Hull-White Research Articles

Monte Carlo Calculation of Exposure Profiles and Greeks for Bermudan and Barrier Options under the Heston Hull-White Model

Valuation of Credit Valuation Adjustment (CVA) has become an important field as its calculation is required in Basel III, issued in 2010, in the wake of the credit crisis. Exposure, which is defined as the potential future loss of a default event without any recovery, is one of the key elementsfor pricing CVA. This paper provides a backward dynamics framework for assessing exposure profiles of European, Bermudan and barrier options under the Heston and Heston Hull-White asset dynamics. We discuss the potential of an efficient and adaptive Monte Carlo approach, the Stochastic Grid Bundling Method}(SGBM), which employs the techniques of simulation, regression and bundling. Greeks of the exposure profiles can be calculated in the same backward iteration with little extra effort. Assuming independence between default event and exposure profiles, we give examples of calculating exposure, CVA and Greeks for Bermudan and barrier options.

ANALYTIC PRICING OF CoCo BONDS

We present a new model for pricing contingent convertible (CoCo) bonds which facilitates the calculation of equity, credit and interest rate risk sensitivities. We assume a lognormal equity process and a Hull–White (normal) short rate process for the conversion intensity with a downward jump in the equity price on conversion. We are able to derive an approximate solution in closed form based on the assumption that the conversion intensity volatility is asymptotically small. The simple first-order approximation is seen to be accurate for a wide range of market conditions, although particularly for longer maturities higher order terms in the asymptotic expansion may be needed.

Pricing perpetual timer option under the stochastic volatility model of Hull–White

Pricing perpetual timer option under the stochastic volatility model of Hull–White

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.

Dual-curve Hull–White interest rate model with stochastic volatility

The sub-prime mortgage crisis of 2008 had a great effect on the financial market; it led to new market features and regulations. In particular, the dual curve feature has appeared in the over-the-counter market. As a result, the yield curve for computing forward values and the yield curve for discounting are no longer the same due to counterparty default risk. In addition, deleveraging regulation encourages financial institutions to invest in widely known simple products, typically at a single short rate. The classical Hull–White model, which is typically useful for simple interest rate products, is inadequate for the dual-curve situation. This study extends the Hull–White model with stochastic volatility to the dual-curve economy. Analytical solutions are derived for vanilla securities such as risk bearing zero bonds, bond options, and caplets. Our formulas facilitate the calibration of parameters by simultaneously fitting to the dual curves and the implied volatility surfaces.

Model Calibration: Global Optimizer vs. Neural Network

Calibration of financial models can have more than one local minima present, requiring the use of global optimization techniques to properly calibrate them. In general, calibrating with a global optimizer will be a slow operation. An artificial neural network, properly trained, can replicate the same behaviour, providing the calibrated parameters in a fraction of the time. In the following, a simulated annealing global optimizer is used to calibrate a two-factor Hull-White model. A neural network is then trained to replicate it, and back-tested out-of-sample with good results.

The Present of Futures: Valuing Eurodollar-Futures Convexity Adjustments in a Multi-Curve World

Futures convexity adjustments in the multi-curve world depend on: i) the distribution of forward LIBORs, ii) the distribution of OIS rates, and iii) the correlation between LIBORs and OIS rates. In this article, we introduce a new multi-curve framework for pricing futures convexity adjustments. We assume that forward LIBORs follow a one-factor shifted-lognormal LMM, and that OIS rates evolve according to a general one-factor Cheyette (1992) model. We will consider two particular cases: 1) OIS rates follow a one-factor Hull-White (1990) model; 2) the LIBOR-OIS basis volatility is minimal, which includes the sub-case of deterministic basis when LIBOR and OIS curves are perfectly correlated. We will derive explicit convexity adjustments in both cases, and analyze the impact of the cap skew in their valuation. We will show that the extent a cap smile affects futures pricing depends on the chosen OIS dynamics, or equivalently on the assumed behavior of the LIBOR-OIS basis.

Modeling the Short Rate as a Levy Process and Option Pricing with the FFT

This paper describes a practical algorithm for modeling interest rate derivatives with the short rate following a Levy process using the fast Fourier transform algorithm (FFT). It can be used with any Levy process for which we have a closed form formula for the characteristic function, this includes a large variety of ‘fat-tailed’ jump diffusion processes. This model allows for the computation of forward rates and can be used to price American and Bermudan exercise options. Pricing algorithms are provided for option bonds and swaptions. The model is effectively equivalent to a tree approach except that diffusion is done by FFT instead of by branching. Pricing algorithms using trees or finite difference method can be easily adapted. Under the Normal distribution it can replicate the single factor Hull-White and Black-Karasinski models. The model supports mean reversion of interest rates. Monte Carlo simulations can be efficiently performed as well.

Correlations between the Market Price of Interest Rate Risk and Bond Yields

This paper examines empirical properties of the market price of interest rate risk, focusing on the relation between the price and interest rates. We briefly summarize how the market price of risk is estimated, and introduce the positive slope model to explain our empirical observation. The market price of risk is estimated for the U.S. Treasury market, 1970-2014, using the Hull–White model. We test the correlation between the market price of interest rate risk and bond yields. The results are that the yield change and term spreads are significantly correlated with the market price of risk, but the initial yields are not correlated with that. These results are theoretically interpreted by a mathematical model, and serve as a valuable reference for risk management as well as for study of financial policy.

Asset pricing under general collateralization

We consider the valuation of collateralized derivative contracts such as bond option or Caplet contracts. We allow for posting different collaterals such as securities or cash for the derivatives and its hedges. The pricing is based on modeling the joint evolution of collateral rate and the spread between collaterals. The Hull–White models are applied to collateral rate and spread to generate the closed pricing formula for zero coupon bond option. We also derive the pricing formula for Caplet under the Libor Market model and SABR model framework.

A unified approach to Bermudan and barrier options under stochastic volatility models with jumps

Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unified approach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull–White, Stein–Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α-Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.

A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps

After the recent financial crisis, the market for volatility derivatives has expanded rapidly to meet the demand from investors, risk managers and speculators seeking diversification of the volatility risk. In this paper, we develop a novel and efficient transform-based method to price swaps and options related to discretely-sampled realized variance under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection method combined with a novel continuous-time Markov chain (CTMC) weak approximation scheme of the underlying variance process. Contracts considered include discrete variance swaps, discrete variance options, and discrete volatility options. Models considered include several popular stochastic volatility models with a general jump size distribution: Heston, Scott, Hull–White, Stein–Stein, α-Hypergeometric, 3/2 and 4/2 models. Our framework encompasses and extends the current literature on discretely sampled volatility derivatives, and provides highly efficient and accurate valuation methods. Numerical experiments confirm our findings.

PRICING PERPETUAL TIMER OPTION UNDER THE STOCHASTIC VOLATILITY MODEL OF HULL–WHITE

The valuation of perpetual timer options under the Hull–White stochastic volatility model is discussed here. By exploring the connection between the Hull–White model and the Bessel process and using time-change techniques, the triple joint distribution for the instantaneous volatility, the cumulative reciprocal volatility and the cumulative realized variance is obtained. An explicit analytical solution for the price of perpetual timer call options is derived as a Black–Scholes–Merton-type formula.

Asset Pricing Under General Collateralization

We consider the valuation of collateralized derivative contracts such as bond option or Caplet contracts. We allow for posting different collaterals such as securities or cash for the derivatives and its hedges. The pricing is based on modeling the joint evolution of collateral rate and the spread between collaterals. The Hull-White models are applied to collateral rate and spread to generate the closed pricing formula for zero coupon bond option. We also derive the pricing formula for Caplet under the Libor Market model and SABR model framework.

Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps

In this paper, we develop a novel and efficient transform-based method to price equity-linked annuities (ELAs), including equity-indexed annuities (EIAs) and cliquet-style payoff structures popular in the insurance market under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection combined with a continuous-time Markov chain (CTMC) weak approximation scheme and spectral filtering. Contracts considered include EIAs with return guarantees of a cliquet style. Models considered include exponential Lévy processes, regime-switching Lévy processes, and stochastic volatility models with a general jump size distribution including Heston, Scott’s, Hull–White, Schöbel–Zhu, and the 3/2 models. We also consider some recently proposed stochastic volatility models in the literature such as the α-Hypergeometric model, and the 4/2 model. Our framework encompasses and extends the current literature on EIAs with highly efficient and accurate valuation methods. Numerical experiments confirm our findings.

The martingale approach for vulnerable binary option pricing under stochastic interest rate

We consider the vulnerable option pricing problem when the stochastic interest rate is driven by a Hull-White model. Based on the firm value model, we suppose that the stock prices, assets and liabilities of a company follow the relevant O-U processes. We adopt the martingale approach to determine the equivalent martingale measure for pricing the vulnerable binary option, the analytical pricing formula of the vulnerable binary options is derived.

Short-term traffic flow prediction using time-varying Vasicek model

This paper provides a two-step approach based on the stochastic differential equations (SDEs) to improve short-term prediction. In the first step of this framework, a Hull-White (HW) model is applied to obtain a baseline prediction model from previous days. Then, the extended Vasicek model (EV) is employed for modeling the difference between observations and baseline predictions (residuals) during an individual day. The parameters of this time-varying model are estimated at each sample using the residuals in a short duration of time before the time point of prediction; so it provides a real time prediction. The extracted model recovers the valuable local variation information during each day. The performance of our method in comparison with other methods improves significantly in terms of root mean squared error (RMSE), mean absolute error (MAE) and mean relative error (MRE) for real data from Tehran’s highways and the open-access PeMS database. We also demonstrate that the proposed model is appropriate for imputing the missing data in traffic dataset and it is more efficient than the probabilistic principal component analysis (PPCA) and k-Nearest neighbors (k-NN) methods.

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.

Simulation in the Real World

In this paper, we propose two practicable approaches for consistently modelling the realworld and risk-neutral measures within cross-asset Monte-Carlo frameworks. We go on to explore the necessity of supporting the real-world measure and consider its calibration with the aid of an explicit example of a 3-factor Hull-White model exhibiting the volatility hump and roll-down effect. This example parallels the 1-factor lattice model considered in Hull, Sokol and White, however, allows risk-premia which are stable through time and illustrates the necessity of having a sufficiently rich correlation structure in order to produce realistic dynamic risk-premia.

LSM Algorithm for Pricing American Option Under Heston–Hull–White’s Stochastic Volatility Model

In this paper, we present American option pricing under Heston---Hull---White's stochastic volatility and stochastic interest rate model. To do this, we first discretize the stochastic processes with Euler discretization scheme. Then, we price American option by using least-squares Monte Carlo algorithm. We also compare the numerical results of our model with the Heston-CIR model. Finally, numerical results show the efficiency of the proposed algorithm for pricing American option under the Heston---Hull---White model.