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Stochastic approximation algorithm is a useful technique which has been exploited successfully in probability theory and statistics for a long time. The step sizes used in stochastic approximation are generally taken to be deterministic and…
Stochastic processes are often used to model complex scientific problems in fields ranging from biology and finance to engineering and physical science. This paper investigates rate-optimal estimation of the volatility matrix of a…
Agents' heterogeneity is recognized as a driver mechanism for the persistence of financial volatility. We focus on the multiplicity of investment strategies' horizons, we embed this concept in a continuous time stochastic volatility…
Rough volatility models are known to reproduce the behavior of historical volatility data while at the same time fitting the volatility surface remarkably well, with very few parameters. However, managing the risks of derivatives under…
We solve the first-passage problem for the Heston random diffusion model. We obtain exact analytical expressions for the survival and hitting probabilities to a given level of return. We study several asymptotic behaviors and obtain…
We study the dependence of volatility on the stock price in the stochastic volatility framework on the example of the Heston model. To be more specific, we consider the conditional expectation of variance (square of volatility) under fixed…
This paper provides several statistical estimators for the drift and volatility parameters of an Ornstein-Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time…
It is a market practice to express market-implied volatilities in some parametric form. The most popular parametrizations are based on or inspired by an underlying stochastic model, like the Heston model (SVI method) or the SABR model (SABR…
We derive the short-maturity asymptotics for option prices in the local volatility model in a new short-maturity limit $T\to 0$ at fixed $\rho = (r-q) T$, where $r$ is the interest rate and $q$ is the dividend yield. In cases of practical…
Let $\sigma_t(x)$ denote the implied volatility at maturity $t$ for a strike $K=S_0 e^{xt}$, where $x\in\bbR$ and $S_0$ is the current value of the underlying. We show that $\sigma_t(x)$ has a uniform (in $x$) limit as maturity $t$ tends to…
In this paper we establish limit theorems for power variations of stochastic processes controlled by fractional Brownian motions with Hurst parameter $H\leq 1/2$. We show that the power variations of such processes can be decomposed into…
The finite sample effect on the Hurst exponent (HE) of realized volatility time series is examined using Bitcoin data. This study finds that the HE decreases as the sampling period $\Delta$ increases and a simple finite sample ansatz…
In this paper we study short-time behavior of the at-the-money implied volatility for Inverse European options with fixed strike price. The asset price is assumed to follow a general stochastic volatility process. Using techniques of the…
In this short note, we show how to use concentration inequalities in order to build exact confidence intervals for the Hurst parameter associated with a one-dimensional fractional Brownian motion
This paper discusses the short-maturity behavior of Asian option prices and hedging portfolios. We consider the risk-neutral valuation and the delta value of the Asian option having a H\"older continuous payoff function in a local…
High precision analytical approximation is proposed for variance-covariance based risk allocation in a portfolio of risky assets. A general case of a single-period multi-factor Merton-type model with stochastic recovery is considered. The…
Stochastic volatility models are the backbone of financial engineering. We study both continuous time diffusions as well as discrete time models. We propose two novel approaches to estimating stochastic volatility diffusions, one using…
Here we demonstrate how we can use Small Volatility Approximation in calibration of Multi-Factor HJM model with deterministic correlations, factor volatilities and mean reversals. It is noticed that quality of this calibration is very good…
Correlations between asset returns are important in many financial applications. In recent years, multivariate volatility models have been used to describe the time-varying feature of the correlations. However, the curse of dimensionality…
Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price…