Andrea Macrina

Beyond Hazard Rates: A New Framework for Credit-Risk Modelling

A new approach to credit risk modelling is introduced that avoids the use of inaccessible stopping times. Default events are associated directly with the failure of obligors to make contractually agreed payments. Noisy information about impending cash flows is available to market participants. In this framework, the market filtration is modelled explicitly, and is assumed to be generated by one or more independent market information processes. Each such information process carries partial information about the values of the market factors that determine future cash flows. For each market factor, the rate at which true information is provided to market participants concerning the eventual value of the factor is a parameter of the model. Analytical expressions that can be readily used for simulation are presented for the price processes of defaultable bonds with stochastic recovery. Similar expressions can be formulated for other debt instruments, including multi-name products. An explicit formula is derived for the value of an option on a defaultable discount bond. It is shown that the value of such an option is an increasing function of the rate at which true information is provided about the terminal payoff of the bond. One notable feature of the framework is that it satisfies an overall dynamic consistency condition that makes it suitable as a basis for practical modelling situations where frequent recalibration may be necessary.

Dam Rain and Cumulative Gain

We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process. The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process. An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow–Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions and rainfall.

Pricing Fixed-Income Securities in an Information-Based Framework

Abstract The purpose of this article is to introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous-time processes that describe the flow of information concerning market factors in a monetary economy. The nominal pricing kernel is assumed to be given at any specified time by a function of the values of information processes at that time. Using a change-of-measure technique, we derive explicit expressions for the prices of nominal discount bonds and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. An example that shows how the model can be calibrated to an arbitrary initial yield curve is presented. We proceed to model the price level, which is also taken at any specified time to be given by a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of the nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply.

Rational multi-curve models with counterparty-risk valuation adjustments

We develop a multi-curve term structure set-up in which the modelling ingredients are expressed by rational functionals of Markov processes. We calibrate to London Interbank Offer Rate swaptions data and show that a rational two-factor log-normal multi-curve model is sufficient to match market data with accuracy. We elucidate the relationship between the models developed and calibrated under a risk-neutral measure and their consistent equivalence class under the real-world probability measure . The consistent -pricing models are applied to compute the risk exposures which may be required to comply with regulatory obligations. In order to compute counterparty-risk valuation adjustments, such as credit valuation adjustment, we show how default intensity processes with rational form can be derived. We flesh out our study by applying the results to a basis swap contract.

HEAT KERNEL MODELS FOR ASSET PRICING

A heat kernel approach is proposed for the development of a novel method for asset pricing over a finite time horizon. We work in an incomplete market setting and assume the existence of a pricing kernel that determines the prices of financial instruments. The pricing kernel is modeled by a weighted heat kernel driven by a multivariate Markov process. The heat kernel is chosen so as to provide enough freedom to ensure that the resulting model can be calibrated to appropriate data, e.g. to the initial term structure of bond prices. A class of models is presented for which the prices of bonds, caplets, and swaptions can be computed in closed form. The dynamical equations for the price processes are derived, and explicit formulae are obtained for the short rate of interest, the risk premium, and for the stochastic volatility of prices. Several of the closed-form models presented are driven by combinations of Markovian jump processes with different probability laws. Such models provide a basis for consistent applications in various market sectors, including equity markets, fixed-income markets, commodity markets, and insurance. The flexible multidimensional and multivariate structure on which the resulting price models are based lends itself well to the modeling of dependence across asset classes. As an illustration, the impact of spiraling debt, a typical feature of a financial crisis, is modeled explicitly, and the contagion effects can be readily observed in the dynamics of the associated asset returns.

HEAT KERNEL INTEREST RATE MODELS WITH TIME-INHOMOGENEOUS MARKOV PROCESSES

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

Security Pricing with Information-Sensitive Discounting

In this paper incomplete-information models are developed for the pricing of securities in a stochastic interest rate setting. In particu- lar we consider credit-risky assets that may include random recovery upon default. The market filtration is generated by a collection of information processes associated with economic factors, on which in- terest rates depend, and information processes associated with mar- ket factors used to model the cash flows of the securities. We use information-sensitive pricing kernels to give rise to stochastic interest rates. Semi-analytical expressions for the price of credit-risky bonds are derived, and a number of recovery models are constructed which take into account the perceived state of the economy at the time of default. The price of European-style call bond options is deduced, and it is shown how examples of hybrid securities, like inflation-linked credit-risky bonds, can be valued. Finally, a cumulative information process is employed to develop pricing kernels that respond to the amount of aggregate debt of an economy.

Real-Time Risk Management: An AAD-PDE Approach

We apply adjoint algorithmic differentiation (AAD) to the risk management of securities when their price dynamics are given by partial differential equations (PDE). We show how AAD can be applied to forward and backward PDEs in a straightforward manner. In the context of one-factor models for interest rates or default intensities, we show how price sensitivities are computed reliably and orders of magnitude faster than with a standard finite-difference (FD) approach. This significantly increased efficiency is obtained by combining (i) the adjoint forward PDE for calibrating model parameters, (ii) the adjoint backward PDE for derivatives pricing, and (iii) the implicit function theorem to avoid iterating the calibration procedure.

Discrete-time interest rate modelling

This paper presents an axiomatic scheme for interest rate models in discrete time. We take a pricing kernel approach, which builds in the arbitrage-free property and provides a link to equilibrium economics. We require that the pricing kernel be consistent with a pair of axioms, one giving the inter-temporal relations for dividend-paying assets, and the other ensuring the existence of a money-market asset. We show that the existence of a positive-return asset implies the existence of a previsible money-market account. A general expression for the price process of a limited-liability asset is derived. This expression includes two terms, one being the discounted risk-adjusted value of the dividend stream, the other characterising retained earnings. The vanishing of the latter is given by a transversality condition. We show (under the assumed axioms) that, in the case of a limited-liability asset with no permanently-retained earnings, the price process is given by the ratio of a pair of potentials. Explicit examples of discrete-time models are provided.

Rational Models for Inflation-Linked Derivatives

We construct models for the pricing and risk management of inflation-linked derivatives. The model is rational in the sense that affine payoffs written on the consumer price index have prices that are rational functions of the state variables. The nominal pricing kernel is constructed in a multiplicative manner that allows for closed-form pricing of vanilla inflation products suchlike zero-coupon swaps, caps and floors, year-on-year swaps, caps and floors, and the exotic limited price index swap. The model retains the attractive features of a nominal multi-curve interest rate model such as closed-form pricing of nominal swaptions. We conclude with examples of how the model can be calibrated to EUR data.