Stochastic integrals driven by fractional Brownian motion and arbitrage: a tale of two integrals

Recent research suggests that fractional Brownian motion can be used to model the long-range dependence structure of the stock market. Fractional Brownian motion is not a semi-martingale and arbitrage opportunities do exist, however. Hu and Øksendal [Infin. Dimens. Anal., Quant. Probab. Relat. Top., 2003, 6, 1–32] and Elliott and van der Hoek [Math. Finan., 2003 Elliott, RJ and van de Hoek, J. 2003. A general fractional white noise theory applications to finance. Math. Finan., 13: 301–330. [Crossref], [Web of Science ®] , [Google Scholar], 13, 301–330] propose the use of the white noise calculus approach to circumvent this difficulty. Under such a setting, they argue that arbitrage does not exist in the fractional market. To unravel this discrepancy, we examine the definition of self-financing strategies used by these authors. By refining their definitions, a new notion of continuously rebalanced self-financing strategies, which is compatible with simple buy and hold strategies, is given. Under this definition, arbitrage opportunities do exist in fractional markets.     

Ruin probability at a given time for a model with liabilities of the fractional Brownian motion type: A partial differential equation approach

In this paper we study the ruin probability at a given time for liabilities of diffusion type, driven by fractional Brownian motion with Hurst exponent in the range (0.5, 1). Using fractional Itô calculus we derive a partial differential equation the solution of which provides the ruin probability. An analytical solution is found for this equation and the results obtained by this approach are compared with the results obtained by Monte-Carlo simulation.     

Premium and reinsurance control of an ordinary insurance system with liabilities driven by a fractional Brownian motion

This paper investigates the problem of premium and reinsurance control of an ordinary insurance system when liabilities are driven by a fractional Brownian motion. The reserve equation is considered using two alternative routes: the first with no reinsurance option, and the second with some controllable proportional reinsurance coverage. Recent results from the theory of fractional linear-quadratic control (fractional calculus) are discussed, partially extended and utilized to derive compact analytical formulae for the optimal functionals of the safety loading (consequently for the respective premium rate), and the volume of the retained risk (or equivalently, for the proportion of the reinsurance coverage).     

A note on Wick products and the fractional Black-Scholes model

In some recent papers (Elliott and van der Hoek 2003; Hu and Øksendal 2003) a fractional Black-Scholes model has been proposed as an improvement of the classical Black-Scholes model (see also Benth 2003; Biagini et al. 2002; Biagini and Øksendal 2004). Common to these fractional Black-Scholes models is that the driving Brownian motion is replaced by a fractional Brownian motion and that the Itô integral is replaced by the Wick integral, and proofs have been presented that these fractional Black-Scholes models are free of arbitrage. These results on absence of arbitrage complelety contradict a number of earlier results in the literature which prove that the fractional Black-Scholes model (and related models) will in fact admit arbitrage. The objective of the present paper is to resolve this contradiction by pointing out that the definition of the self-financing trading strategies and/or the definition of the value of a portfolio used in the above papers does not have a reasonable economic interpretation, and thus that the results in these papers are not economically meaningful. In particular we show that in the framework of Elliott and van der Hoek 2003, a naive buy-and-hold strategy does not in general qualify as self-financing. We also show that in Hu and Øksendal 2003, a portfolio consisting of a positive number of shares of a stock with a positive price may, with positive probability, have a negative value.Received: August 2004, Mathematics Subject Classification (2000): 91B28, 60H05JEL Classification: G10Support of the first author from the Jan Wallander and Tom Hedelius foundation is gratefully acknowledged. The research of the second author is supported by the Swedish Research Council.     

Drawdowns preceding rallies in the Brownian motion model

We study drawdowns and rallies of Brownian motion. A rally is defined as the difference of the present value of the Brownian motion and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. This paper determines the probability that a drawdown of a units precedes a rally of b units. We apply this result to examine stock market crashes and rallies in the geometric Brownian motion model.     

A new closed-form formula for pricing European options under a skew Brownian motion

In this paper, we present a new pricing formula based on a modified Black–Scholes (B-S) model with the standard Brownian motion being replaced by a particular process constructed with a special type of skew Brownian motions. Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options.” The European Journal of Finance 13 (6): 523–544] have worked on this model, the results they obtained are incorrect. In this paper, not only do we identify precisely where the errors in Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options”. The European Journal of Finance 13 (6): 523–544] are, we also present a new closed-form pricing formula based on a newly proposed equivalent martingale measure, called ‘endogenous risk neutral measure’, by which only endogenous risks should and can be fully hedged. The newly derived option pricing formula takes the B-S formula as a special case and it does not induce any significant additional burden in terms of numerically computing option values, compared with the effort involved in computing the B-S formula.     

Non-ideal Brownian motion,generalized Langevin Equation and its application to the security market

Brownian motion has been extensively applied in the field of mathematical finance in modeling the stochastic processes of returns on securities. In this paper basic and generalized Langevin Equations with memory are used to augment Brownian motion to capture the well stylized facts of the financial market that frictions and imperfect information exist. The operator method of Fourier-Laplace transform with an appropriate kernel (influence function) is used to circumvent the difficulty associated with solving a time dependent nonlinear differential Equation, and a practical computational method is proposed.From the Langevin Equation, autocorrelation of the return process and the deviation of the return distribution from an ideal Brownian motion are extracted. It is also proven that the time-dependent differential Equation has a unique solution and that it is much more generalized than a martingale Brownian motion functional.     

Skew Brownian Motion and Pricing European Options

Abstract

The volatility smile and systematic mispricing of the Black–Scholes option pricing model are the typical motivation for examining stochastic processes other than geometric Brownian motion to describe the underlying stock price. In this paper a new stochastic process is presented, which is a special case of the skew-Brownian motion of Itô and McKean. The process in question is the sum of a standard Brownian motion and an independent reflecting Brownian motion that is similar in construction to the stochastic representation of a skew-normal random variable. This stochastic process is taken in its exponential form to price European options. The derived option price nests the Black–Scholes equation as a special case and is flexible enough to accommodate stochastic volatility as well as stochastic skewness.     

Profit Possibilities in Currency Markets: Arbitrage, Hedging, and Speculation

This paper reviews and extends the existing literature on covered arbitrage, delineates the conditions for profitable arbitrage with the hedging instruments of forward and options contracts in the foreign exchange markets, and defines the maximum possible profits out of a given market environment. Next, the simple rules on speculation are articulated with and without transaction costs, and then we show how speculation can be covered with options and forwards. Finally, speculation is integrated with arbitrage and hedging, and further compounding of profit possibilities is illustrated.     

Double boundary crossing result for the brownian motion

Abstract

In a recent paper a general bounded crossing result for the Brownian motion is obtained for a linear upper boundary based on the method of Kac (1951). Based on his main results we are able to develop in the present paper some simple expressions for crossing probabilities in case of a lower and an upper linear boundary. We will consider a Brownian motion process for the surplus of an insurance portfolio. This surplus must stay between two given bounds. If the surplus will cross the upper boundary, we can pay a dividend. The lower boundary can reflect the influence of control authorities and regulation measurements.     

Arbitrage and Viability in Insurance Markets

Insurance markets are subject to transaction costs and constraints on portfolio holdings. Therefore, unlike the frictionless asset markets case, viability is not equivalent to absence of arbitrage possibilities. We use the concept of unbounded arbitrage to characterize viable prices on a complete and an incomplete insurance market. In the complete market, there is an insurance contract for every possible event. In the incomplete market, risk can be insured through proportional and excess of loss like insurance contracts. We show how the the structure of viable prices is affected by the portfolio constraints, the transaction costs, and the structure of marketed contracts.     

Buy rough,sell smooth

Recent work has documented roughness in the time series of stock market volatility and investigated its implications for option pricing. We study a strategy for trading stocks based on measures of their implied and realized roughness. A strategy that goes long the roughest-volatility stocks and short the smoothest-volatility stocks earns statistically significant excess annual returns of 6% or more, depending on the time period and strategy details. The profitability of the strategy is not explained by standard factors. We compare alternative measures of roughness in volatility and find that the profitability of the strategy is greater when we sort stocks based on implied rather than realized roughness. We interpret the profitability of the strategy as compensation for near-term idiosyncratic event risk.     

贷款证券化、监管资本套利与资本监管改进

巴塞尔协议的资本充足率指标可以反映银行部门吸收风险损失的能力,但是无法监测和控制银行体系外的贷款总额和累积的信用风险。20世纪70年代的贷款证券化创新导致银行进行监管资本套利,并使得资本充足率监管趋于失效。本文基于贷款证券化下银行贷款余额与社会贷款余额的差异,分析银行监管资本套利的微观机制并提出改进资本监管的建议。     

Statistical arbitrage in the Black–Scholes framework

In this study, we prove the existence of statistical arbitrage opportunities in the Black–Scholes framework by considering trading strategies that consist of borrowing at the risk-free rate and taking a long position in the stock until it hits a deterministic barrier level. We derive analytical formulas for the expected value, variance and probability of loss for the discounted cumulative trading profits. The statistical arbitrage condition is derived in the Black–Scholes framework, which imposes a constraint on the Sharpe ratio of the stock. Furthermore, we verify our theoretical results via extensive Monte Carlo simulations.     

Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.