On value preserving and growth optimal portfolios

In a discrete-time financial market setting, the paper relates various concepts introduced for dynamic portfolios (both in discrete and in continuous time). These concepts are: value preserving portfolios, numeraire portfolios, interest oriented portfolios, and growth optimal portfolios. It will turn out that these concepts are all associated with a unique martingale measure which agrees with the minimal martingale measure only for complete markets.     

The MEMMs for Markov-Modulated GBMs

In this paper, we consider the option pricing problem when the risky underlying assets are driven by Markov-modulated geometric Brownian motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the risky asset, depend on unobservable states of the economy which are modeled by a continuous-time hidden Markov chain. The market described by the Markov-modulated GBM model is incomplete in general, and, hence, the martingale measure is not unique. We adopt the minimal relative entropy martingale measure (MEMM) for the Markov-modulated GBM model as the suitable martingale measure and we obtain the MEMM for the market in general sense.     

Comparing the minimal Hellinger martingale measure of order q to the q-optimal martingale measure

This paper investigates the relationship between the minimal Hellinger martingale measure of order q$q$ (MHM measure hereafter) and the q$q$-optimal martingale measure for any q≠1$q\ne 1$. First, we provide more results for the MHM measure; in particular we establish its complete characterization in two manners. Then we derive two equivalent conditions for both martingale measures to coincide. These conditions are in particular fulfilled in the case of markets driven by Lévy processes. Finally, we analyze the MHM measure as well as its relationship to the q$q$-optimal martingale measure for the case of a discrete-time market model.     

Risk Minimization for a Filtering Micromovement Model of Asset Price

Abstract

The classical option hedging problems have mostly been studied under continuous-time or equally spaced discrete-time models, which ignore two important components in the actual price: random trading times and market microstructure noise. In this paper, we study optimal hedging strategies for European derivatives based on a filtering micromovement model of asset prices with the two commonly ignored characteristics. We employ the local risk-minimization criterion to develop optimal hedging strategies under full information. Then, we project the hedging strategies on the observed information to obtain hedging strategies under partial information. Furthermore, we develop a related nonlinear filtering technique under the minimal martingale measure for the computation of such hedging strategies.     

The relative entropy in CGMY processes and its applications to finance

The CGMY market model generates infinite equivalent martingale measures (EMM). In order to price options, we need an adequate method to choose one EMM. This paper presents the relative entropy for CGMY processes, and apply it to choosing an EMM called the model preserving minimal entropy martingale measure.      

基于指数效用无差异价值过程的不完备市场下的可违约期权的定价模型

本文研究不完备市场情况下的可违约期权的动态指数效用无差异定价。不同于大多数的可违约期权定价文献,本文没有假定鞅的不变性,即通常的H 假设,而是通过信息流的扩张和测度的变换,将信用风险敏感的资产转换为一个G 局部鞅,其后引入一个具体的倒向随机微分方程（BSDE）,并证明该方程解的存在性与唯一性;然后利用无差异价值过程C_t（B,α）在最小熵鞅测度下对一般的投资策略为上鞅,而在最优投资策略下为鞅的事实,证明无差异价值过程C_t（B,α）就是BSDE 的解,从而给出可违约期权的定价。     

Option Valuation with a Discrete-Time Double Markovian Regime-Switching Model

Abstract

This article develops an option valuation model in the context of a discrete-time double Markovian regime-switching (DMRS) model with innovations having a generic distribution. The DMRS model is more flexible than the traditional Markovian regime-switching model in the sense that the drift and the volatility of the price dynamics of the underlying risky asset are modulated by two observable, discrete-time and finite-state Markov chains, so that they are not perfectly correlated. The states of each of the chains represent states of proxies of (macro)economic factors. Here we consider the situation that one (macro)economic factor is caused by the other (macro)economic factor. The market model is incomplete, and so there is more than one equivalent martingale measure. We employ a discrete-time version of the regime-switching Esscher transform to determine an equivalent martingale measure for valuation. Different parametric distributions for the innovations of the price dynamics of the underlying risky asset are considered. Simulation experiments are conducted to illustrate the implementation of the model and to document the impacts of the macroeconomic factors described by the chains on the option prices under various different parametric models for the innovations.     

三叉树模型下的等价鞅测度

本文研究了三叉树模型下的等价鞅测度刻划问题，得到了三叉树模型的最小熵鞅测度，逆相对熵鞅测度，方差最优鞅测度和极小鞅测度的精确表达式。     

Equivalent martingale measures for large financial markets in discrete time

We show that in a discrete-time large financial market the absence of certain asymptotic arbitrage opportunities is equivalent to the existence of martingale measures in a strong sense. We also consider the Arbitrage Pricing Model with stable random variables where we are able to give explicit necessary and sufficient conditions using market parameters.     

Optimal martingale measures for defaultable assets

We model a defaultable asset as solution to a stochastic differential equation driven by both a Brownian motion and the counting process martingale associated to the one-jump process. We discuss in this framework the minimal entropy martingale measure as well as the linear Esscher and the minimal martingale measure. In particular we deal with some rather delicate verification issues.     

Unit-linked life insurance policies: Optimal hedging in partially observable market models

In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract, that is a combination of a term insurance policy and a pure endowment, whose final value depends on the trend of a stock market where the premia the policyholder pays are invested. To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the policyholder. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk–Kunita–Watanabe decomposition of the insurance claim with respect to the minimal martingale measure and the available information flow. We provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure. Finally, we discuss applications in a Markovian setting via filtering.     

A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market

In [Riesner, M., 2006. Hedging life insurance contracts in a Lévy process financial market. Insurance Math. Econom. 38, 599–608] the (locally) risk-minimizing hedging strategy for unit-linked life insurance contracts is determined in an incomplete financial market driven by a Lévy process. The considered risky asset is not a martingale under the original measure and therefore, a change of measure to the minimal martingale measure is performed.The goal of this paper is to show that the risk-minimizing hedging strategy under the new martingale measure which is found in the paper cited above is not the locally risk-minimizing strategy under the original measure. Finally, the real locally risk-minimizing strategy is derived and a relationship between the number of risky assets held in the proposed portfolio cited in the above-mentioned paper and the one proposed here is given.     

Application of Moore-Penrose inverse in deciding the minimal martingale measure

The Moore-Penrose inverse is an important tool in algebra.This paper shows that the MoorePenrose inverse is also an effcient technique in determining the minimal martingale measure if a security price follows a semi-martingale which satisfies some structure condition.We extend a result of Dzhaparidze and Spreij concerning the Moore-Penrose inverse to the case that the Moore-Penrose inverse of any matrix-valued predictable process is still predictable.Furthermore,we obtain an explicit formula of the minimal martingale measure by employing the Moore-Penrose inverse.Specifically,the minimal martingale measure in a generalized Black-Scholes model is found.     

A note on convergence of option prices and their Greeks for Lévy models

We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

Risk Minimizing Option Pricing in a Regime Switching Market

Abstract

We study option pricing in a regime switching market where the risk free interest rate, growth rate and the volatility of a stock depends on a finite state Markov chain. Using a minimal martingale measure we show that the risk minimizing option price satisfies a system of Black–Scholes partial differential equations with weak coupling.     

Discrete-time local risk minimization of payment processes and applications to equity-linked life-insurance contracts

We develop a theory of local risk minimization for payment processes in discrete time, and apply this theory to the pricing and hedging of equity-linked life-insurance contracts. Thus, we extend the work of Møller (2001a) in several directions: from risk minimization (which is done under a martingale measure) to local risk minimization (which is done under an arbitrary measure), from single claims to payment processes, from complete financial markets to possibly incomplete financial markets, from a single risky asset to several risky assets, and from finite state spaces to general state spaces.Moreover, we show that, when tradable financial assets are independent of mortality, a locally risk-minimizing hedging strategy for most claims in the combined financial and mortality market (such as those arising from equity-indexed annuities) may be expressed as the product of two simpler locally risk-minimizing hedging strategies: one for a purely financial claim, the other for a traditional (i.e. non-equity-linked) life-insurance claim.Finally, we also show, under general assumptions, that the minimal measure for the combined market is the product of the minimal measure for the financial market and the physical measure for the mortality.     

General dynamic term structures under default risk

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath et al. (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability. In this framework, we derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes.     

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.