The characteristic function of rough Heston models

It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non‐Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing. Using an original link between nearly unstable Hawkes processes and fractional volatility models, we compute the characteristic function of the log‐price in rough Heston models. In the classical Heston model, the characteristic function is expressed in terms of the solution of a Riccati equation. Here, we show that rough Heston models exhibit quite a similar structure, the Riccati equation being replaced by a fractional Riccati equation.     

Arbitrage with Fractional Brownian Motion

Fractional Brownian motion has been suggested as a model for the movement of log share prices which would allow long–range dependence between returns on different days. While this is true, it also allows arbitrage opportunities, which we demonstrate both indirectly and by constructing such an arbitrage. Nonetheless, it is possible by looking at a process similar to the fractional Brownian motion to model long–range dependence of returns while avoiding arbitrage.     

Volatility of volatility is (also) rough

Using high-frequency data for major volatility indexes, we compute the volatility of volatility and show that its logarithm follows a fractional Brownian motion with Hurst parameter smaller than 1/2 thereby extending to the volatility asset class the recent findings obtained for the equity index markets. The results confirm that the volatility of volatility is a rough process and it possesses the long memory property. We also show that the correlation between the volatility and the volatility of volatility is positive, consistent with observations in the volatility option market. Lastly, a robustness check using volatility futures confirms the findings.     

NO ARBITRAGE UNDER TRANSACTION COSTS, WITH FRACTIONAL BROWNIAN MOTION AND BEYOND

We establish a simple no-arbitrage criterion that reduces the absence of arbitrage opportunities under proportional transaction costs to the condition that the asset price process may move arbitrarily little over arbitrarily large time intervals.
We show that this criterion is satisfied when the return process is either a strong Markov process with regular points, or a continuous process with full support on the space of continuous functions. In particular, we prove that proportional transaction costs of any positive size eliminate arbitrage opportunities from geometric fractional Brownian motion for H ∈ (0, 1) and with an arbitrary continuous deterministic drift.     

MULTIFRACTIONAL STOCHASTIC VOLATILITY MODELS

The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. mBm is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for varying local regularity, and to decouple it from long‐range dependence properties. While we believe that mBm is potentially useful in a variety of applications in finance, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in Comte and Renault. Using the stochastic calculus with respect to mBm developed in Lebovits and Lévy Véhel, we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization‐based cubature methods, to get accurate approximations. This allows us to test the behavior of our model (as well as the one in Comte and Renault) with respect to its parameters, and in particular its ability to explain some features of the implied volatility surface. An advantage of our model is that it is able both to fit smiles at different maturities, and to take volatility persistence into account in a more precise way than Comte and Renault.     

The Feldstein–Horioka puzzle in South Africa: A fractional cointegration approach

The Feldstein–Horioka (FH) puzzle, that is the strong correlation between saving and investment in a world where obstacles to capital mobility are limited, has been studied extensively since it was exposed in 1980. Even though the theoretical and empirical literature has examined many of its potential causes, the puzzle persists. This paper aims at shedding further light on the issue by investigating the relationship between saving and investment in South Africa since 1946 using fractional integration and cointegration techniques to account for high persistence in the series. We find evidence of fractional cointegration between saving and investment, indicating some degree of persistence in the gap between the two variables. We also find a structural break in saving and investment ratios to GDP around 1980, which roughly coincides with the start of a financial deregulation process in South Africa. While fractional cointegration holds before the break, it does not thereafter. In other words, while the FH puzzle is observed before the start of financial deregulation, it subsequently disappears. This suggests that financial deregulation may have loosened the link between saving and investment.     

一种无线异构网络中的分步协作基站分簇方法

为抑制小区间干扰提高系统性能,基于异构网络提出了一种分步基站分簇方法。该方法首先依据期望用户受到的除主服务基站之外的其余基站的信号干扰强度,确定待分簇基站集合;然后根据期望用户接收到待选集合中各个基站的信号强度,确定最终参与协作的基站集合。分步基站分簇方法通过缩小用户选择范围,同时综合考虑用户所在小区的信号强度和周围基站的干扰信号强度进行协作基站分簇。仿真结果表明:相对于现有功率最大法和阈值法,基于分步的基站分簇方法具有更高的用户端传输速率,更好地提升了系统性能。     

Optimal portfolio under fractional stochastic environment

Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non‐Markovian) fractional stochastic environment (for all values of the Hurst index ). We rigorously establish a first‐order approximation of the optimal value, when the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein–Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth‐ order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.     

Is There Long Memory in Indian Stock Market Returns? An Empirical Search

The issue of long memory, though has important theoretical and practical implications, has not received much attention in India. This article examines the issue of long memory in mean of the stock returns by employing a set of sophisticated time-series tests including a bias reduced log periodogram test of Andrews and Guggenberger. The study used daily values of 29 major indices including sectoral indices traded on the National Stock Exchange and Bombay Stock Exchange from April 2003 to March 2012, which provide insights into relation between composition of indices and long memory. The findings of the study suggest significant presence of long memory in mean returns of the medium- and small-sized indices and weaker evidences for large cap indices. Further, the study identifies a relationship between presence of long memory and market structure variables. The use of linear models in the presence of long memory would result in incorrect inferences, and this calls for investigation of appropriate long memory model to generate profits in Indian stock market.     

基于聚类分析的分数阶Fourier变换信号分离与检测

针对应用分数阶Fourier变换检测多分量信号计算量大、效率低的问题,提出在分数 阶Fourier域采取混沌-多步拟牛顿法和聚类分析相结合的方法分离与检测信号。在保证全局 快速搜索最强信号的条件下,实现一次检测多个较强信号,并将已检信号逐次消去,减小对 剩余信号检测的影响。仿真结果表明,该方法能快速分离检测多种调制的频谱混叠信号。     

利用分数阶Fourier变换抑制高频地波雷达中线性调频干扰

高频地波雷达(HFGWR)受到严重的射频干扰影响。单频射频干扰在接收信号中体现为高强度的线性调频信号,从而污染所有距离元。为抑制射频干扰,通过分析其频率特征,使用分数阶傅里叶变换（FRFT）将原始信号转换到分数阶傅里叶域,对射频干扰对应的谱峰置零,达到抑制干扰的目的。该方法的优点在于抑制射频干扰的同时无损干扰位置处的回波信号,无需重构信号。实测数据分析表明：FRFT不仅能有效抑制射频干扰,信噪比提高可达10 dB以上,而且其计算复杂度较小,满足雷达实时工作要求。     

PRICING AND HEDGING DOUBLE-BARRIER OPTIONS: A PROBABILISTIC APPROACH

Barrier options have become increasingly popular over the last few years. Less expensive than standard options, they may provide the appropriate hedge in a number of risk management strategies. In the case of a single-barrier option, the valuation problem is not very difficult (see Merton 1973 and Goldman, Sosin, and Gatto 1979). the situation where the option gets knocked out when the underlying instrument hits either of two well-defined boundaries is less straightforward. Kunitomo and Ikeda (1992) provide a pricing formula expressed as the sum of an infinite series whose convergence is studied through numerical procedures and suggested to be rapid. We follow a methodology which proved quite successful in the case of Asian options (see Geman and Yor 1992,1993) and which has its roots in some fundamental properties of Brownian motion. This methodology permits the derivation of a simple expression of the Laplace transform of the double-barrir price with respect to its maturity date. the inversion of the Laplace transform using techniques developed by Geman and Eydeland (1995), is then fairly easy to perform.     

Return predictability of variance differences: A fractionally cointegrated approach

This paper examines the fractional cointegration between downside (upside) components of realized and implied variances. A positive association is found between the strength of their cofractional relation and the return predictability of their differences. That association is established via the common long-memory component of the variances that are fractionally cointegrated, which represents the volatility-of-volatility factor that determines the variance premium. Our results indicate that market fears play a critical role not only in driving the long-run equilibrium relationship between implied-realized variances but also in understanding the return predictability. A simulation study further verifies these claims.     

低信噪比下LFM- PRBC信号参数的快速估计

为了解决目前算法中线性调频-伪码（LFM- PRBC）信号参数估计计算量较大的问题,提出了一种快速估计算法。该算法采用解线调与分数阶傅里叶变换（FRFT）进行参数的估计。首先对信号进行解线调估计出调频斜率的粗略值,然后由调频斜率确定旋转角,通过FRFT估计出码元宽度的粗略值。根据延时再进行解线调估计出调频斜率的精确值,再通过FRFT估计出码元宽度的精确值与起始频率。该算法不仅计算量较低,同时具有很高的估计精度与很强的抗噪性,仿真实验验证了该算法的有效性。     

一种基于分数阶Fourier变换的雷达运动目标检测算法

针对雷达回波为多分量LFM信号时,时频分析存在的交叉项干扰问题,提出了一种基于分数阶Fourier变换(Fractional Fourier Transform,FRFT)的伪Wigner分布(PWD).该方法通过在参数平面按阈值进行峰值搜索确定变换域阶次,再在相应的分数阶Fourier域计算PWD,有效地抑制了交叉项的干扰,有利于更好地提取信号的时频信息.仿真实验证明了在强背景噪声下该算法的有效性.     

一种Farrow结构数字延时滤波器的设计

普通数字延时滤波器虽然结构简单，但系数计算过程复杂，在延时参数快速变化时，系数更新速度无法满足实时性要求，在工程应用上受限制。采用Farrow结构数字延时滤波器能够更加灵活高效地进行分数延时滤波，延时参数改变时，无需重新计算滤波器系数，更容易在现场可编程门阵列(FPGA)上实现。介绍了一种Farrow结构数字延时滤波器，提出采用基于对称结构的滤波器系数求解方法，并经过加权优化，获得最终Farrow滤波器的系数。系数计算过程中，通过对设计所得Farrow滤波器延时精度和误差的分析，调整加权因子的取值和滤波器阶数，进而提高延时精度。计算机仿真结果证明了加权对称系数求解Farrow滤波器系数方法的有效性和实用性。     

A note on the long rate in factor models of the term structure

In this paper, we consider factor models of the term structure based on a Brownian filtration. We show that the existence of a nondeterministic long rate in a factor model of the term structure implies, as a consequence of the Dybvig–Ingersoll–Ross theorem, that the model has an equivalent representation in which one of the state variables is nondecreasing. For two‐dimensional factor models, we prove moreover that if the long rate is nondeterministic, the yield curve flattens out, and the factor process is asymptotically nondeterministic, then the term structure is unbounded. Finally, we provide an explicit example of a three‐dimensional affine factor model with a nondeterministic yet finite long rate in which the volatility of the factor process does not vanish over time.     

Option pricing under fast‐varying long‐memory stochastic volatility

Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long‐range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the option price in the regime where the volatility process is fast mean reverting. The volatility process is modeled as a smooth and bounded function of a fractional Ornstein–Uhlenbeck process. We give the expression for the implied volatility, which has a fractional term structure.     

THE TRACKING ERROR RATE OF THE DELTA‐GAMMA HEDGING STRATEGY

We analyze the convergence rate of the quadratic tracking error, when a Delta‐Gamma hedging strategy is used at N discrete times. The fractional regularity of the payoff function plays a crucial role in the choice of the trading dates, in order to achieve optimal rates of convergence.     

Bond returns and betas as dependent upon conditioned Brownian motion

This research uses a conditioned Brownian motion process, the Brownian bridge, to analyze bond return dynamics and systematic risk. The results are contrasted with prior results that assume that yields follow unconditioned Brownian motion and reveal significant differences. Expected returns and the variance of expected returns can be given in much more concise terms that are more easily interpreted. Also, bond betas are demonstrated to depend on returns above an initially stated deterministic yield and shown to fluctuate proportionately with this excess return, which can often be negative. Finally, empirical tests show that bond excess returns do follow a Brownian bridge process or at least do so up to a multiple of the conditioned drift term.