arXiv subjects
Publications and source records attributed to Masaaki Fukasawa.
A constant weight asset allocation is a popular investment strategy and is optimal under a suitable continuous model. We study the tracking error for the target continuous rebalancing strategy by a feasible discrete-in-time rebalancing under a general multi-dimensional Brownian semimartingale model of asset prices. In a high-frequency asymptotic framework, we derive an asymptotically efficient sequence of simple predictable strategies.
We consider Geometric Mean Market Makers -- a special type of Decentralized Exchange -- with two types of users: liquidity takers and arbitrageurs. Liquidity takers trade at prices that can create arbitrage opportunities, while arbitrageurs align the exchange's price with the external market price. We show that in Geometric Mean Market Makers charging proportional transaction fees, Impermanent Loss can be super-hedged by a model-free rebalancing strategy. Moreover, we demonstrate that in such a DEX, the exchange rate is of finite variation, so that loss-versus-rebalancing (the shortfall of providing liquidity versus the corresponding constant-weights portfolio) vanishes.
We study the weak convergence rate in the discretization of rough volatility models. After showing a lower bound $2H$ under a general model, where $H$ is the Hurst index of the volatility process, we give a sharper bound $H + 1/2$ under a linear model.
Following-up Fukasawa and Gatheral (Frontiers of Mathematical Finance, 2022), we prove that the BBF formula, the SABR formula, and the rough SABR formula provide asymptotically arbitrage-free approximations of the implied volatility under, respectively, the local volatility model, the SABR model, and the rough SABR model.
Our study aims to specify the asymptotic error distribution in the discretization of a stochastic Volterra equation with a fractional kernel. It is well-known that for a standard stochastic differential equation, the discretization error, normalized with its rate of convergence $1/\sqrt{n}$, converges in law to the solution of a certain linear equation. Similarly to this, we show that a suitably normalized discretization error of the Volterra equation converges in law to the solution of a certain linear Volterra equation with the same fractional kernel.
Following an approach originally suggested by Balland in the context of the SABR model, we derive an ODE that is satisfied by normalized volatility smiles for short maturities under a rough volatility extension of the SABR model that extends also the rough Bergomi model. We solve this ODE numerically and further present a very accurate approximation to the numerical solution that we dub the rough SABR formula.
In this chapter we first briefly review the existing approaches to hedging in rough volatility models. Next, we present a simple but general result which shows that in a one-factor rough stochastic volatility model, any option may be perfectly hedged with a dynamic portfolio containing the underlying and one other asset such as a variance swap. In the final section we report the results of a back-test experiment using real data, where VIX options are hedged with a forward variance swap. In this experiment, using a rough volatility model allows to almost completely remove the bias and reduce the overall hedging error by a factor of 27% compared to traditional diffusion-based models.
We formulate a superhedging theorem in the presence of transaction costs and model uncertainty. Asset prices are assumed continuous and uncertainty is modelled in a parametric setting. Our proof relies on a new topological framework in which no Krein-Smulian type theorem is available.
Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials. For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al. (2019) and Friz et al. (2020).
We study sufficient conditions for a local asymptotic mixed normality property of statistical models. We develop a scheme with the $L^2$ regularity condition proposed by Jeganathan [\textit{Sankhya Ser. A} \textbf{44} (1982) 173--212] so that it is applicable to high-frequency observations of stochastic processes. Moreover, by combining with Malliavin calculus techniques by Gobet [\textit{Bernoulli} \textbf{7} (2001) 899--912, 2001], we introduce tractable sufficient conditions for smooth observations in the Malliavin sense, which do not require Aronson-type estimates of the transition density function. Our results, unlike those in the literature, can be applied even when the transition density function has zeros. For an application, we show the local asymptotic mixed normality property of degenerate (hypoelliptic) diffusion models under high-frequency observations, in both complete and partial observation frameworks. The former and the latter extend previous results for elliptic diffusions and for integrated diffusions, respectively.
A stochastic hybrid system, also known as a switching diffusion, is a continuous-time Markov process with state space consisting of discrete and continuous parts. We consider parametric estimation of theQmatrix for the discrete state transitions and of the drift coefficient for the diffusion part. First, we derive the likelihood function under the complete observation of a sample path in continuous-time. Then, extending a finite-dimensional filter for hidden Markov models developed by Elliott et al. (Hidden Markov Models, Springer, 1995) to stochastic hybrid systems, we derive the likelihood function and the EM algorithm under a partial observation where the continuous state is monitored continuously in time, while the discrete state is unobserved.
We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of a SDE with a time changed Brownian motion, dated back to Doeblin (1940). In cases where the diffusion coefficient is bounded and $β$-Hölder continuous with $0 < β\leq 1$, we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat a SDE with no strong solution. Our scheme is the first to achieve the strong convergence for the case $0 < β< 1/2$.
First, we give an asymptotic expansion of short-dated at-the-money implied volatility that refines the preceding works and proves in particular that non-rough volatility models are inconsistent to a power law of volatility skew. Second, we show that given a power law of volatility skew in an option market, a continuous price dynamics of the underlying asset with non-rough volatility admits an arbitrage opportunity. The volatility therefore has to be rough in a viable market of the underlying asset of which the volatility skew obeys a power law.
The aim of this study is to find a generic method for generating a path of the solution of a given stochastic differential equation which is more efficient than the standard Euler-Maruyama scheme with Gaussian increments. First we characterize the asymptotic distribution of pathwise error in the Euler-Maruyama scheme with a general partition of time interval and then, show that the error is reduced by a factor (d+2)/d when using a partition associated with the hitting times of sphere for the driving d-dimensional Brownian motion. This reduction ratio is the best possible in a symmetric class of partitions. Next we show that a reduction which is close to the best possible is achieved by using the hitting time of a moving sphere which is easier to implement.
Rough volatility models are continuous time stochastic volatility models where the volatility process is driven by a fractional Brownian motion with the Hurst parameter smaller than half, and have attracted much attention since a seminal paper titled "Volatility is rough" was posted on SSRN in 2014 showing that the log realized volatility time series of major stock indices have the same scaling property as such a rough fractional Brownian motion has. We however find by simulations that the impressive approach tends to suggest the same roughness irrespectively whether the volatility is actually rough or not; an overlooked estimation error of latent volatility often results in an illusive scaling property. Motivated by this preliminary finding, here we develop a statistical theory for a continuous time fractional stochastic volatility model to examine whether the Hurst parameter is indeed estimated smaller than half, that is, whether the volatility is really rough. We construct a quasi-likelihood estimator and apply it to realized volatility time series. Our quasi-likelihood is based on the error distribution of the realized volatility and a Whittle-type approximation to the auto-covariance of the log-volatility process. We prove the consistency of our estimator under high frequency asymptotics, and examine by simulations its finite sample performance. Our empirical study suggests that the volatility is indeed rough; actually it is even rougher than considered in the literature.
A small-time Edgeworth expansion of the density of an asset price is given under a general stochastic volatility model, from which asymptotic expansions of put option prices and at-the-money implied volatilities follow. A limit theorem for at-the-money implied volatility skew and curvature is also given as a corollary. The rough Bergomi model is treated as an example.
We study how trading costs are reflected in equilibrium returns. To this end, we develop a tractable continuous-time risk-sharing model, where heterogeneous mean-variance investors trade subject to a quadratic transaction cost. The corresponding equilibrium is characterized as the unique solution of a system of coupled but linear forward-backward stochastic differential equations. Explicit solutions are obtained in a number of concrete settings. The sluggishness of the frictional portfolios makes the corresponding equilibrium returns mean-reverting. Compared to the frictionless case, expected returns are higher if the more risk-averse agents are net sellers or if the asset supply expands over time.
We study a Edgeworth-type refinement of the central limit theorem for the discretizacion error of Itô integrals. Towards this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula.