ngfei Li

Optimal Stopping and Early Exercise: An Eigenfunction Expansion Approach

This paper proposes a new approach to solve finite-horizon optimal stopping problems for a class of Markov processes that includes one-dimensional diffusions, birth--death processes, and jump diffusions and continuous-time Markov chains obtained by time-changing diffusions and birth-and-death processes with Levy subordinators. When the expectation operator has a purely discrete spectrum in the Hilbert space of square-integrable payoffs, the value function of a discrete optimal stopping problem has an expansion in the eigenfunctions of the expectation operator. The Bellmans dynamic programming for the value function then reduces to an explicit recursion for the expansion coefficients. The value function of the continuous optimal stopping problem is then obtained by extrapolating the value function of the discrete problem to the limit via Richardson extrapolation. To illustrate the method, the paper develops two applications: American-style commodity futures options and Bermudan-style abandonment and capacity expansion options in commodity extraction projects under the subordinate Ornstein-Uhlenbeck model with mean-reverting jumps with the value function given by an expansion in Hermite polynomials.

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.

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.

Discretely Monitored First Passage Problems and Barrier Options: An Eigenfunction Expansion Approach

This paper develops an eigenfunction expansion approach to solve discretely monitored first passage time problems for a rich class of Markov processes, including diffusions and subordinate diffusions with jumps, whose transition or Feynman–Kac semigroups possess eigenfunction expansions in L2

Option Pricing in Some Non-Levy Jump Models

L^{2}

An efficient algorithm based on eigenfunction expansions for some optimal timing problems in finance

-spaces. Many processes important in finance are in this class, including OU, CIR, (JD)CEV diffusions and their subordinate versions with jumps. The method represents the solution to a discretely monitored first passage problem in the form of an eigenfunction expansion with expansion coefficients satisfying an explicitly given recursion. A range of financial applications is given, drawn from across equity, credit, commodity, and interest rate markets. Numerical examples demonstrate that even in the case of frequent barrier monitoring, such as daily, approximating discrete first passage time problems with continuous solutions may result in unacceptably large errors in financial applications. This highlights the relevance of the method to financial applications.

Analytical representations for the basic affine jump diffusion

This paper considers pricing European options in a large class of one-dimensional Markovian jump processes known as subordinate diffusions, which are obtained by time changing a diffusion process with an independent Levy or additive random clock. These jump processes are non-Levy in general, and they can be viewed as a natural generalization of many popular Levy processes used in finance. Subordinate diffusions offer richer jump behavior than Levy processes and they have found a variety of applications in financial modeling. The pricing problem for these processes presents unique challenges, as existing numerical PIDE schemes fail to be efficient and the applicability of transform methods to many subordinate diffusions is unclear. We develop a novel method based on a finite difference approximation of spatial derivatives and matrix eigendecomposition, and it can deal with diffusions that exhibit various types of boundary behavior. Since financial payoffs are typically not smooth, we apply a smoothing tech...

Error Analysis of Finite Difference and Markov Chain Approximations for Option Pricing

This paper considers the optimal switching problem and the optimal multiple stopping problem for one-dimensional Markov processes in a finite horizon discrete time framework. We develop a dynamic programming procedure to solve these problems and provide easy-to-verify conditions to characterize connectedness of switching and exercise regions. When the transition or Feynman-Kac semigroup of the Markov process has discrete spectrum, we develop an efficient algorithm based on eigenfunction expansions that explicitly solves the dynamic programming problem. We also prove that the algorithm converges exponentially in the series truncation level. Our method is applicable to a rich family of Markov processes which are widely used in financial applications, including many diffusions as well as jump-diffusions and pure jump processes that are constructed from diffusion through time change. In particular, many of these processes are often used to model mean-reversion. We illustrate the versatility of our method by considering three applications: valuation of combination shipping carriers, interest-rate chooser flexible caps and commodity swing options. Numerical examples show that our method is highly efficient and has significant computational advantages over standard numerical PDE methods that are typically used to solve such problems.

Equivalent Measure Changes for Subordinate Diffusions

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.