Determination of risk pricing measures from market prices of risk

A new insurance provider or a regulatory agency may be interested in determining a risk measure consistent with observed market prices of a collection of risks. Using a relationship between distorted coherent risk measures and spectral risk measures, we provide a method for reconstructing distortion functions from the observed prices of risk. The technique is based on an appropriate application of the method of maximum entropy in the mean, which builds upon the classical method of maximum entropy.     

Good Deals and compatible modification of risk and pricing rule: a regulatory treatment

This work studies Good Deals in a scenario in which a firm uses decision-making tools based on a coherent risk measure, and in which the market prices are determined with a sub-linear pricing rule. The most important observation of this work is that the existence of a Good Deal is equivalent to the incompatibility between the pricing rule and the risk measure. In this paper, we look into this situation from a regulatory point of view to rule out Good Deals with the purpose of stabilizing financial markets. We propose some practical ways of modifying a risk measure so a regulator can set appropriate levels of capital requirements for a financial institution.     

Event risk, contingent claims and the temporal resolution of uncertainty

We study the pricing and hedging of contingent claims that are subject to Event Risk which we define as rare and unpredictable events whose occurrence may be correlated to, but cannot be hedged perfectly with standard marketed instruments. The super-replication costs of such event sensitive contingent claims (ESCC), in general, provide little guidance for the pricing of these claims. Instead, we study utility based prices under two scenarios of resolution of uncertainty for event risk: when the event is continuously monitored, or when it is revealed only at the payment date. In both cases, we transform the incomplete market optimal portfolio choice problem of an agent endowed with an ESCC into a complete market problem with a state and possibly path-dependent utility function. For negative exponential utility, we obtain an explicit representation of the utility based prices under both information resolution scenarios and this in turn leads us to a simple characterization of the early resolution premium. For constant relative risk aversion utility functions we propose a simple numerical scheme and study the impact of size of the position, wealth and expected return on these prices.     

Risk-averse order policies with random prices in complete market and retailers’ private information

We consider a retailer who orders products before the price for them becomes known. The price is an outcome of perfect competition in a complete market. Since the demand is price sensitive, the uncertainty in prices induces uncertain profits and associated risks. In this paper we show that if the retailer is risk averse and, as a result, selects a utility function of profit to maximize, then his subjective assessment of future prices is affected by the risk attitude. This, in turn, introduces a bias in retailer’s ordering policies. By considering coordinated pricing and ordering policies we derive a relationship between risk aversion, retailer’s subjective (private) assessment and the market implied, risk neutral forecast. This relationship and the induced bias are then illustrated for two typical operations management strategies which involve either inventory considerations or promotions avoiding accumulation of stocks.     

基于鞅测度的流动性风险溢价的测算

研究了在一般市场条件下流动性风险的定价问题.首先借助金融数学和金融工程的无套利思想在鞅测度下对市场风险和流动性风险进行定价,通过等价测度变换,使可交易资产的贴现价值过程转化为鞅过程,得到了市场风险和流动性风险的市场价格,进而给出了流动性风险溢价的计算公式.得到的风险的市场价格在同一市场中对于所有可交易资产都是相同的,并且这一价格对于所有投资者也都是相同的,不会因投资者的风险厌恶水平的不同而不同.     

债券定价及利率风险的市场价格的重构方法

假设利率变化的模型是由随机微分方程给出,则可以用推导Black-Scholes方程的方法来推出债券价格满足的偏微分方程,得到一个抛物型的偏微分方程.但是,在债券定价的方程中隐含有一个参数λ称为利率风险的市场价格.所谓债券定价的反问题,就是由不同到期时间的债券的现在价格来得到利率风险的市场价格.对随机利率模型下债券定价的正问题先给予介绍和差分数值求解方法,并介绍了反问题,且对反问题给出了数值方法.     

Coupling and option price comparisons in a jump-diffusion model

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures.

Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.     

无套利期权定价模型在一般均衡框架下的一致性研究

期权定价有无套利方法和一般均衡方法两种.本文在一般均衡框架下构造了一个允许连续消费的简单经济模型,并将基于无套利方法的期权定价模型中所假定的标的证券的价格变化动态过程内生化于理性预期均衡中.在常数相对风险厌恶(CRRA)的效用函数的条件下,我们推导出Merton(1973)期权定价公式,从而证明无套利方法与均衡方法的内在一致性,而CRRA这种类型的效用函数是无套利定价模型在一般均衡框架中成立的充分条件.本文进一步将此模型在一个简单经济中扩展到m种证券的情况,也得到相似的结论.     

Pricing Defaultable Bonds in a Markov Modulated Market

We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low risk-free interest rate, high payout rate and high volatility.     

Option pricing when the regime-switching risk is priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.     

A Convenient Way to Characterize Equivalent Martingale Measures in Incomplete Markets

It is an empirical fact that the (empirically) relevant models for asset prices often describe markets that are incomplete in terms of their underlying assets, yielding many possible equivalent martingale measures under the no-arbitrage assumption. By using actual derivative prices, i.e., prices as observed in the market, additional information about the empirically relevant equivalent martingale measures might be obtained. In order to be able to process such information easily one needs a convenient way to represent all possible equivalent martingale measures in relation to derivative prices. In this paper we present such a convenient characterization. Conceptually, our characterization is not different from existing characterizations using, for example, Radon–Nikodym derivatives of martingale measures with respect to objective probabilities, but our characterization offers some advantages. The main advantage is that pricing derivatives is split up into two steps. The first step is solving a related complete markets pricing problem. This is a well-studied problem, so that it can easily be solved generally. In the second step a weighted average of the first step complete markets price must be calculated. Pricing under different equivalent martingale measures in the original market only differs with respect to the second step. The empirically relevant weighting can be determined by confronting the theoretical with the actually observed prices. As a byproduct we obtain a new and natural definition of idiosyncratic risk, which we show to be in line with the use of this term in the literature.To illustrate the ideas we discuss several examples. Among others we obtain the Hull–White formula for options on assets with stochastic volatility under close to minimal conditions that (for example) do not rely on a specification of the processes in terms of Itô diffusion.we relax the assumption of no-correlation between asset prices and volatilities in the Hull–White framework; we consider the case where the stochastic volatility does bear a risk-premium; we discuss pricing under stochastic interest rates; and we consider square-root type processes. All these pricing problems, and many more, can conveniently be handled using the approach based on our characterization of the equivalent martingale measures in continuous time markets that are incomplete in the underlying assets.     

Securitization, structuring and pricing of longevity risk

Pricing and risk management for longevity risk have increasingly become major challenges for life insurers and pension funds around the world. Risk transfer to financial markets, with their major capacity for efficient risk pooling, is an area of significant development for a successful longevity product market. The structuring and pricing of longevity risk using modern securitization methods, common in financial markets, have yet to be successfully implemented for longevity risk management. There are many issues that remain unresolved for ensuring the successful development of a longevity risk market. This paper considers the securitization of longevity risk focusing on the structuring and pricing of a longevity bond using techniques developed for the financial markets, particularly for mortgages and credit risk. A model based on Australian mortality data and calibrated to insurance risk linked market data is used to assess the structure and market consistent pricing of a longevity bond. Age dependence in the securitized risks is shown to be a critical factor in structuring and pricing longevity linked securitizations.     

A pricing model for secondary market yield based floating rate notes subject to default risk

The purpose of this article is to price secondary market yield based floating rate notes (SMY-FRNs) subject to default risk. SMY-FRNs are derivatives on the default-free term structure of interest rates, on the term structures for default-risky credit classes, and on the structure of a determined pool of bonds. The main problem in SMY-FRN pricing (as compared to the pricing of standard interest rate or credit derivatives) is market incompleteness, which makes traditional no-arbitrage pricing by replication fail. In general, SMY-FRNs are subject to two types of default risk. First, the SMY-FRN issuer may go bankrupt (direct default risk). Second, the possibility of the bankruptcy of the issuers in the underlying pool has an influence on the SMY-FRN coupons (indirect default risk). This article is the first one which provides a no-arbitrage pricing model for SMY-FRNs with direct and indirect default risks. It is also the first article applying incomplete market pricing methodology to SMY-FRNs.     

Risk-neutral economy and zero price of risk

The paper investigates the equilibrium in an economy in which all participants are indifferent to risk. The mechanism of asset and derivative pricing in such economy is identified. It is shown that no economy in equilibrium with stochastic interest rates can be simultaneously risk-neutral and have zero market price of risk. On the other hand, there exist equilibrium economies with risk-averse participants and zero prices of risk.     

Pricing longevity risk with the parametric bootstrap: A maximum entropy approach

In recent years, there has been significant development in the securitization of longevity risk. Various methods for pricing longevity risk have been proposed. In this paper we present an alternative pricing method, which is based on the maximization of the Shannon entropy in physics. Specifically, we propose implementing this pricing method with the parametric bootstrap (Brouhns et al., 2005), which is highly flexible and can be performed under different model assumptions. Through this pricing method we also quantify the impact of cohort effects and parameter uncertainty on prices of mortality-linked securities. Numerical illustrations based on longevity bonds with different maturities are provided.     

带有重置条款的可转换债券定价

可转换债券是中国证券市场的热点之一.本文主要研究如何给带有重置条款的可转换债券进行定价.文中采用了等价鞅测度的思想将标的物从风险世界转换到风险中性世界中,然后在风险中性世界中应用鞅评价方法对带有重置条款的可转换债券进行定价.     

基于战略顾客行为的最优定价与容量选择模型

研究了当市场中同时存在战略顾客和短视顾客时零售商的最优定价与容量选择问题。零售商在正常销售阶段和出清销售阶段制定不同的销售价格,同时通过容量选择影响战略顾客的购买行为,而战略顾客则根据零售商的定价和容量选择确定最优购买时机。分别分析了零售商在无限容量时的定价决策、固定价格时的容量选择、固定容量时的定价决策以及有限容量下的定价与容量选择四种情形。研究结果表明,零售商在无容量限制时的最优定价决策是制定两阶段定价策略,在固定价格时的最优容量选择依赖于模型的参数,而当零售商的容量固定时,部分满足出清销售阶段的顾客需求始终优于完全满足出清销售阶段的顾客需求。     

Pricing Fixed-Income Securities in an Information-Based Framework

Abstract

The purpose of this article is to introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous-time processes that describe the flow of information concerning market factors in a monetary economy. The nominal pricing kernel is assumed to be given at any specified time by a function of the values of information processes at that time. Using a change-of-measure technique, we derive explicit expressions for the prices of nominal discount bonds and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. An example that shows how the model can be calibrated to an arbitrary initial yield curve is presented. We proceed to model the price level, which is also taken at any specified time to be given by a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of the nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply.     

Efficient option risk measurement with reduced model risk

Options require risk measurement that is also computationally efficient as it is important to derivatives risk management. There are currently few methods that are specifically adapted for efficient option risk measurement. Moreover, current methods rely on series approximations and incur significant model risks, which inhibit their applicability for risk management.In this paper we propose a new approach to computationally efficient option risk measurement, using the idea of a replicating portfolio and coherent risk measurement. We find our approach to option risk measurement provides fast computation by practically eliminating nonlinear computational operations. We reduce model risk by eliminating calibration and implementation risks by using mostly observable data, we remove internal model risk for complex option portfolios by not admitting arbitrage opportunities, we are also able to incorporate liquidity or model misspecification risks. Additionally, our method enables tractable and convex optimisation of portfolios containing multiple options. We conduct numerical experiments to test our new approach and they validate it over a range of option pricing parameters.     

A note on utility based pricing and asymptotic risk diversification

In principle, liabilities combining both insurancial risks (e.g. mortality/longevity, crop yield,...) and pure financial risks cannot be priced neither by applying the usual actuarial principles of diversification, nor by arbitrage-free replication arguments. Still, it has been often proposed in the literature to combine these two approaches by suggesting to hedge a pure financial payoff computed by taking the mean under the historical/objective probability measure on the part of the risk that can be diversified. Not surprisingly, simple examples show that this approach is typically inconsistent for risk adverse agents. We show that it can nevertheless be recovered asymptotically if we consider a sequence of agents whose absolute risk aversions go to zero and if the number of sold claims goes to infinity simultaneously. This follows from a general convergence result on utility indifference prices which is valid for both complete and incomplete financial markets. In particular, if the underlying financial market is complete, the limit price corresponds to the hedging cost of the mean payoff. If the financial market is incomplete but the agents behave asymptotically as exponential utility maximizers with vanishing risk aversion, we show that the utility indifference price converges to the expectation of the discounted payoff under the minimal entropy martingale measure.