Option pricing in incomplete markets

Expected utility maximization is a very useful approach for pricing options in an incomplete market. The results from this approach contain many important features observed by practitioners. However, under this approach, the option prices are determined by a set of coupled nonlinear partial differential equations in high dimensions. Thus, it represents numerous significant difficulties in both theoretical analysis and numerical computations. In this paper, we present accurate approximate solutions for this set of equations.     

Convex bounds on multiplicative processes, with applications to pricing in incomplete markets

Extremal distributions have been extensively used in the actuarial literature in order to derive bounds on functionals of the underlying risks, such as stop-loss premiums or ruin probabilities, for instance. In this paper, the idea is extended to a dynamic setting. Specifically, convex bounds on multiplicative processes are derived. Despite their relative simplicity, the extremal processes are shown to produce reasonably accurate bounds on option prices in the classical trinomial model for incomplete markets.     

带基差风险和交易费用的不完全市场下的期权定价方法

本文研究了同时带有基差风险和交易费用的不安全市场中的权证定价方法。把[1]的模型推广到了考虑基差风险的情况[2]。期权的价格以一个三维自由边界问题的解给出,并含有两个相关的股票价格变量的相关系数。     

Valuing risky income streams in incomplete markets

A model for pricing and hedging in incomplete markets is proposed. This model is derived from expected utility theory, and a connection with the traditional no‐arbitrage framework is noted. It is shown that the CGM model can be implemented to value risky assets in incomplete markets.     

Price bounds of mortality-linked security in incomplete insurance market

This study investigates reasonable price bounds for mortality-linked securities when the issuer has only a partial hedging ability. The price bounds are established by minimizing the difference between the benchmark price and the replicating portfolio cost subject to the gain–loss ratio of excess payoff of the mortality-linked securities. In contrast to the previous studies, the assumptions of no-arbitrage pricing and utility-based pricing are not fully employed in this study because of the incompleteness of the insurance securitization market. Instead, a framework including three insurance basis assets is constructed to search for the price bounds of mortality-linked securities and use the Swiss Re mortality catastrophe bond, issued in 2003, as a numerical example. The proposed price bounds are valuable for setting bid–asked spreads and coupon premiums, and establishing trading strategies in the raising mortality securitization markets.     

The structure of optimal consumption streams in general incomplete markets

We prove that for any incomplete market and any concave utility function the marginal propensities to consume and to save are always positive. Furthermore, we introduce a class of incomplete markets that includes almost all well known examples of market incompleteness in finance and macroeconomics. Two concrete examples are idiosyncratic income shocks and general, diffusion driven incompleteness. For all markets in our class we explicitly solve the associated utility maximization problem by a recursive construction and derive many important properties. For example, precautionary savings and the diminishing marginal propensity to consume. Effectively, the class is characterized by these two economic properties. We also prove that the growth rate of consumption is always larger when markets are incomplete and that precautionary savings are monotone increasing in the size of idiosyncratic risk. Our construction can be implemented computationally by an efficient, robust numerical scheme. We thank two anonymous referees for useful comments and remarks.     

Geometry of polar wedges in Riesz spaces and super-replication prices in incomplete financial markets

This paper is devoted to a further generalisation of the main results in [5] including the representation of the weak super-replication price (cf. equation (1.6)). In addition to the already established weakening of the technical assumptions in [5] (cf. [24] and [25]), the main results in [5] can be still generalised by considering the geometric structure of the underlying problem (based on the properties of Riesz spaces and polar wedges therein). In Section 5 we show under which geometric conditions of the relevant sets the results still hold (cf. Theorem 5.3 and Corollary 5.5). In particular, we can completely remove the restrictive admissibility assumption and carry forward equation (1.4) to a larger class of wedges (cf. Corollary 5.5). The authors gratefully acknowledge support from EPSRC grant no. GR/S80202/01     

Incomplete financial markets and contingent claim pricing in a dual expected utility theory framework

This paper investigates the price for contingent claims in a dual expected utility theory framework, the dual price, considering arbitrage-free financial markets. A pricing formula is obtained for contingent claims written on n underlying assets following a general diffusion process. The formula holds in both complete and incomplete markets as well as in constrained markets. An application is also considered assuming a geometric Brownian motion for the underlying assets and the Wang transform as the distortion function.     

A Model with Interacting Assets Driven by Poisson Processes

Abstract

An extension with noise given by Poisson processes of a model of financial market with several assets that are interacting, i.e., influencing each other (even in the absence of noise) is given. We present explicit formulae for the stock price process as well as for the prices of European multi-asset contingent claims based on a residual risk minimization approach. We also provide an explicit hedging formula.     

Pricing early exercise contracts in incomplete markets

We present a utility-based methodology for the valuation of early exercise contracts in incomplete markets. Incompleteness stems from nontraded assets on which the contracts are written. This methodology takes into account the individuals attitude towards risk and yields nonlinear pricing rules. The early exercise indifference prices solve a quasilinear variational inequality with an obstacle term. They are also shown to satisfy an optimal stopping problem with criterion given by their European indifference price counterpart. A class of numerical schemes are developed for the variational inequalities and a general approach for solving numerically nonlinear equations arising in incomplete markets is discussed.Accepted: May 2003, AMS Classification: 93E20, 60G40, 60J75The second author acknowledges partial support from NSF Grants DMS 0102909 and DMS 0091946.     

Designing and pricing guarantee options in defined contribution pension plans

The shift from defined benefit (DB) to defined contribution (DC) is pervasive among pension funds, due to demographic changes and macroeconomic pressures. In DB all risks are borne by the provider, while in plain vanilla DC all risks are borne by the beneficiary. However, for DC to provide income security some kind of guarantee is required. A minimum guarantee clause can be modeled as a put option written on some underlying reference portfolio and we develop a discrete model that selects the reference portfolio to minimize the cost of a guarantee. While the relation DB–DC is typically viewed as a binary one, the model shows how to price a wide range of guarantees creating a continuum between DB and DC. Integrating guarantee pricing with asset allocation decision is useful to both pension fund managers and regulators. The former are given a yardstick to assess if a given asset portfolio is fit-for-purpose; the latter can assess differences of specific reference funds with respect to the optimal one, signaling possible cases of moral hazard. We develop the model and report numerical results to illustrate its uses.     

EXISTENCE OF STOCHASTIC EQUILIBRIUM WITH INCOMPLETE FINANCIAL MARKETS

This paper analyzes the aritrage-tree security markets and the general equilibrium ex-istence problem for a stochastic economy with incomplete financial markets. Information structure is given by an event tree. This paper restricts attention to puraly financial securities. It isassume that trading takes place in the sequence of spot markets and futures markets for securi-ties payable in units of account. Unlimited short-selling in securities is allowed. Financial markets may be incomplete, some consumption streams may be impossible to obtain by any tradingstrategy. Securities may be individually precluded from trade at arbitrary states and dates. Thesecurity price process is arbitrage-free the dividend process if and only if there exists a stochaticstate price (present value) process : the present value of the security prices at every vertex isthe present value of their dividend and capital values over the set of immediate successors ; thecurrent value of each security at every vertex is the present value of its future dividend streamover all succeeding vertices. The existence of such an equilibrium is proved under the followingcondition: continuous, weakly convex, strictly monotone and complete preferences, strictlypositive endowmenta and dividends processes.     

The equity risk premium and the riskfree rate in an economy with borrowing constraints

Our objective is to study analytically the effect of borrowing constraints on asset returns. We explicitly characterize the equilibrium for an exchange economy with two agents who differ in their risk aversion and are prohibited from borrowing. In a representative-agent economy with CRRA preferences, the Sharpe ratio of equity returns and the riskfree rate are linked by the risk aversion parameter. We show that allowing for preference heterogeneity and imposing borrowing constraints breaks this link. We find that an economy with borrowing constraints exhibits simultaneously a relatively high Sharpe ratio of stock returns and a relatively low riskfree interest rate, compared to both representative-agent and unconstrained heterogeneous-agent economies.      

Indifference pricing of a life insurance portfolio with risky asset driven by a shot-noise process

In this paper, we investigate the pricing problem for a portfolio of life insurance contracts where the life contingent payments are equity-linked depending on the performance of a risky stock or index. The shot-noise effects are incorporated in the modeling of stock prices, implying that sudden jumps in the stock price are allowed, but their effects may gradually decline over time. The contracts are priced using the principle of equivalent utility. Under the assumption of exponential utility, we find the optimal investment strategy and show that the indifference premium solves a non-linear partial integro-differential equation (PIDE). The Feynman–Kač form solutions are derived for two special cases of the PIDE. We further discuss the problem for the asymptotic shot-noise process, and find the probabilistic representation of the indifference premium. We also provide some numerical examples and analyze parameter sensitivities for the results obtained in this paper.     

Pricing and hedging of American contingent claims in incomplete markets

0.IntroductionandSummaryThecelebratedpapersof[2]and[3],pavedthewayforpricingoptionsonstocks,onthebasisofthefollowingprinciple:inacompletemarket(suchastheoneinSection1.5),everycontingentclaimcanbeattainedexactlybyinvestinginthemarketandstartingwithala...     

Optimal strategies for hedging portfolios of unit-linked life insurance contracts with minimum death guarantee

In this paper, we are interested in hedging strategies which allow the insurer to reduce the risk to their portfolio of unit-linked life insurance contracts with minimum death guarantee. Hedging strategies are developed in the Black and Scholes model and in the Merton jump-diffusion model. According to the new frameworks (IFRS, Solvency II and MCEV), risk premium is integrated into our valuations. We will study the optimality of hedging strategies by comparing risk indicators (Expected loss, volatility, VaR and CTE) in relation to transaction costs and costs generated by the re-hedging error. We will analyze the robustness of hedging strategies by stress-testing the effect of a sharp rise in future mortality rates and a severe depreciation in the price of the underlying asset.     

Option pricing and hedging in incomplete market driven by Normal Tempered Stable process with stochastic volatility

This paper develops a distribution class, termed Normal Tempered Stable, by subordinating a drifted Brownian motion through a strictly increasing Tempered Stable process that generalizes the Variance Gamma and the Normal Inverse Gaussian and is used to model the logarithm asset returns. The newly added parameter is to create subclasses for all the distributions discovered in financial market. The empirical test suggests that time series of Technology stock returns in US market reject both the Variance Gamma distribution and the Normal Inverse Gaussian distribution and admit instead another subclass of the Normal Tempered Stable distribution. Furthermore, we introduce stochastic volatilities into the Normal Tempered Stable process and derive explicit formulae for option pricing and hedging by means of the characteristic function based methods. To answer the question of how well different models work in practice, we investigate four models adopting data on daily equity option prices and obtain several findings from the numerical results. To sum up, the Normal Tempered Stable process with stochastic volatility is able to adequately capture implied volatility dynamics and seen as a superior model relative to the jump-diffusion stochastic volatility model, based on the construction methodology that incorporates more sophisticated and flexible jump structure and the systematic and realistic treatment of volatility dynamics. The Normal Tempered Stable model turns out to have the competitive performance in an efficient manner given that it only requires three parameters.     

An extension of the Euler Laplace transform inversion algorithm with applications in option pricing

We show that the Euler algorithm for Laplace transform inversion can be extended to functions defined on the entire real line, if they have specific decay features. Our objective is to apply the method to option pricing problems, specifically when inverting Laplace transforms of the option price in the logarithm of the strike.     

On the stability of evolutionary dynamics in games with incomplete information

In an interaction it is possible that one agent has features it is aware of but the opponent is not. These features (e.g. cost, valuation or fighting ability) are referred to as the agent’s type. The paper compares two models of evolution in symmetric situations of this kind. In one model the type of an agent is fixed and evolution works on strategies of types. In the other model every agent adopts with fixed probabilities both types, and type-contingent strategies are exposed to evolution. It is shown that the dynamic stability properties of equilibria may differ even when there are only two types and two strategies. However, in this case the dynamic stability properties are generically the same when the payoff of a player does not depend directly on the type of the opponent. Examples illustrating these results are provided.     

Time-invariant portfolio strategies in structured products with guaranteed minimum equity exposure

We introduce a new exotic option to be used within structured products to address a key disadvantage of standard time-invariant portfolio protection: the well-known cash-lock risk. Our approach suggests enriching the framework by including a threshold in the allocation mechanism so that a guaranteed minimum equity exposure (GMEE) is ensured at any point in time. To be able to offer such a solution still with hard capital protection, we apply an option-based structure with a dynamic allocation logic as underlying. We provide an in-depth analysis of the prices of such new exotic options, assuming a Heston–Vasicek-type financial market model, and compare our results with other options used within structured products. Our approach represents an interesting alternative for investors aiming at downsizing protection via time-invariant portfolio protection strategies, meanwhile being also afraid to experience a cash-lock event triggered by market turmoils.