledger technologies.

Yield curves from different bond data sets

It is well known that zero coupon rates are not observable variables. Their estimation process may be cumbersome and time consuming. We explore the extent to which the set of security prices used in the yield curve construction of three popular interest rate datasets (from the Federal Reserve Board, the US Department of the Treasury, and Bloomberg) may determine the results of different analyses. Using the same US Treasury prices from GovPX and applying the same fitting technique, we estimate zero coupon rates using different baskets of assets, i.e., including/excluding bills, on-the-run, and off-the-run bonds, attempting to mimic those used by each data providers. To illustrate the uncertainty surrounding these alternatives representations of the underlying yield curve, we examine common uses of these data sets in pricing, risk management and macroeconomic purposes. We find significant and sometime overwhelming differences in the volatility term structure, the pricing of interest rate derivatives, and the correlations among different forward rates particularly in both ends of the yield curve. Relevant implications are also observed on a classic test of the expectations hypothesis. The simplest asset basket, which only includes the on-the-run bills and bonds, is probably the one with the best results.

Pricing Interest Rate Derivatives after Ibor Fallback

In this paper, I introduce and define interest rate models for pricing interest rate derivatives in ibor fallback environment. July 2017 Andrew Bailey, the CEO of the U.K. Financial Conduct Authority (FCA), speech on The future of LIBOR contains that there is an increased expectation that some IBOR benchmarks will be discontinued in the near future. On July 12, 2018, ISDA published a consultation on benchmark fallbacks to market participants, and on the end of 2018, published the final results of new benchmark fallbacks for interest rate derivatives contracts that reference certain IBORs. As a result, quantitative analysts, risk managers and financial system engineers have been forced to develop fallback rate dynamics that model new benchmark fallbacks of the final results published by ISDA and evaluate interest rate derivatives corresponding to the fallback rate. Furthermore, they have to evaluate both ibor based interest rate derivatives contracted before ibor fallback and ones after it for ibor fallback environment. I show that fallback rate dynamics can be described with extending the classic interest rate models, and the dynamics can be available for pricing interest rate derivatives in ibor fallback environment.

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.

PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique

SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan and Lesniewsk (2008), Mercurio and Morini (2009) and Rebonato and White (2008). Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method.

A Consistent Stochastic Model of the Term Structure of Interest Rates for Multiple Tenors

Explicitly taking into account the risk incurred when borrowing at a shorter tenor versus lending at a longer tenor (roll-over risk), we construct a stochastic model framework for the term structure of interest rates in which a frequency basis (i.e. a spread applied to one leg of a swap to exchange one floating interest rate for another of a different tenor in the same currency) arises endogenously. This roll-over risk consists of two components, a credit risk component due to the possibility of being downgraded and thus facing a higher credit spread when attempting to roll over short-term borrowing, and a component reflecting the (systemic) possibility of being unable to roll over short-term borrowing at the reference rate (e.g., LIBOR) due to an absence of liquidity in the market. The modelling framework is of reduced form in the sense that (similar to the credit risk literature) the source of credit risk is not modelled (nor is the source of liquidity risk). However, the framework has more structure than the literature seeking to simply model a different term structure of interest rates for each tenor frequency, since relationships between rates for all tenor frequencies are established based on the modelled roll-over risk. We proceed to consider a specific case within this framework, where the dynamics of interest rate and roll-over risk are driven by a multifactor Cox/Ingersoll/Ross-type process, show how such model can be calibrated to market data, and used for relative pricing of interest rate derivatives, including bespoke tenor frequencies not liquidly traded in the market.

A novel partial integrodifferential equation-based framework for pricing interest rate derivatives under jump-extended short-rate models

Interest rate derivatives under jump-extended short-rate models have commonly been valued using lattice methods. Unfortunately, lattice methods have pitfalls, mainly in terms of accuracy, efficiency and ease of programming. This paper proposes a much faster and more accurate valuation method based on partial integrodifferential equations (PIDEs). Spatial differential and integral terms are discretized by fourthorder finite differences and Simpson's rule, respectively. For the time-stepping, we investigate a Crank-Nicolson scheme, an implicit-explicit (IMEX) scheme and an exponential time integration scheme. We demonstrate our method by pricing bonds as well as European and American options on both zero-coupon and coupon bonds under jump-extended constant-elasticity-of-variance processes. Through a variety of test problems, we demonstrate that our PIDE-based approach is remarkably superior to lattice methods in terms of accuracy, speed and ease of implementation. By way of example, pricing a one-year zero-coupon bond under the Vasicek model extended with exponential jumps, our method attains an accuracy of 104 in 0:09 seconds with the IMEX scheme, whereas a truncated tree takes 78 seconds to reach the same accuracy.

Pricing interest rate derivatives with model risk

This paper studies an interest rate derivative when there is the model risk in an interest rate model. We consider a mean reverting interest rate process whose volatility model is not known. Most of prices of interest rate derivatives cannot be determined uniquely, based on this interest rate model. We study the price bounds of a derivative and propose how to calculate the price bounds by a trinomial model. Further, we analyze the model risk of derivatives and their portfolios numerically.

A Comparison of Control Variate Methods for Pricing Interest Rate Derivatives in the LIBOR Market Model

AbstractThis paper studies the control variate method for pricing interest rate derivatives driven by the LIBOR market model. Several control variates are constructed based on distinctive approximations for the LIBOR market model. Numerical results show the great efficiency of our methods. The idea in this paper can also be extended to price other interest rate derivatives under the LIBOR market model, such asSwaptions, Caps, some path dependent interest rate derivatives, and so forth.

An Arbitrage-Free Nelson-Siegel Model with Unspanned Stochastic Volatility for the Pricing of Interest Rate Derivatives

We propose a generalized arbitrage-free Nelson-Siegel model under the HJM framework. It features unspanned stochastic volatility factors while maintaining a Nelson-Siegel factor loading structure. The price of the interest rate derivatives, including European options, Caps and Swaptions are then obtained in semi closed-form. We calibrate the model using an extensive panel data, including the US Libor rates, Swap rates, caps and swaptions. By estimating our model via the extended Kalman filter, we find strong evidence that our model prices interest rates and their derivatives accurately.

Pricing Interest Rate Derivatives with Multilinear Interpolations and Transition Densities

We present a general and flexible numerical procedure for pricing European interest-rate derivatives within multifactor affine term structure models by means of piecewise multilinear interpolations. Our procedure relies to the maximum extent on the true density of the state process and solves the pricing problem in quasi-closed form. Then, we show how to generalize our approach for pricing American-style options. As an illustration, we price European interest-rate swaptions and Eurodollar futures options, which cannot be analytically evaluated. We use nine affine models and show that our approach converges rapidly and competes well against Monte Carlo simulation. We also demonstrate that our approach remains well-behaved for pricing deeply out-of-the money interest-rate derivatives.

Pricing Interest Rate Derivatives and Computing Pathwise Greeks in the Extended LIBOR Market Model

Classical interest rate models were formulated under the no-arbitrage assumption of a unique yield curve. However, in the outbreak of the 2007 global financial crisis, market interest rates witnessed unprecedented segmentation, and this instigated the modelling of a basis spread, whose non-negligible widening could no longer be neglected. One extension of the LIBOR market model (LMM) to incorporate basis spread was proposed by Mercurio (2010a), and in this paper, we examine how his theoretical description is translated into practical implementation to price interest rate derivatives and compute pathwise Greeks. Specifically, our contribution is to develop and implement the theory of the extended LMM in the framework of a displaced diffusion process in the spot LIBOR measure. The adjoint mode from algorithmic differentiation is premiered in this new implementation to compute pathwise deltas and vegas before the accumulated knowledge is applied in the pricing of interest rate caps and one-way floaters.