[Geometric Lévy Process & MEMM] Pricing Model and Related Estimation Problems

In this article the [Geometric Lévy Process & MEMM] pricingmodel is proposed. This model is an option pricing model for theincomplete markets, and this model is based on the assumptions that theprice processes are geometric Lévy processes and that the pricesof the options are determined by the minimal relative entropy methods.This model has many good points. For example, the theoretical part ofthe model is contained in the framework of the theory of Lévyprocess (additive process). In fact the price process is also aLévy process (with changed Lévy measure) under the minimalrelative entropy martingale measure (MEMM), and so the calculation ofthe prices of options are reduced to the computation of functionals ofLévy process. In previous papers, we have investigated thesemodels in the case of jump type geometric Lévy processes. In thispaper we extend the previous results for more general type of geometricLévy processes. In order to apply this model to real optionpricing problems, we have to estimate the price process of theunderlying asset. This problem is reduced to the estimation problem ofthe characteristic triplet of Lévy processes. We investigate thisproblem in the latter half of the paper.     

From the Minimal Entropy Martingale Measures to the Optimal Strategies for the Exponential Utility Maximization: the Case of Geometric Lévy Processes

In this article, we will consider a multi-dimensional geometric L'evy process as a financial market model. We will first determine the minimal entropy martingale measure (MEMM); we will next derive the optimal strategy for the exponential utility maximization of terminal wealth concretely from the representation of the MEMM. JEL Classification: D46, D52, G12 AMS (2000) Subject Classification: 60G44, 60G51, 60G52,60H20, 60J75, 91B16, 91B28, 94A17     

The Minimal Entropy Martingale Measure (MEMM) for a Markov-Modulated Exponential Lévy Model

This paper deals with the characterization problem of the minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Lévy model. This model is characterized by the presence of a background process modulating the risky asset price movements between different regimes or market environments. This allows to stress the strong dependence of financial assets price with structural changes in the market conditions. Our main results are obtained from the key idea of working conditionally on the modulator-factor process. This reduces the problem to studying the simpler case of processes with independent increments. Our work generalizes some previous works in the literature dealing with either the exponential Lévy case or the exponential-additive case.     

Evaluation of the MEMM, Parameter Estimation and Option Pricing for Geometric Lévy Processes

In this paper, we shall propose a useful approach to evaluate concretely the MEMM (minimal entropy martingale measure) for the typical geometric Lévy processes such as compound Poisson, stable, VG (Variance Gamma), CGMY (Carr-Geman-Madan-Yor), NIG (Normal Inverse Gaussian), etc. In addition, we shall estimate the parameters of geometric Lévy processes and value the European call option and Asian call option using the Nikkei financial data.     

Option pricing and Esscher transform under regime switching

Summary 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 underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

Stochastic orders in dynamic reinsurance markets

We consider a dynamic reinsurance market, where the traded risk process is driven by a compound Poisson process and where claim amounts are unbounded. These markets are known to be incomplete, and there are typically infinitely many martingale measures. In this case, no-arbitrage pricing theory can typically only provide wide bounds on prices of reinsurance claims. Optimal martingale measures such as the minimal martingale measure and the minimal entropy martingale measure are determined, and some comparison results for prices under different martingale measures are provided. This leads to a simple stochastic ordering result for the optimal martingale measures. Moreover, these optimal martingale measures are compared with other martingale measures that have been suggested in the literature on dynamic reinsurance markets.Received: March 2004, Mathematics Subject Classification (2000): 62P05, 60J75, 60G44JEL Classification: G10     

Esscher transforms and the minimal entropy martingale measure for exponential Lévy models

In this paper we offer a systematic survey and comparison of the Esscher martingale transform for linear processes, the Esscher martingale transform for exponential processes, and the minimal entropy martingale measure for exponential Lévy models, and present some new results in order to give a complete characterization of those classes of measures. We illustrate the results with several concrete examples in detail.     

The Minimal Entropy Martingale Measures for Exponential Additive Processes

In this paper, we will consider exponential additive processes as a financial market model. Under a mild condition, we will determine the minimal entropy martingale measures (MEMMs) for the exponential additive processes. To this end, we will prepare several results on the exponential moment of additive processes and integrals based on them. As an application of our result, we will deduce optimal strategy for exponential utility maximization problem. We will also investigate our result through several examples, such as time-dependent versions of double Poisson model, Merton model and Kou model.     

完全市场中的资产定价——有限离散时间情形

研究完全市场中有限离散时间情形下的资产定价问题。首先,给出了无风险收益的概念,借助无风险收益定义了一种风险中性概率。基于这个概率,得到了资产的价格等于随机现金流与随机贴现因子乘积的期望,而且资产的价格还等于资产支付关于q的期望对无风险收益的贴现值。其次,借助无风险概率考虑了资产在多期情形下的资产定价,得出了相应的股票期权公式,尤其作为推论给出了欧式看涨期权的定价公式,并对资产价格过程的鞅性作了讨论。     

On <Emphasis Type="Italic">q</Emphasis>-optimal martingale measures in exponential Lévy models

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Lévy models. As a consequence, we find that in the one-dimensional case, the q-optimal equivalent martingale measures may exist only if the tails for upward jumps are extraordinarily light. Moreover, we derive the convergence of q-optimal signed, resp. absolutely continuous, martingale measures to the minimal entropy martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed. C.N. gratefully acknowledges financial support by UniCredit, Markets and Investment Banking. However, this paper does not reflect the opinion of UniCredit, Markets and Investment Banking, it is the personal view of the authors.     

Generalised Sharpe Ratios and Asset Pricing in Incomplete Markets

The paper presents an incomplete market pricingmethodology generating asset pricebounds conditional on the absence of attractiveinvestment opportunities in equilibrium.The paper extends and generalises the seminal article ofCochrane and Saá-Requejowho pioneered option pricing based on the absenceof arbitrage and high Sharpe Ratios. Ourcontribution is threefold:We base the equilibrium restrictions on an arbitrary utility function, obtaining theCochrane and Saá-Requejo analysis as a special case with truncated quadratic utility. We extend the definition of Sharpe Ratio from quadratic utility to the entire family of CRRA utility functions and restate the equilibrium restrictions in terms ofGeneralised Sharpe Ratios which, unlike the standard Sharpe Ratio, provide aconsistent ranking of investment opportunities even when asset returns are highlynon-normal. Last but not least, we demonstrate that for Itô processes theCochrane and Saá-Requejo price bounds are invariant to the choice of the utilityfunction, and that in the limit they tend to a unique price determined by theminimal martingale measure.     

Fast estimation of true bounds on Bermudan option prices under jump-diffusion processes

Fast pricing of American-style options has been a difficult problem since it was first introduced to the financial markets in 1970s, especially when the underlying stocks’ prices follow some jump-diffusion processes. In this paper, we extend the ‘true martingale algorithm’ proposed by Belomestny et al. [Math. Finance, 2009, 19, 53–71] for the pure-diffusion models to the jump-diffusion models, to fast compute true tight upper bounds on the Bermudan option price in a non-nested simulation manner. By exploiting the martingale representation theorem on the optimal dual martingale driven by jump-diffusion processes, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal–dual algorithm, therefore significantly improving the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our algorithm.     

Minimal Hellinger martingale measures of order <Emphasis Type="Italic">q</Emphasis>

This paper proposes an extension of the minimal Hellinger martingale measure (MHM hereafter) concept to any order q≠1 and to the general semimartingale framework. This extension allows us to provide a unified formulation for many optimal martingale measures, including the minimal martingale measure of Föllmer and Schweizer (here q=2). Under some mild conditions of integrability and the absence of arbitrage, we show the existence of the MHM measure of order q and describe it explicitly in terms of pointwise equations in ? ^d . Applications to the maximization of expected power utility at stopping times are given. We prove that, for an agent to be indifferent with respect to the liquidation time of her assets (which is the market’s exit time, supposed to be a stopping time, not any general random time), she is forced to consider a habit formation utility function instead of the original utility, or equivalently she is forced to consider a time-separable preference with a stochastic discount factor.     

A liquidity-based model for asset price bubbles

We provide a new liquidity-based model for financial asset price bubbles that explains bubble formation and bubble bursting. The martingale approach to modeling price bubbles assumes that the asset's market price process is exogenous and the fundamental price, the expected future cash flows under a martingale measure, is endogenous. In contrast, we define the asset's fundamental price process exogenously and asset price bubbles are endogenously determined by market trading activity. This enables us to generate a model that explains both bubble formation and bubble bursting. In our model, the quantity impact of trading activity on the fundamental price process—liquidity risk—is what generates price bubbles. We study the conditions under which asset price bubbles are consistent with no arbitrage opportunities and we relate our definition of the fundamental price process to the classical definition.     

A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.As a concrete example, we specialize to a variant of the Hull-White or Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q.JEL Classification: D52, G13