arXiv subjects

Yaozhong Hu

Publications and source records attributed to Yaozhong Hu.

113 records · Page 7Linked to original sources

arXiv2006

A Delayed Black and Scholes Formula I

In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic differential delay equation (sdde). We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form representation of the option price. Furthermore, the model maintains the no-arbitrage property and the completeness of the market. The derivation of the option-pricing formula is based on an equivalent martingale measure.

math.PR↗

arXiv2006

A Delayed Black and Scholes Formula II

This article is a sequel to [A.H.M.P]. In [A.H.M.P], we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic delay equation with fixed delays in the drift and diffusion terms. In this article, we look at models of the stock price described by stochastic functional differential equations with variable delays. We present a class of examples of stock dynamics with variable delays that permit an explicit form for the option pricing formula. As in [A.H.M.P], the market is complete with no arbitrage. This is achieved through the existence of an equivalent martingale measure. In subsequent work, the authors intend to test the models in [A.H.M.P] and the present article against real market data.

math.PR↗

arXiv2006

Rough Path Analysis Via Fractional Calculus

Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued Hölder continuous functions of order $\displaystyle β\in (\frac13, \frac12)$ and $f$ is a continuously differentiable function such that $f'$ is $λ$-Höldr continuous for some $λ>\frac1β-2$. Under some further smooth conditions on $f$ the integral is a continuous functional of $x$, $y$, and the tensor product $x\otimes y$ with respect to the Hölder norms. We derive some estimates for these integrals and we solve differential equations driven by the function $y$. We discuss some applications to stochastic integrals and stochastic differential equations.

math.PR↗

arXiv2005

Renormalized self-intersection local time for fractional Brownian motion

Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter H\in(0,1). Assume d\geq2. We prove that the renormalized self-intersection local time\ell=\int_0^T\int_0^tδ(B_t^H-B_s^H) ds dt -E\biggl(\int_0^T\int_0^tδ(B_t^H-B_s^H) ds dt\biggr) exists in L^2 if and only if H<3/(2d), which generalizes the Varadhan renormalization theorem to any dimension and with any Hurst parameter. Motivated by a result of Yor, we show that in the case 3/4>H\geq\frac{3}{2d}, r(ε)\ell_ε converges in distribution to a normal law N(0,Tσ^2), as εtends to zero, where \ell_ε is an approximation of \ell, defined through (2), and r(ε)=|\logε|^{-1} if H=3/(2d), and r(ε)=ε^{d-3/(2H)} if 3/(2d)<H.

math.PR↗

arXiv2004

Some Processes Associated with Fractional Bessel Processes

Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=% \sqrt{(B_{t}^{1})^{2}+... +(B_{t}^{d})^{2}}$ be the fractional Bessel process. Itô's formula for the fractional Brownian motion leads to the equation $ R_{t}=\sum_{i=1}^{d}\int_{0}^{t}\frac{B_{s}^{i}}{R_{s}}% dB_{s}^{i}+H(d-1)\int_{0}^{t}\frac{s^{2H-1}}{R_{s}}ds . $ In the Brownian motion case ($H=1/2$), $X_{t}=\sum_{i=1}^{d}\int_{0}^{t} frac{B_{s}^{i}}{% R_{s}}dB_{s}^{i}$ is a Brownian motion. In this paper it is shown that $X_{t}$ is \underbar{not} a fractional Brownian motion if $H\not=1/2$. We will study some other properties of this stochastic process as well.

math.PR↗