Intensity-based Framework Research Articles

Valuation of guaranteed minimum maturity benefits under mean reversion and jump models with surrender risk

We propose an efficient valuation approach for guaranteed minimum benefits embedded in variable annuity (VA) contracts, where the price of underlying asset is modeled by a mean reversion process with jumps. We consider the case in which early surrender is permitted, and an intensity-based framework is used to capture surrender behavior and other asset price volatilities. To analyze the contract within this complex stochastic setting, we rely on the Fourier cosine series expansion method to approximate the expectation of the value function, which serves as an estimation of the value of VA benefits. We conduct numerical examples to demonstrate the efficiency and accuracy of the proposed method. Sensitivity analysis of involved parameters is given.

The role of interest rate environment in mortgage pricing

Contributing to the literature on the relationship between interest rates and real economy, this paper examines the effect of interest rate environment on mortgage rates with and without prepayment and default risk. We model a mortgage in an intensity-based framework and show that how high interest rate environment leads to a larger mortgage rate spread, especially, in the presence of higher prepayment or default probabilities. Our results also show that in low interest rate environment, negative spreads can be observed. Another contribution of the paper is the divergence from conventional structural models which helps in overcoming the irrationality of decision-making and its adaptability to different economic environments. We also provide case studies on the effects (and interdependence) of the prepayment and default risks on the mortgage rates.

Efficient valuation of guaranteed minimum maturity benefits in regime switching jump diffusion models with surrender risk

We present an efficient valuation approach for guaranteed minimum maturity benefits (GMMBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We allow early surrender of the VA contract and impose surrender charges, which are important in practice to discourage early termination/lapse of the contract. We consider both continuously-monitored and discretely-monitored surrender behaviors before maturity, and utilize an intensity-based framework. Based on the continuous-time Markov chain (CTMC) approximation combined with the Fourier cosine series expansion method, we find that the valuation problem can be solved under a regime-switching jump diffusion framework. Both error analysis and numerical experiments demonstrate the accuracy and efficiency of the proposed method.

Intensity-based framework for surrender modeling in life insurance

In this paper, we propose an intensity-based framework for surrender modeling. We model the surrender decision under the assumption of stochastic intensity and use, for comparative purposes, the affine models of Vasicek and Cox–Ingersoll–Ross for deriving closed-form solutions of the policyholder’s probability of surrendering the policy. The introduction of a closed-form solution is an innovative aspect of the model we propose. We evaluate the impact of dynamic policyholders’ behavior modeling the dependence between interest rates and surrendering (affine dependence) with the assumption that mortality rates are independent of interest rates and surrendering. Finally, using experience-based decrement tables for both surrendering and mortality, we explain the calibration procedure for deriving our model’s parameters and report numerical results in terms of best estimate of liabilities for life insurance under Solvency II.

Correlated intensity, counter party risks, and dependent mortalities

In this paper we use an intensity-based framework to analyze and compute the correlated default probabilities, both in finance and actuarial sciences, following the idea of “change of measure” initiated by Collin-Dufresne et al. (2004). Our method is based on a representation theorem for joint survival probability among an arbitrary number of defaults, which works particularly effectively for certain types of correlated default models, including the counter-party risk models of Jarrow and Yu (2001) and related problems such as the phenomenon of “flight to quality”. The results are also useful in studying the recently observed dependent mortality for married couples involving spousal bereavement. In particular we study in details a problem of pricing Universal Variable Life (UVL) insurance products. The explicit formulae for the joint-life status and last-survivor status (or equivalently, the probability distribution of first-to-default and last-to-default in a multi-firm setting) enable us to derive the explicit solution to the indifference pricing formula without using any advanced results in partial differential equations.

Modelling default contagion using multivariate phase-type distributions

We model dynamic credit portfolio dependence by using default contagion in an intensity-based framework. Two different portfolios (with ten obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CDS-correlations. After the calibration, which are perfect for the banking portfolio, and good for the auto case, we study several quantities of importance in active credit portfolio management. For example, implied multivariate default and survival distributions, multivariate conditional survival distributions, implied default correlations, expected default times and expected ordered default times. The default contagion is modelled by letting individual intensities jump when other defaults occur, but be constant between defaults. This model is translated into a Markov jump process, a so called multivariate phase-type distribution, which represents the default status in the credit portfolio. Matrix-analytic methods are then used to derive expressions for the quantities studied in the calibrated portfolios.

Valuation of Residential Mortgage-Backed Securities with Proportional Hazard Model: Cumulant Expansion Approach to Pricing RMBS

This paper proposes a pricing formula for residential mortgage-backed securities (RMBS) with the proportional hazard model. First, we develop basic models of mortgage contracts with prepayment risk in the intensity-based framework. Next, assuming the proportional hazard model to describe prepayment risk, which is used as a typical prepayment model both academically and in practice; a general pricing formula for not only RMBS, but also IO and PO is derived by using the cumulant expansion method. Furthermore, it is also shown that the formula is applicable to various types of the proportional hazard models. Finally, numerical examples based on Japanese RMBS market data demonstrate that the formula is very accurate and useful in practice.

Credit cycles and macro fundamentals

Abstract We use an intensity-based framework to study the relation between macroeconomic fundamentals and cycles in defaults and rating activity. Using Standard and Poor's U.S. corporate rating transition and default data over the period 1980–2005, we directly estimate the default and rating cycle from micro data. We relate this cycle to the business cycle, bank lending conditions, and financial market variables. In line with earlier studies, the macro variables appear to explain part of the default cycle. However, we strongly reject the correct dynamic specification of these models. The problem is solved by adding an unobserved dynamic component to the model, which can be interpreted as an omitted systematic credit risk factor. By accounting for this latent factor, many of the observed macro variables loose their significance. There are a few exceptions, but the economic impact of the observed macro variables for credit risk remains low. We also show that systematic credit risk factors differ over transition types, with risk factors for downgrades being noticeably different from those for upgrades. We conclude that portfolio credit risk models based only on observable systematic risk factors omit one of the strongest determinants of credit risk at the portfolio level. This has obvious consequences for current modeling and risk management practices.

Macro Factors in the Term Structure of Credit Spreads

We estimate arbitrage-free term structure models of US Treasury yields and spreads on BBB and Brated corporate bonds in a doubly-stochastic intensity-based framework. A novel feature of our analysis is the inclusion of macroeconomic variables - indicators of real activity, inflation and financial conditions - as well as latent factors, as drivers of term structure dynamics. Our results point to three key roles played by macro factors in the term structure of spreads: they have a significant impact on the level, and particularly the slope, of the curves; they are largely responsible for variation in the prices of systematic risk; and speculative grade spreads exhibit greater sensitivity to macro shocks than high grade spreads. In addition to estimating risk-neutral default intensities, we provide estimates of physical default intensities using data on Moody's KMV EDFs™ as a forward-looking proxy for default risk. We find that the real and financial activity indicators, along with filtered estimates of the latent factors from our term structure model, explain a large portion of the variation in EDFs™ across time. Furthermore, measures of the price of default event risk implied by estimates of physical and risk-neutral intensities indicate that compensation for default event risk is countercyclical, varies widely across the cycle, and is higher on average and more variable for higher-rated bonds.

Modelling Default Contagion Using Multivariate Phase-Type Distributions

We model dynamic credit portfolio dependence by using default contagion in an intensity-based framework. Two different portfolios (with 10 obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CDS-correlations. After the calibration, which are perfect for the banking portfolio, and good for the auto case, we study several quantities of importance in active credit portfolio management. For example, implied multivariate default and survival distributions, multivariate conditional survival distributions, implied default correlations, expected default times and expected ordered defaults times. The default contagion is modelled by letting individual intensities jump when other defaults occur, but be constant between defaults. This model is translated into a Markov jump process, a so called multivariate phase type distribution, which represents the default status in the credit portfolio. Matrix-analytic methods are then used to derive expressions for the quantities studied in the calibrated portfolios.

Intensity-based framework and penalty formulation of optimal stopping problems

Financial derivatives commonly contain premature termination clauses, which are embedded rights held by the holder or writer. Well known examples of these stopping rights include the early exercise right in American options, the callable right in callable securities and the prepayment right in mortgage loans. In this paper, we show how to model the mortgagor's prepayment in mortgage loans and the issuer's call in the American warrant as an event risk using the intensity based approach, where the propensity of prepayment or calling is modeled by the intensity of a Poisson process. We illustrate that the corresponding pricing formulation resembles the penalty approximation approach commonly used in the solution of the linear complementarity formulation of an optimal stopping problem. We obtain several theoretical results on the prepayment strategies of mortgage loans and calling policies of American warrants. We also propose robust second order accurate numerical schemes for solving the penalty formulation of an optimal stopping problem.

Intensity-Based Framework and Penalty Formulation of Optimal Stopping Problems

Financial derivatives commonly contain pre-mature termination clauses, which are embedded rights held by the holder or writer. Well known examples of these stopping rights include the early exercise right in American options, callable right in callable securities and pre-payment right in mortgage loans. In this paper, we show how to model the mortgagor's prepayment in mortgage loans and issuer's call in American warrant as event risks using the intensity based approach, where the propensity of prepayment or calling is modeled by the intensity of a Poisson process. We illustrate that the corresponding pricing formulation resembles the penalty approximation approach commonly used in the solution of the linear complementarity formulation of an optimal stopping problem. We obtain several theoretical results on the prepayment strategies of mortgage loans and calling polices of American warrants. We also propose robust second order accurate numerical schemes for solving the penalty formulation of an optimal stopping problem.

Affine Models, Credit Spreads and the Delivery Option of a Multi-Issuer Bond Future

The contribution of this paper is two-fold: Firstly we use the intensity-based framework of Duffie/Singleton (1999) to analyse empirically the credit spread between Italian and German government bonds in an affine model framework. Secondly we value the delivery option of a multi-issuer bond-future contract. This contract allows the short to deliver bonds of different issuers and, therefore, depends on the credit spread between these bonds. The empirical part studies the valuation of German and Italian bonds using two-factor affine models. Similar to Pearson/Sun (1994) we combine cross-sectional and time-series information to estimate the process parameters with maximum likelihood. From our results we conclude that two-factor models produce about the same pricing errors for German and Italian bonds. However, both models explain only imperfectly the observed convex shape and the volatility smile of the default-spread structure. In the second part we analyse the delivery option of a bond future that is similar to the German Bund Future but allows additionally to deliver Italian government bonds. In contrast to our results for the valuation of coupon bonds we find that the option values in both models differ considerably. Our results provide useful insights for empirical work concerning affine intensity based models for the valuation of default-risky bonds and options that depend on a credit spread.

The Valuation of Executive Stock Options in an Intensity-Based Framework *

This paper presents a general intensity-based framework to value executive stock options (ESOs). It builds upon the recent advances in the credit risk modeling arena. The early exercise or forfeiture due to voluntary or involuntary employment termination and the early exercise due to the executive’s desire for liquidity or diversification are modeled as an exogenous point process with random intensity dependent on the stock price. Two analytically tractable specifications are given where the ESO value, expected time of exercise or forfeiture, and the expected stock price at the time of exercise or forfeiture are calculated in closed-form. JEL classification: G13, G39, M41.