Martingale Method Research Articles

Optimal investment with a corporate bond

The present paper analyzes the optimal investment strategy in a corporate (defaultable) bond, a stock and a bank account in a continuous time model. We model the corporate bond price through a reduced-form approach and solve the dynamics of its price. The optimal investment process will be worked out first with a general risk-averse utility function, and then an optimal strategy with CARA utility will be presented using martingale methods. The optimal investment strategy is analyzed numerically for the CARA utility.

Martingale methods for pricing inventory penalties under continuous replenishment and compound renewal demands

This paper addresses the problem of inventory penalty pricing under the risk-neutral valuation principle. The underlying production-inventory system has a constant replenishment rate and a compound renewal demand stream (i.e., iid demand interarrival times are independent of iid demand sizes), and is subject to underage and overage penalties. Our pricing approach treats the penalties as a series of perpetual American options, and constructs auxiliary martingale processes in term of the inventory process. We provide a necessary and sufficient martingale condition for general compound renewal demands. Explicit expressions of penalty functions for underage and overage are obtained for the case where demand arrivals follow a Poisson process.

The quantile process under random censoring

In this paper we discuss the asymptotical properties of quantile processes under random censoring. In contrast to most work in this area we prove weak convergence of an appropriately standardized quantile process under the assumption that the quantile regression model is only linear in the region, where the process is investigated. Additionally, we also discuss properties of the quantile process in sparse regression models including quantile processes obtained from the Lasso and adaptive Lasso. The results are derived by a combination of modern empirical process theory, classical martingale methods and a recent result of Kato (2009).

Voluntary retirement and portfolio selection: Dynamic programming approaches

I consider a continuous-time optimal consumption and portfolio selection problem with voluntary retirement. When the agent’s utility of consumption and leisure are of Cobb–Douglas form, I use the dynamic programming method to derive the value function and optimal strategies in closed-form. These coincide with the solutions of Farhi and Panageas (2007) [7] , who have solved the problem using a martingale method.

Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of ideas related to the Marcinkiewicz-Zygmund strong law of large numbers, as well as more recent work of Kesten and Maller. Our proofs are robust and use martingale methods. We demonstrate the benefit of the generality of our results by applications to some non-classical models, including random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes.

A Martingale Approach to Optimal Portfolios with Jump-diffusions

This paper investigates optimal investment-consumption strategies that maximize the expected utility of consumption and/or terminal wealth under jump-diffusion models using a martingale method. We characterize the optimal trading strategy and the optimal martingale measure in terms of a system of equations and obtain explicit solutions for the power and logarithmic utility case.

Some Results on a Double Compound Poisson-Geometric Risk Model with Interference

In this paper, we study the actual operating of an insurance company with random income. A double compound Poisson-Geometric risk model with interference was established. By using the martingale method, the adjustment coefficient equation, the formula and the upper bound of ruin probability, the time to reach a given level in this new risk mo- del were obtained.

Nonparametric estimation for a two-dimensional renewal process

This paper introduces nonparametric estimation methods for spatial point processes satisfying a particular renewal property. Martingale methods yield a unified approach for renewal processes in both one and two dimensions, and can be used for both synchronous and asynchronous data. In each case, we obtain martingale estimators of the avoidance probabilities that characterize the renewal process. Asymptotic properties of the estimators are studied analytically and empirically.

The martingale method of shortfall risk minimization in a discrete time market

The martingale method of shortfall risk minimization in a discrete time market

Mini-Workshop: Random Trees, Information and Algorithms

The subject of this Mini-Workshop is the probabilistic analysis of random tree models that originate from applications in Computer Science. Emphasis is put on their connections to algorithms and information theory. Trees with a stochastic growth dynamic appear in Computer Science as data structures, in the context of coding schemes as well as connected to fundamental algorithms such as sorting, searching and selecting. The focus of this Mini-Workshop is on probabilistic and analytic techniques that have been developed recently in the asymptotic analysis of random trees such as martingale methods, connections to branching random walks, the contraction method, the method of moments as well as various techniques based on generating functions.

A Dynamic Programming Approach to Constrained Portfolios

This paper studies constrained portfolio problems that may involve constraints on the probability or the expected size of a shortfall of wealth or consumption. Our first contribution is that we solve the problems by dynamic programming, which is in contrast to the existing literature that applies the martingale method. More precisely, we construct the non-separable value function by formalizing the optimal constrained terminal wealth to be a (conjectured) contingent claim on the optimal non-constrained terminal wealth. This is relevant by itself, but also opens up the opportunity to derive new solutions to constrained problems. As a second contribution, we thus derive new results for non-strict constraints on the shortfall of intermediate wealth and/or consumption.

Pricing of Some Exotic Options under Jump Diffusion and Stochastic Interest Rates Model

This paper assumes that jump process in underlying assets-stock price is more common than Poisson process and derive the pricing formulas of some exotic options under the stochastic interest rates by martingale method with the risk-neutral hypothesis.

OPTIMAL CONSUMPTION AND INVESTMENT IN INCOMPLETE MARKETS WITH GENERAL CONSTRAINTS

We study an optimal consumption and investment problem in a possibly incomplete market with general, not necessarily convex, stochastic constraints. We provide explicit solutions for investors with exponential, logarithmic as well as power utility and show that they are unique if the constraints are convex. Our approach is based on martingale methods that rely on results on the existence and uniqueness of solutions to BSDEs with drivers of quadratic growth.

A delay financial model with stochastic volatility; martingale method

In this paper, we extend a delayed geometric Brownian model by adding a stochastic volatility term, which is driven by a hidden process of fast mean reverting diffusion, to the delayed model. Combining a martingale approach and an asymptotic method, we develop a theory for option pricing under this hybrid model. The core result obtained by our work is a proof that a discounted approximate option price can be decomposed as a martingale part plus a small term. Subsequently, a correction effect on the European option price is demonstrated both theoretically and numerically for a good agreement with practical results.

Pricing Option on Jump Diffusion and Stochastic Interest Rates Model

This paper assumed that the stock price jump process for a special kind of renewal jump process, that is incident time interval for independent and subordinate to Gamma distribution random variable sequence. We obtain the European bi-direction option pricing formulas on jump diffusion model under the stochastic interest rates by simply mathematical induce by means of martingale method.

Pricing Equity Swaps in an Economy with Jumps

Empirical evidence confirms that asset price processes exhibit jumps and that asset returns are not Gaussian. We provide a pricing model for equity swaps including quanto equity swaps for a non-Gaussian market. The market is driven by a general marked point process as well as by a standard multidimensional Wiener process. In order to obtain closed-form solutions of the swap values, we assume that all parameters in the asset price processes are deterministic, but possibly functions of time. We derive swap prices using martingale methods rather than replicating portfolios, and we show how to calculate the convexity correction term analytically. Our results are an extension of the results of Liao and Wang (2003; Pricing models of equity swaps, The Journal of Futures Markets, 23(8), pp. 751–772). The martingale method is the key that enables the extension.

Option Pricing and Hedging under Stochastic Verhulst-Gompertz Equation

Emphasis is placed on the valuation of plain vanilla option when the price process of underlying asset is described by the stochastic Verhulst-Gompertz Equation with network externality effects in a complete market. The method is based on the change of measure, Girsanov theorem and martingale valuation techinique. The application to an exchange option is made attempt and the valuation formula for this option like the Black-Scholes one is derived. A simple relationship similar to the put-call-parity between the exchange call option and the put option is provided. Our results will be useful to analyze and to hedge the price evolution at a sudden rise or crash of stock and commodity markets. 1 Introduction. The prices of some assets traded in the financial market have the ten- dency raised more when the prices begin to rise, it is because most traders expect that a price rises more and the purchase orders of other traders are induced as a result. Conversely, when the prices begin to fall, selling orders of other traders are induced. In the actual mar- ket, we often observe such a phenomenon called network externality price effects. These effects have been recognized as very significant concept in the field of marketing. Nerlove and Arrow(8) , Bass(2), Vidale and Wolfe(11) and Gould(4) made use of this deterministic version to study the dynamic behavior of new product and advertising policy. Their models were stimulated by a classical theory of logistic curve by Verhulst(10). Gompertz extended Verhulst's model to so-called a deterministic Verhulst-Gompertz model in order to investi- gate the growth of population. In the fields of finance, Schwartz(9) gave a stochastic version of Verhulst-Gompertz model and developed the random behavior of commodity price. His model was based on the biological problem studied by Goel and Richer-Dyn(3). The main purpose of this paper is devoted to describe the network externality effects by a generalized stochastic Verhulst-Gompertz equation and to derive pricing formulae for a plain vanilla and an exchange options when the price of underlying asset shows the network externality price effects. The organization of this paper is as follows. The next section briefly introduces the generalized stochastic Verhulst-Gompertz equation to express the network externality effects and in Section 3 we develope the martingale measure to evaluate the value of option. In Section 4 we derive a hedging strategy and a pricing formula of an exchange option that gives the holder the right which exchanges the underlying asset with the stochastic Verhulst-Gompertz equation for the asset with the geometric Brownian motion (Black-Scholes model) by means of martingale pricing method(7). If we regard the former as the spot price of energy commodity and the latter as the stock price of business firm that trades the energy commodity, such an exchang option will be very useful to hedge the sudden rise or heavy fall in price of energy and make a contribution to the stabilized price of energy. The paper concludes with a brief summary.

Utility maximization in models with conditionally independent increments

We consider the problem of maximizing expected utility from terminal wealth in models with stochastic factors. Using martingale methods and a conditioning argument, we determine the optimal strategy for power utility under the assumption that the increments of the asset price are independent conditionally on the factor process.

Generalized control variate methods for pricing Asian options

The conventional control variate method proposed by Kemna and Vorst (1990) to evaluate Asian options under the Black-Scholes model can be interpreted as a particular selection of linear martingale controls. We generalize the constant control parameter into a control process to gain more reduction on variance. By means of an option price approximation, we construct a martingale control variate method, which outperforms the conventional control variate method. It is straightforward to extend such linear control to a nonlinear situation such as the American Asian option problem. From the variance analysis of martingales, the performance of control variate methods depends on the distance between the approximate martingale and the optimal martingale. This measure becomes helpful for the design of control variate methods for complex problems such as Asian option under stochastic volatility models. We demonstrate multiple choices of controls and test them under MC/QMC (Monte Carlo/ Quasi Monte Carlo)- simulations. QMC methods work signicantly well after adding a control, the variance reduction ratios increase to 260 times for randomized QMC compared with 60 times for MC simulations with a control.

Option pricing with quadratic volatility: a revisit

This paper considers the pricing of European options on assets that follow a stochastic differential equation with a quadratic volatility term. We correct errors in the existing literature, extend the pricing formulas to arbitrary root configurations, and list alternative representations of option pricing formulas to improve computational performance. Our exposition is based entirely on probabilistic arguments, adding a fresh perspective and new intuition to the existing PDE-dominated literature on the subject. Our main tools are martingale methods and shift of probability measure; the fact that the underlying process is typically a strict local martingale is carefully considered throughout the paper.