Squared Bessel Processes and Their Applications to the Square Root Interest Rate Model

We study the Bessel processes withtime-varying dimension and their applications to the extended Cox-Ingersoll-Rossmodel with time-varying parameters. It is known that the classical CIR model is amodified Bessel process with deterministic time and scale change. We show thatthis relation can be generalized for the extended CIR model with time-varyingparameters, if we consider Bessel process with time-varying dimension. Thisenables us to evaluate the arbitrage free prices of discounted bonds and theircontingent claims applying the basic properties of Bessel processes. Furthermorewe study a special class of extended CIR models which not only enables us to fitevery arbitrage free initial term structure, but also to give the extended CIRcall option pricing formula.     

No arbitrage conditions for simple trading strategies

Strict local martingales may admit arbitrage opportunities with respect to the class of simple trading strategies. (Since there is no possibility of using doubling strategies in this framework, the losses are not assumed to be bounded from below.) We show that for a class of non-negative strict local martingales, the strong Markov property implies the no arbitrage property with respect to the class of simple trading strategies. This result can be seen as a generalization of a similar result on three dimensional Bessel process in Delbaen and Schachermayer (Math Finance 4:343–348, 1994). We also provide no arbitrage conditions for stochastic processes within the class of simple trading strategies with shortsale restriction.     

Pricing errors and estimates of risk premia in factor models

In this paper, we consider the effect of the theoretical pricing error in the arbitrage pricing model on estimates of risk premia implied by the model. Under arbitrage pricing, the pricing error satisfies a strong bounding condition where for an infinite set of assets, the sum of squared pricing errors is bounded. We characterize the pricing error in terms of orders of probability and estimate an expected returns model which allows for pricing errors less than order one in probability. The principal finding of the paper is that misspecification of the pricing error and misspecification of the factor structure has no effect on the bias or mean squared error of the dominant risk premium. This implies that an exact form of arbitrage pricing can be used to estimate risk premia.     

On option pricing models in the presence of heavy tails

We propose a modification of the option pricing framework derived by Borland which removes the possibilities for arbitrage within this framework. It turns out that such arbitrage possibilities arise due to an incorrect derivation of the martingale transformation in the non-Gaussian option models which are used in that paper. We show how a similar model can be built for the asset price processes which excludes arbitrage. However, the correction causes the pricing formulas to be less explicit than the ones in the original formulation, since the stock price itself is no longer a Markov process. Practical option pricing algorithms will therefore have to resort to Monte Carlo methods or partial differential equations and we show how these can be implemented. An extra parameter, which needs to be specified before the model can be used, will give market makers some extra freedom when fitting their model to market data.     

On arbitrages arising with honest times

In the context of a general continuous financial market model, we study whether the additional information associated with an honest time τ gives rise to arbitrage profits. By relying on the theory of progressive enlargement of filtrations, we explicitly show that no kind of arbitrage profit can ever be realised strictly before τ, whereas classical arbitrage opportunities can be realised exactly at τ as well as after τ. Moreover, arbitrages of the first kind can only be obtained by starting to trade as soon as τ occurs. We carefully study the behavior of local martingale deflators and consider no-arbitrage-type conditions weaker than no free lunch with vanishing risk.     

An Interest Rate Model with Upper and Lower Bounds

We propose a new interest rate dynamicsmodel where the interest rates fluctuate in a bounded region. The model ischaracterised by five parameters which are sufficiently flexible to reflect theprediction of the future interest rates distribution. The interest rate convergesin law to a Beta distribution and has transition probabilities which arerepresented by a series of Jacobi polynomials. We derive the moment evaluationformula of the interest rate. We also derive the arbitrage free pure discountbond price formula by a weighted series of Jacobi polynomials. Furthermore wegive simple lower and upper bounds for the arbitrage free discount bond pricewhich are tight for the narrow interest rates region case. Finally we show thatthe numerical evaluation procedure converges to the exact value in the limitand evaluate the accuracy of the approximation formulas for the discount bondprices.     

The principle of not feeling the boundary for the SABR model

The stochastic alpha–beta–rho (SABR) model is widely used in fixed income and foreign exchange markets as a benchmark. The underlying process may hit zero with a positive probability and therefore an absorbing boundary at zero should be specified to avoid arbitrage opportunities. However, a variety of numerical methods choose to ignore the boundary condition to maintain the tractability. This paper develops a new principle of not feeling the boundary to quantify the impact of this boundary condition on the distribution of underlying prices. It shows that the probability of the SABR hitting zero decays to 0 exponentially as the time horizon shrinks. Applying this principle, we further show that conditional on the volatility process, the distribution of the underlying process can be approximated by that of a time-changed Bessel process with an exponentially negligible error. This discovery provides a theoretical justification for many almost exact simulation algorithms for the SABR model in the literature. Numerical experiments are also presented to support our results.     

Asset pricing theory for two price economies

We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction. Furthermore the developed valuation operators call into question the current practice of Debt Valuation Adjustments.     

A jump to default extended CEV model: an application of Bessel processes

We develop a flexible and analytically tractable framework which unifies the valuation of corporate liabilities, credit derivatives, and equity derivatives. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link between default and equity volatility, we assume that the hazard rate of default is an increasing affine function of the instantaneous variance of returns on the underlying stock. To capture the negative link between volatility and stock price, we assume a constant elasticity of variance (CEV) specification for the instantaneous stock volatility prior to default. We show that deterministic changes of time and scale reduce our stock price process to a standard Bessel process with killing. This reduction permits the development of completely explicit closed form solutions for risk-neutral survival probabilities, CDS spreads, corporate bond values, and European-style equity options. Furthermore, our valuation model is sufficiently flexible so that it can be calibrated to exactly match arbitrarily given term structures of CDS spreads, interest rates, dividend yields, and at-the-money implied volatilities.     

The stochastic collocation Monte Carlo sampler: highly efficient sampling from ‘expensive’ distributions

In this article, we propose an efficient approach for inverting computationally expensive cumulative distribution functions. A collocation method, called the Stochastic Collocation Monte Carlo sampler (SCMC sampler), within a polynomial chaos expansion framework, allows us the generation of any number of Monte Carlo samples based on only a few inversions of the original distribution plus independent samples from a standard normal variable. We will show that with this path-independent collocation approach the exact simulation of the Heston stochastic volatility model, as proposed in Broadie and Kaya [Oper. Res., 2006, 54, 217–231], can be performed efficiently and accurately. We also show how to efficiently generate samples from the squared Bessel process and perform the exact simulation of the SABR model.     

Rating model arbitrage in CDO markets: An empirical analysis

We analyze whether information asymmetry between issuers and investors leads to rating model arbitrage in Collateralized Debt Obligation markets. Rating model arbitrage is defined as the issuer's deliberate capitalization of information asymmetry at the investor's cost on the basis of different rating processes. Using data from CDO transactions grouped by both rating agencies and underlying rating methodologies, we test for homogeneity of characteristic transaction features within the group and heterogeneity between the different groups. We find that the hypothesis stating non-existence of rating model arbitrage on the basis of information asymmetry does not hold as individual patterns of transaction characteristics within each group could be identified.     

Arbitrage risk and market efficiency

This article explores arbitràge risk and models a testable hypothesis for studies in the treasury bill futures market efficiency. The modern mean-variance theory applied to a hedged arbitrage portfolio is used for the analysis. For a given expected arbitrage profit, we derive minimum variance arbitrage (MVA) conditions. A minimum variance arbitrage line (MVAL) is then derived to show the risk-return tradeoff for arbitrage. Market efficiency conditions are discussed by taking into account arbitrage risk along with bid-ask spreads. The analysis in this study helps explain the puzzle of inefficiencies in the T-bill futures market. Because refinancing and variation margin (due to marking-to-market) are required for arbitrage using futures trading in general, our ex ante arbitrage model using the case of T-bill futures can be applied to other futures markets.     

MEASURING LIMITS OF ARBITRAGE IN FIXED‐INCOME MARKETS

An emerging literature relies on an index of limits of arbitrage in fixed‐income markets. We analyze the benefits of an index that is model‐free, robust, and intuitive. This new index strengthens the evidence that limits of arbitrage proxy for risks priced in the cross‐section of returns. Trading simulations show that the new index improves identification of limits of arbitrage because it bypasses a noisy estimation step. Relative value indices in the United States, United Kingdom, Japan, Germany, Italy, France, Switzerland, and Canada exhibit strong commonality and high correlations with local volatility and funding conditions. The indices are updated regularly and available publicly.     

Statistical properties of demand fluctuation in the financial market

Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either directly given by a mean-reverting CEV process or as a transformed Ornstein–Uhlenbeck process. For the latter, we introduce a new model based on a simple hyperbolic transformation. Various numerical methods for integrating mean-reverting CEV processes are analysed and compared with respect to positivity preservation and efficiency. Moreover, we develop a simple and robust integration scheme for the two-dimensional system using the strong convergence behaviour as an indicator for the approximation quality. This method, which we refer to as the IJK (137) scheme, is applicable to all types of stochastic volatility models and can be employed as a drop-in replacement for the standard log-Euler procedure.     

Overcoming limits of arbitrage: Theory and evidence

Limits to arbitrage arise because financial intermediaries may face funding constraints when mispricing worsens. Using a model with limits to arbitrage, where we allow arbitrageurs to secure capital even in case of underperformance, we show that arbitrageurs that are more protected from withdrawals have more mean-reverting and volatile returns. Using data on hedge fund performance, we find robust support for these hypotheses: Funds with contractual impediments to withdrawals, and funds with performance-insensitive outflows, recover more quickly after a bad year and have more volatile returns. Our evidence is consistent with the idea that some hedge funds overcome the limits to arbitrage.     

A jump telegraph model for option pricing

In this paper we introduce a financial market model based on continuous time random motions with alternating constant velocities and jumps occurring when the velocities are switching. This model is free of arbitrage if jump directions are in a certain correspondence with the velocities of the underlying random motion. Replicating strategies for European options are constructed in detail. Exact formulae for option prices are derived.     

高频数据下基于组合预测思想的统计套利策略创新设计与实证研究

采用两只股票的日数据和5种高频数据,借鉴组合预测思想,综合利用协整模型和新卡尔曼滤波模型,与统计套利策略具体目标相结合,设计出新统计套利组合策略,实证分析数据频率、策略选择对统计套利效果的影响。结果表明：运用高频数据及引入卡尔曼滤波模型均有效,但卡尔曼滤波模型与协整模型不存在明显优劣之分,选择组合策略是必要的;组合策略收益性显著优于采取单一模型的套利策略;组合策略下的套利组合随数据频率提高,收益率波动性更小、更稳定;组合策略接近市场中性,能很好地免疫市场风险。     

Term Structure Movements and Pricing Interest Rate Contingent Claims

This paper derives an arbitrage-free interest rate movements model (AR model). This model takes the complete term structure as given and derives the subsequent stochastic movement of the term structure such that the movement is arbitrage free. We then show that the AR model can be used to price interest rate contingent claims relative to the observed complete term structure of interest rates. This paper also studies the behavior and the economics of the model. Our approach can be used to price a broad range of interest rate contingent claims, including bond options and callable bonds.     

The relationship between conditional value at risk and option prices with a closed-form solution

Options and CVaR (conditional value at risk) are significant areas of research in their own right; moreover, both are important to risk management and understanding of risk. Despite the importance and the overlap of interests in CVaR and options, the literature relating the two is virtually non-existent. In this paper we derive a model-free, simple and closed-form analytic equation that determines the CVaR associated with a put option. This relation is model free and is applicable in complete and incomplete markets. We show that we can account for implied volatility effects using the CVaR risk of options. We show how the relation between options and CVaR has important risk management implications, particularly in terms of integrated risk management and preventing arbitrage opportunities. We conduct numerical experiments to demonstrate obtaining CVaR from empirical options data.     

Liquidity and Arbitrage in Options Markets: A Survival Analysis Approach

This paper examines the determinants of the time it takes foran index options market to return to no arbitrage values afterput-call parity deviations, using intraday transactions datafrom the French index options market. We employ survival analysisto characterize how limits to arbitrage influence the expectedduration of arbitrage deviations. After controlling for conventionallimits to arbitrage, we show that liquidity-linked variablesare associated with a faster reversion of arbitrage profits.The introduction of an Exchange Traded Fund also affects thesurvival rates of deviations, but this impact essentially stemsfrom the reduction in the level of potential arbitrage profits.