ANALYTICAL COMPARISONS OF OPTION PRICES IN STOCHASTIC VOLATILITY MODELS

This paper gives an ordering on option prices under various well-known martingale measures in an incomplete stochastic volatility model. Our central result is a comparison theorem that proves convex option prices are decreasing in the market price of volatility risk, the parameter governing the choice of pricing measure. The theorem is applied to order option prices under q -optimal pricing measures. In doing so, we correct orderings demonstrated numerically in Heath, Platen, and Schweizer ( Mathematical Finance , 11(4), 2001) in the special case of the Heston model.     

APPROXIMATE COMPLETENESS WITH MULTIPLE MARTINGALE MEASURES

We construct a financial market with countably many securities for which there are two equivalent martingale measures under which the market is approximately complete. Thus, approximate completeness does not in general guarantee unique consistent prices for nonmarketed claims. the construction also produces an economy with two agents and infinitely many traded goods which is in equilibrium but has no equilibrium when a new good (recognized by all as redundant) is tentatively traded.     

No Arbitrage in Discrete Time Under Portfolio Constraints

In frictionless securities markets, the characterization of the no-arbitrage condition by the existence of equivalent martingale measures in discrete time is known as the fundamental theorem of asset pricing. In the presence of convex constraints on the trading strategies, we extend this theorem under a closedness condition and a nondegeneracy assumption. We then provide connections with the superreplication problem solved in Föllmer and Kramkov (1997).     

Option Pricing in ARCH-type Models

ARCH models have become popular for modeling financial time series. They seem, at first, however, to be incompatible with the option pricing approach of Black, Scholes, Merton et al., because they are discrete-time models and possess too much variability. We show that completeness of the market holds for a broad class of ARCH-type models defined in a suitable continuous-time fashion. As an example we focus on the GARCH(1,1)-M model and obtain, through our method, the same pricing formula as Duan, who applied equilibrium-type arguments.     

A NOTE ON ARBITRAGE AND CLOSED CONVEX CONES

We give an example of a subspace K of     such that     , where     denotes the closure with respect to convergence in probablity. On the other hand, the cone   C ≔ K − L ^∞₊  is dense in   L ^∞  with respect to the weak-star topology  σ( L ^∞, L ¹)  . This example answers a question raised by I. Evstigneev. The topic is motivated by the relation of the notion of no arbitrage and the existence of martingale measures in Mathematical Finance.     

BILATERAL COUNTERPARTY RISK UNDER FUNDING CONSTRAINTS—PART I: PRICING

This and the follow‐up paper deal with the valuation and hedging of bilateral counterparty risk on over‐the‐counter derivatives. Our study is done in a multiple‐curve setup reflecting the various funding constraints (or costs) involved, allowing one to investigate the question of interaction between bilateral counterparty risk and funding. The first task is to define a suitable notion of no arbitrage price in the presence of various funding costs. This is the object of this paper, where we develop an “additive, multiple curve” extension of the classical “multiplicative (discounted), one curve” risk‐neutral pricing approach. We derive the dynamic hedging interpretation of such an “additive risk‐neutral” price, starting by consistency with pricing by replication in the case of a complete market. This is illustrated by a completely solved example building over previous work by Burgard and Kjaer.     

WIENER CHAOS: A NEW APPROACH TO OPTION HEDGING

This paper addresses the problem of estimating and analyzing the residual risk that is not hedged by a discrete hedging strategy. the use of die chaotic representation allows an elegant decomposition of the residual risk to be hedged by adequate assets. Alternative strategies to the classical delta hedging and optimization under the risk-neutral and historical probabilities are discussed.     

Optimal investment,derivative demand,and arbitrage under price impact

This paper studies the optimal investment problem with random endowment in an inventory‐based price impact model with competitive market makers. Our goal is to analyze how price impact affects optimal policies, as well as both pricing rules and demand schedules for contingent claims. For exponential market makers preferences, we establish two effects due to price impact: constrained trading and nonlinear hedging costs. To the former, wealth processes in the impact model are identified with those in a model without impact, but with constrained trading, where the (random) constraint set is generically neither closed nor convex. Regarding hedging, nonlinear hedging costs motivate the study of arbitrage free prices for the claim. We provide three such notions, which coincide in the frictionless case, but which dramatically differ in the presence of price impact. Additionally, we show arbitrage opportunities, should they arise from claim prices, can be exploited only for limited position sizes, and may be ignored if outweighed by hedging considerations. We also show that arbitrage‐inducing prices may arise endogenously in equilibrium, and that equilibrium positions are inversely proportional to the market makers' representative risk aversion. Therefore, large positions endogenously arise in the limit of either market maker risk neutrality, or a large number of market makers.     

Dynamic Arbitrage‐Free Asset Pricing with Proportional Transaction Costs

This paper studies multiperiod asset pricing theory in arbitrage‐free financial markets with proportional transaction costs. The mathematical formulation is based on a Euclidean space for weakly arbitrage‐free security markets and strongly arbitrage‐free security markets. We establish the weakly arbitrage‐free pricing theorem and the strongly arbitrage‐free pricing theorem.     

基于因子分析的套利定价模型及实证研究

众所周知,建立套利定价模型的关键在于因素的筛选,计算量很大。而因子分析能将为数众多的原始指标变量经过分析综合为少数几个公共因子变量,从而大大减少计算的复杂度。本文利用因子分析的方法对11个因素进行筛选,确定四个能够很好地反映所有因素包含的信息但又互不相关的公共因子变量,并建立套利定价模型,实证检验说明,通过该方法进行因素筛选建立的套利定价模型具有较好的定价效果。     

沪深300股指期货定价实证研究

股指期货是一种发展迅速的金融衍生产品,而合约定价问题是其重要研究方向之一。股指期货定价的基本方法是利用无套利定价原理得出的持有成本模型;而如果综合交易费用、融资成本、存贷款利差、保证金等市场因素,则可以得到股指期货的无套利定价区间。使用这两种模型对中国金融期货交易所的沪深300股指期货仿真交易合约进行实证分析,结果发现,实际交易价格和理论价格有较大偏差,市场中存在大量套利机会,定价效率有待提高。为此可以考虑的建议包括允许融资融券交易、推出沪深300指数ETF等。     

Razor-and-Blades pricing revisited

From razors and blades to printers and ink cartridges to smartphones and monthly usage charges to media devices and content, razor-and-blades pricing is commonplace. The argument for such a business model is compelling: entice consumers to adopt with a low initial price for the ‘razor,’ build up an installed base, and more than make up for the initial subsidy by charging a high price for replacement ‘blades.’ The problem is, many consumer enticement, customer lock-in, and competitive lock-out mechanisms look less and less tenable given modern-day developments such as the Internet, Google searches, social media, the hacker revolution, the ‘maker movement,’ rapidly improving technology, leaky supply chains, and global markets. This article characterizes the what, why, and how of razor-and-blades pricing; then examines the present-day tenability of such a pricing practice; and concludes with an impetus and a call for innovation—innovation in, perhaps, the pricing of and the purchasing arrangement for the initial razor; the value proposition from the razor and the razor-and-blades system; the architecture of the razor-and-blades system; and the delivery, especially in terms of customer experience, of value from the razor-and-blades system.     

OPTION PRICING USING THE TERM STRUCTURE OF INTEREST RATES TO HEDGE SYSTEMATIC DISCONTINUITIES IN ASSET RETURNS1

This paper demonstrates the use of term-structure-related securities in the design of dynamic portfolio management strategies that hedge certain systematic jump risks in asset return. Option pricing formulas based on the absence of arbitrage opportunities in this context are also developed. the analysis is for the case where assets returns are driven by a finite number of Brownian motions and an m-variate point process. the inclusion of :the additional traded assets in the term structure makes it possible to hedge systematic jumps imbedded in the m variate point process.     

Pricing by Arbitrage Under Arbitrary Information

A substantial applications literature on pricing by arbitrage has effectively restricted information to that arising solely from securities markets; return distributions are then governed solely by past prices. We reconsider pricing by arbitrage in markets rendered incomplete by arbitrary information, which, moreover, may influence required returns. We show that contingent claims depending solely on securities' normalized price histories are priced by arbitrage if and only if all risk-adjusted probabilities agree upon the law of primitive securities' normalized prices. We show how existing diffusion-based results can be preserved, and resolve an issue relating to Merton's (1973) stochastic interest rate model.     

DO ARBITRAGE‐FREE PRICES COME FROM UTILITY MAXIMIZATION?

In this paper we ask whether, given a stock market and an illiquid derivative, there exists arbitrage‐free prices at which a utility‐maximizing agent would always want to buy the derivative, irrespectively of his own initial endowment of derivatives and cash. We prove that this is false for any given investor if one considers all initial endowments with finite utility, and that it can instead be true if one restricts to the endowments in the interior. We show, however, how the endowments on the boundary can give rise to very odd phenomena; for example, an investor with such an endowment would choose not to trade in the derivative even at prices arbitrarily close to some arbitrage price.     

中国概念股指期货跨市场套利研究——基于沪深300指数、H股指数、新华富时A50指数期货的实证分析

针对沪深300指数、H股指数、新华富时A50指数期货,给出确定投资比例、选择投资时机及度量投资风险的方法,对中国概念股指期货的跨市场套利机会进行研究,结论显示:中国沪深300股票指数与周边市场的中国概念股票指数之间存在着普遍关联性,并且这种关联性可以转化为套利机会.实证结果表明:当1∶0.836546作为A50股指期货与H股股指期货的持仓比时,可以得到最优套利结果.     

Robustness of the Black and Scholes Formula

Consider an option on a stock whose volatility is unknown and stochastic. An agent assumes this volatility to be a specific function of time and the stock price, knowing that this assumption may result in a misspecification of the volatility. However, if the misspecified volatility dominates the true volatility, then the misspecified price of the option dominates its true price. Moreover, the option hedging strategy computed under the assumption of the misspecified volatility provides an almost sure one-sided hedge for the option under the true volatility. Analogous results hold if the true volatility dominates the misspecified volatility. These comparisons can fail, however, if the misspecified volatility is not assumed to be a function of time and the stock price. The positive results, which apply to both European and American options, are used to obtain a bound and hedge for Asian options.     

A COMMENT ON MARKET FREE LUNCH AND FREE LUNCH

Frittelli (2004) introduced a market free lunch depending on the preferences of the agents in the market. He characterized no arbitrage and no free lunch with vanishing risk in terms of no market free lunch (the difference comes from the class of utility functions determining the market free lunch). In this note we complete the list of characterizations and show directly (using the theory of Orlicz spaces) that no free lunch is equivalent to the absence of market free lunch with respect to monotone concave utility functions.     

The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time

We prove a version of the Fundamental Theorem of Asset Pricing, which applies to Kabanov's modeling of foreign exchange markets under transaction costs. The financial market is described by a   d × d   matrix-valued stochastic process  (Π _t ) ^T _{t =0}  specifying the mutual bid and ask prices between d assets. We introduce the notion of "robust no arbitrage," which is a version of the no-arbitrage concept, robust with respect to small changes of the bid-ask spreads of  (Π _t ) ^T _{t =0}  . The main theorem states that the bid-ask process  (Π _t ) ^T _{t =0}  satisfies the robust no-arbitrage condition iff it admits a strictly consistent pricing system. This result extends the theorems of Harrison-Pliska and Kabanov-Stricker pertaining to the case of finite Ω, as well as the theorem of Dalang, Morton, and Willinger and Kabanov, Rásonyi, and Stricker, pertaining to the case of general Ω. An example of a  5 × 5  -dimensional process  (Π _t )² _{t =0}  shows that, in this theorem, the robust no-arbitrage condition cannot be replaced by the so-called strict no-arbitrage condition, thus answering negatively a question raised by Kabanov, Rásonyi, and Stricker.     

RISK INDIFFERENCE PRICING IN JUMP DIFFUSION MARKETS

We study the risk indifference pricing principle in incomplete markets: The (seller's)  risk indifference price        is the initial payment that makes the  risk  involved for the seller of a contract equal to the risk involved if the contract is not sold, with no initial payment. We use stochastic control theory and PDE methods to find a formula for       and similarly for      . In particular, we prove that  where    p _low   and    p _up   are the lower and upper hedging prices, respectively.