Black-Derman-Toy Research Articles

Zero Black-Derman-Toy Model in Catastrophic Events: COVID-19 Performance Analysis

In this article, we continue the research of our recent interest rate tree model, called the Zero Black-Derman-Toy (ZBDT) model, which includes the possibility of a jump at each step to a practically zero interest rate. This approach allows a better match with the risk of financial slowdown caused by catastrophic events. We present how to valuate a wide range of financial derivatives using such a model. The classical Black-Derman-Toy (BDT) model and a novel ZBDT model are described, and analogies in their calibration methodology are established. Finally, two cases of applications of the novel ZBDT model are introduced. The first is the hypothetical case of an S-shaped term structure and decreasing volatility of yields. The second case is an application in the structure of US sovereign bonds in the 2020 economic slowdown caused by the coronavirus pandemic. The objective of this study is to understand the differences presented by the valuation in both models for exotic derivatives.

Zero Black–Derman–Toy Interest Rate Model

We propose a modification of the classical Black–Derman–Toy (BDT) interest rate tree model, which includes the possibility of a jump with a small probability at each step to a practically zero interest rate. The corresponding BDT algorithms are consequently modified to calibrate the tree containing zero interest rate scenarios. This modification is motivated by the recent 2007–2008 crisis in the United States, and it quantifies the risk of future crises in bond prices and derivatives. The proposed model can be useful to price derivatives. A comparison of option prices and implied volatilities on US Treasury bonds computed with both the proposed and the classical tree model is provided in six different scenarios along the different periods comprising the years 2002–2017.

Bermudan option in Singapore Savings Bonds

The Singapore Savings Bonds (SSB) is a unique investment program offered by the Singapore government whereby retail investors can earn risk-free tax-free step-up interest closely matched to Treasury bond rates for up to 10 years and can redeem on any business day prior to maturity without any early redemption penalty. This study analyses the unique design of the SSB and provides a valuation of the Bermudan option for early redemption that is embedded in the SSB. The Black–Derman–Toy model is used to build the interest rate tree, and an iterative method is employed to avoid arbitrary specification of the pre-determined short rate volatility function. This bespoke Bermudan option can have changing strike prices over time. It also has a novel characteristic whereby the value of exercise to a buyer need not equal to the cost of being exercised to a seller. Better understanding of embedded options within government savings bonds leads to innovative designs that may encourage effective citizens’ savings.

Invariant approaches for the analytic solution of the stochastic Black-Derman toy model

We work on the analytical solution of the stochastic differential equations (SDE) via invariant approaches. In particularly, we focus on the stochastic Black-Derman Toy (BDT) interest rate model, among others. After we present corresponding (1+1) parabolic linear PDE for BDT-SDE, we use theoretical framework about the invariant approaches for the (1+1) linear PDE being done in the literature. We show that it is not possible to reduce BDT-PDE into the first and second Lie canonical forms. On the other hand, we success to find transformations for reducing it to the third Lie canonical form. After that, we obtain analytical solution of BDT-PDE by using these transformations. Moreover, we conclude that it can be reduced to the fourth Lie canonical form but, to the best of our knowledge, its analytical solution in this form is hard to find yet.

Implementation of Short-Rate Models - A Case Study of the Black-Derman-Toy Model of Interest Rate

In order to value a bond with one or more embedded options, one must consider the volatility of interest rates, as changes in volatility will affect the possibility of the option being exercised. The valuation of bonds with embedded options can be achieved by constructing a binomial or trinomial interest rate tree that models the random evolution of future interest rates and then by using backward-induction to generate the value of the bond at each node. No-arbitrage models such as the Black-Derman-Toy (BDT) model are more commonly used in practice to price financial instruments with embedded options or contingent claims. In this paper therefore, we explore how one of the no-arbitrage short rate models, the BDT model, can be implemented in a practical and reproducible manner. This is the first in our series of discussions on short-rate models. In future essays, we will discuss the implementation of other short-rate models namely the Hull-White, Ho-Lee, Kalotay-Fabozzi-Williams and the Black-Karansinski models of interest rates.

Pricing Zero-Coupon Catastrophe Bonds Using EVT with Doubly Stochastic Poisson Arrivals

The frequency and severity of climate abnormal change displays an irregular upward cycle as global warming intensifies. Therefore, this paper employs a doubly stochastic Poisson process with Black Derman Toy (BDT) intensity to describe the catastrophic characteristics. By using the Property Claim Services (PCS) loss index data from 2001 to 2010 provided by the US Insurance Services Office (ISO), the empirical result reveals that the BDT arrival rate process is superior to the nonhomogeneous Poisson and lognormal intensity process due to its smaller RMSE, MAE, MRPE, and U and larger E and d. Secondly, to depict extreme features of catastrophic risks, this paper adopts the Peak Over Threshold (POT) in extreme value theory (EVT) to characterize the tail characteristics of catastrophic loss distribution. And then the loss distribution is analyzed and assessed using a quantile-quantile (QQ) plot to visually check whether the PCS index observations meet the generalized Pareto distribution (GPD) assumption. Furthermore, this paper derives a pricing formula for zero-coupon catastrophe bonds with a stochastic interest rate environment and aggregate losses generated by a compound doubly stochastic Poisson process under the forward measure. Finally, simulation results verify pricing model predictions and show how catastrophic risks and interest rate risk affect the prices of zero-coupon catastrophe bonds.

Hogan–Weintraub singularity and explosive behaviour in the Black–Derman–Toy model

We consider the simulation of the Black, Derman, Toy model with log-normally distributed rates in the spot measure, simulated in discrete time and with a continuous state variable. We note an explosive behaviour in the Eurodollar futures convexity adjustment at a critical value of the volatility, which depends on the maturity, the rate tenor and simulation time step size. In the limit of a very small simulation time step , this singularity appears for any volatility and reproduces the Hogan-Weintraub singularity, which is generic for short rate interest rate models with log-normally distributed rates. The singular behaviour arises from a region in the state space which is usually truncated off in finite difference and grid methods, and poorly sampled in Monte Carlo methods, and thus is not observed under usual simulation methods. We study the conditions under which this transition appears and give upper and lower bounds on the critical volatility.

An empirical study of the pricing of reverse floating interest rate‐linked products

PurposeReverse floating interest rate‐linked structured products are important innovative products for investors to achieve a relatively high yield at low interest rates, and the reasonable pricing of such products is an important factor to influence investors' needs and issuers' profits. The purpose of this paper is to empirically analyze the rationality of the pricing of reverse floating interest rate‐linked products.Design/methodology/approachThis paper combines the Itô's Lemma and introduces the Black‐Derman‐Toy (BDT) model into the time‐varying volatility to build a binary tree interest rates BDT model under the time‐varying volatility, and to establish the pricing model of reverse floating interest rate‐linked products. Dozens of product data of ABN AMRO Bank and other world‐renowned banks or financial institutions are empirically analyzed.FindingsThe results show that the average pricing of these products is high, and the expected rate of return of the product is lower than the same period of the Five‐year US Treasury Bill rate.Originality/valueThis paper has combined the theory and practice together. The research method described in this paper is of significance to the pricing of interest rate‐linked structured products, and the pricing method of binary tree BDT model to solve the term structure of interest rates and estimation problem of volatility term structure of interest rates.

Using the Black–Derman–Toy interest rate model for portfolio optimization

Abstract No-arbitrage interest rate models are designed to be consistent with the current term structure of interest rates. The diffusion of the interest rates is often approximated with a tree, in which the scenario-dependent fair price of any security is calculated as the present value of the risk-neutral expectation by backward induction. To use this tree in a portfolio optimization context it is necessary to account for the so-called “market price of risk”. In this paper we present a method to change the conditional probabilities in the Black–Derman–Toy model to the physical (or real) measure, including the market price of risk, and explore the economic implications for expected spot rates and for expected bond returns.

Pricing multicallable range accruals with the Libor Market Model

PurposeThe paper aims to propose a consistent and robust pricing/hedging methodology for callable fixed income structures with embedded caplet‐linked options.Design/methodology/approachA range of recently published (1997‐2003) works about the Libor Market Model (LMM) tackle the problems of modelling the forward curve with more than two factors and calibrating it to caps either/or to swaps. Other articles involve the pricing of Bermudan options using Monte Carlo simulation. In the form of case study, the very popular structure of multicallable range accrual bonds is used. A complete calibration methodology is described in detail, which links the structure's price to the market caps and swaptions prices as well as to the historical correlations between forward rates. We present the direct implementation of the Monte Carlo technique for this particular problem. Furthermore, we explore the application of the Longstaff–Schwartz least squares algorithm and its variations for the estimation of the expected value of continuation.FindingsThis paper suceeds in producing a consistent and robust pricing/hedging methodology for callable fixed income structures with embedded caplet‐linked options.Practical implicationsThe increased complexity of similar fixed income structures makes traditional approaches like Black–Derman–Toy or Hull‐White trees inadequate for the task of consistent pricing and hedging. Therefore, care must be taken to ensure consisted hedging across the different volatility markets.Originality/valueThis article explores variations and settings of the popular LMM and the Longstaff‐Scwartz algorithm that can be relatively consistent with both the cap and swaption volatility market. The framework is built using as a benchmark the most liquid fixed income structure so that it can be tested for robustness.

From data to model and back to data: A bond portfolio management problem

The bond portfolio management problem is formulated as a multiperiod stochastic program using interest rate scenarios. The scenarios are sampled from the binomial lattice from a Black–Derman–Toy model. The paper analyzes the sensitivity of the solution of the resulting large-scale mathematical program with respect to the model inputs. The numerical results are for the Italian bond market.

Calibrating the Black-Derman-Toy model: some theoretical results

The Black-Derman-Toy (BDT) model is a popular one-factor interest rate model that is widely used by practitioners. One of its advantages is that the model can be calibrated to both the current market term structure of interest rate and the current term structure of volatilities. The input term structure of volatility can be either the short term volatility or the yield volatility. Sandmann and Sondermann derived conditions for the calibration to be feasible when the conditional short rate volatility is used. In this paper conditions are investigated under which calibration to the yield volatility is feasible. Mathematical conditions for this to happen are derived. The restrictions in this case are more complicated than when the short rate volatilities are used since the calibration at each time step now involves the solution of two non-linear equations. The theoretical results are illustrated by showing numerically that in certain situations the calibration based on the yield volatility breaks down for apparently plausible inputs. In implementing the calibration from period n to period n + 1, the corresponding yield volatility has to lie within certain bounds. Under certain circumstances these bounds become very tight. For yield volatilities that violate these bounds, the computed short rates for the period (n, n + 1) either become negative or else explode and this feature corresponds to the economic intuition behind the breakdown.

On the form and risk-sensitivity of zero coupon bonds for a class of interest rate models

We consider the form and the comparative static properties of the price of a zero coupon bond with maturity T for a broad class of interest rate models. We first demonstrate that increased volatility increases the price of a T-claim whenever the price is convex as a function of the current short rate. We then present a class of diffusion models (including, for example, the Dothan, the Black–Derman–Toy, and the Merton model of interest rates) for which the positivity of the sign of the relationship between volatility and the price of zero coupon bonds is always unambiguously guaranteed. Consequently, we find that for the considered class of models the price of zero coupon bonds can be completely ordered in terms of the riskiness of the underlying interest rate dynamics. We also show that for the proposed class of interest rate models, increased volatility increases the price of all convex and non-increasing T-claims as well.

Bermudan Swaptions in the LIBOR Market Model

Bermudan swaptions have until recently been valued using only one-factor models such as the Black-Derman-Toy (BDT) or Black-Karasinski (BK) models. The LIBOR Market (LM) model which is a more general multi-factor model is becoming increasingly popular as a benchmark model. Whereas the BDT and BK models can be approximated using a lattice facilitating easy valuation of Bermudan swaption, the LM model doesn't conform to the lattice framework and as such the valuation seems very difficult. Monte-Carlo simulation is a popular alternative to the lattice framework for derivatives valution. In order to facilitate valuation of Bermudan swaptions the Monte-Carlo simulation technique must be extended. A few methods doing this are presently available, eg [And98]. A common feature of these methods is that the estimated option premia are only lower bounds on the true premia. The Stochastic Mesh method proposed by [BG97b] for valuation of Bermudan (equity) options with applications to equity options provides a lower and an upper bound. We have applied this method to the LM model and use this to verify the premia found by Andersen. We will also apply the approach suggested in [LS98] to the LM model and verify the premia found using that approach. As it turns out this approach is a special case of the [And98] approach. Furthermore we also examine the impact on the Bermudan swaption premia when moving from a LM model with only one factor to a LM model with multiple factors and do indeed find a significant---but not dramatic---impact. We find the [And98] and [LS98] approaches to be mutually consistent and in line with results obtained from low-biased Stochastic Mesh estimates.

A User's Guide to Interest Rate Models: Applications for Structured Finance

The advent of derivatives and structured products has coincided with a proliferation of fixed income models used to analyze hedging, pricing, forecasting, and estimation for the term structure of interest rates. This article evaluates five models Ho‐Lee (HL); Black‐Derman‐Toy (BDT); Vasicek; Cox‐Ingersoll‐Ross (CIR); and Heath‐Jarrow‐Morton (HJM) (see Exhibit 1) that are currently used by structured finance practitioners. We suggest which models are most appropriate for assets with different time horizons, interest rate sensitivities and cashflow properties. The authors link model selection to structured financial instruments with the singular focus on the trade‐off between model precision/complexity and calculation costs.