Rafael Mendoza-Arriaga

Time-Changed Markov Processes in Unified Credit-Equity Modeling

This paper develops a novel class of hybrid credit-equity models with state-dependent jumps, local-stochastic volatility and default intensity based on time changes of Markov pro- cesses with killing. We model the defaultable stock price process as a time changed Markov difiusion process with state-dependent local volatility and killing rate (default intensity). When the time change is a Levy subordinator, the stock price process exhibits jumps with state-dependent Levy measure. When the time change is a time integral of an activity rate process, the stock price process has local-stochastic volatility and default intensity. When the time change process is a Levy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state-dependent jumps, local-stochastic volatility and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace trans- form of the time change are both available in closed form, the expectation operator of the time changed process is expressed in closed form as a single integral in the complex plane. If the payofi is square-integrable, the complex integral is further reduced to a spectral ex- pansion. To illustrate our general framework, we time change the jump-to-default extended CEV model (JDCEV) of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local-stochastic volatility and default intensity. These models can be used to jointly price equity and credit derivatives.

Pricing Equity Default Swaps under the Jump to Default Extended CEV Model

Equity default swaps (EDS) are hybrid credit-equity products that provide a bridge from credit default swaps (CDS) to equity derivatives with barriers. This paper develops an analytical solution to the EDS pricing problem under the jump-to-default extended constant elasticity of variance model (JDCEV) of Carr and Linetsky. Mathematically, we obtain an analytical solution to the first passage time problem for the JDCEV diffusion process with killing. In particular, we obtain analytical results for the present values of the protection payoff at the triggering event, periodic premium payments up to the triggering event, and the interest accrued from the previous periodic premium payment up to the triggering event, and we determine arbitrage-free equity default swap rates and compare them with CDS rates. Generally, the EDS rate is strictly greater than the corresponding CDS rate. However, when the triggering barrier is set to be a low percentage of the initial stock price and the volatility of the underlying firm’s stock price is moderate, the EDS and CDS rates are quite close. Given the current movement to list CDS contracts on organized derivatives exchanges to alleviate the problems with the counterparty risk and the opacity of over-the-counter CDS trading, we argue that EDS contracts with low triggering barriers may prove to be an interesting alternative to CDS contracts, offering some advantages due to the unambiguity, and transparency of the triggering event based on the observable stock price.

Modelling Electricity Prices: A Time Change Approach

To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump Cox-Ingersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a time-inhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the time-changed JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss–Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity derivatives, we introduce a class of measure changes that transforms one time-changed JCIR process into another time-changed JCIR process. We derive a closed-form formula for the futures price and obtain the Laplace transform of futures option price in terms of the Laplace transform of the time-changed JCIR process, which can then be efficiently inverted to yield the option price. By fitting our model to two major electricity markets in the US, we show that it is able to capture both the trajectorial and the statistical properties of electricity prices. Comparison with a popular jump-diffusion model is also provided.

Multivariate Subordination of Markov Processes with Financial Applications

This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes Xi with killing and an independent d-dimensional time change T, we construct a new process by time changing each of the Markov processes Xi with a coordinate Ti. When T is a d-dimensional Levy subordinator, the time changed process Yi:=Xi(Ti(t)) is a time-homogeneous Markov process with state-dependent jumps and killing in the product of the state spaces of Xi. The dependence among jumps of its components is governed by the d-dimensional Levy measure of the subordinator. When T is a d-dimensional additive subordinator, Y is a time-inhomogeneous Markov process. When Ti= ∫0tVsi ds with Vi forming a multi-variate Markov process, (Yi,Vi) is a Markov process, where each Vi plays a role of stochastic volatility of Yi. This construction provides a rich modeling architecture for building multivariate models in finance with time-dependent and state-dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multi-name unified credit-equity models and correlated commodity models.

Ornstein-Uhlenbeck Processes Time Changed with Additive Subordinators and Their Applications in Commodity Derivative Models

We characterize Ornstein-Uhlenbeck processes time changed with additive subordinators as time- inhomogeneous Markov semimartingales, based on which a new class of commodity derivative models is developed. Our models are tractable for pricing European, Bermudan and American futures options. Calibration examples show that they can be better alternatives than those developed in Li and Linetsky (2012). Our method can be applied to many other processes popular in various areas besides finance to develop time-inhomogeneous Markov processes with desirable features and tractability.

Additive Subordination and Its Applications in Finance

This paper studies additive subordination, which we show is a useful technique for constructing time-inhomogeneous Markov processes with analytical tractability. This technique is a natural generalization of Bochner’s subordination that has proved to be extremely useful in financial modeling. Probabilistically, Bochner’s subordination corresponds to a stochastic time change with respect to an independent Lévy subordinator, while in additive subordination, the Lévy subordinator is replaced by an additive one. We generalize the classical Phillips theorem for Bochner’s subordination to the additive subordination case, based on which we provide Markov and semimartingale characterizations for a rich class of jump-diffusions and pure jump processes obtained from diffusions through additive subordination, and obtain spectral decomposition for them. To illustrate the usefulness of additive subordination, we develop an analytically tractable cross-commodity model for spread option valuation that is able to calibrate the implied volatility surface of each commodity. Moreover, our model can generate implied correlation patterns that are consistent with market observations and economic intuitions.

Variance Swaps on Defaultable Assets and Market Implied Time-Changes

We compute the value of a variance swap when the underlying is modeled as a Markov diffusion process time changed by a Levy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent Levy measure and local stochastic volatility and have a local stochastic default intensity. Moreover, the Levy subordinator that drives the underlying can be obtained directly by observing European call/put prices. To illustrate our general framework, we provide an explicit formula for the value of a variance swap when the underlying is modeled as a Levy subordinated jump-to-default constant elasticity of variance process (see [Carr and V. Linetsky, Finance Stoch., 10, pp. 303--330, 2005]). In this example, we extend the results of [Mendoza-Arriaga, Carr, and Linetsky, Math. Finance, 20, pp. 527--569, 2010], by allowing for joint valuation of credit and equity derivatives as well as variance swaps.

Marshall-Olkin Distributions, Subordinators, Efficient Simulation, and Applications to Credit Risk

Abstract In the paper we present a novel construction of Marshall–Olkin (MO) multivariate exponential distributions of failure times as distributions of the first-passage times of the coordinates of multidimensional Lévy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Lévy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2 n -1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation effort is linear in n. To illustrate, we present a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

Analytical representations for the basic affine jump diffusion

The Basic Affine Jump Diffusion (BAJD) process is widely used in financial modeling. In this paper, we develop an exact analytical representation for its transition density in terms of a series expansion that is uniformly-absolutely convergent on compacts. Computationally, our formula can be evaluated to high level of accuracy by easily adding new terms which are given explicitly. Furthermore, it can be easily generalized to give an analytical expression for the transition density of the subordinate BAJD process which is more realistic than the BAJD process, while existing approaches cannot.

Valuation of collateralized debt obligations in a multivariate subordinator model

The paper develops valuation of multi-name credit derivatives, such as collateralized debt obligations (CDOs), based on a novel multivariate subordinator model of dependent default (failure) times. The model can account for high degree of dependence among defaults of multiple firms in a credit portfolio and, in particular, exhibits positive probabilities of simultaneous defaults of multiple firms. The paper proposes an efficient simulation algorithm for fast and accurate valuation of CDOs with large number of firms.