arXiv subjects

Qidi Peng

Publications and source records attributed to Qidi Peng.

21 records · Page 2Linked to original sources

arXiv2015

Fractional Hida Malliavin Derivatives and Series Representations of Fractional Conditional Expectations

We represent fractional conditional expectations of a functional of fractional Brownian motion as a convergent series in L^2 space. When the target random variable is some function of a discrete trajectory of fractional Brownian motion, we obtain a backward Taylor series representation; when the target functional is generated by a continuous fractional filtration, the series representation is obtained by applying a "frozen path" operator and an exponential operator to the functional. Three examples are provided to show that our representation gives useful series expansions of ordinary expectations of target random variables.

math.PR↗

arXiv2014

A Bond Option Pricing Formula in the Extended CIR Model, with an Application to Stochastic Volatility

We provide a complete representation of the interest rate in the extended CIR model. Since it was proved in Maghsoodi (1996) that the representation of the CIR process as a sum of squares of independent Ornstein-Uhlenbeck processes is possible only when the dimension of the interest rate process is integer, we use a slightly different representation, valid when the dimension is not integer. Our representation consists in an infinite sum of squares of basic processes. Each basic process can be described as an Ornstein-Uhlenbeck process with jumps at fixed times. In this case, the price of a bond option resembles the Black-Scholes formula, where the normal distribution is replaced by the generalized chi-square distribution. The formula is in closed form, up to the solution of a Riccati equation for the bond price of the option. We then provide a generalization of our representation to an extended CIR model with stochastic volatility. We present a closed form approximation of the price of a bond option, valid when the expiration of the option is small and the speed of mean-reversion of volatility is high. The approximation is in "full" closed form, i.e., it does not require to solve an ordinary differential equation.

math.PR↗

arXiv2012

Hölder regularity and series representation of a class of stochastic volatility models

Let $Φ:\R\rightarrow\R$ be an arbitrary continuously differentiable deterministic function such that $|Φ|+|Φ'|$ is bounded by a polynomial. In this article we consider the class of stochastic volatility models in which ${Z(t)}_{t\in [0,1]}$, the logarithm of the price process, is of the form $Z(t)=\int_{0}^t Φ(X(s)) dW(s)$, where ${X(s)}_{s\in[0,1]}$ denotes an arbitrary centered Gaussian process whose trajectories are, with probability 1, Hölder continuous functions of an arbitrary order $α\in (1/2,1]$, and where ${W(s)}_{s\in[0,1]}$ is a standard Brownian motion independent on ${X(s)}_{s\in [0,1]}$. First we show that the critical Hölder regularity of a typical trajectory of ${Z(t)}_{t\in[0,1]}$ is equal to 1/2. Next we provide for such a trajectory an expression as a random series which converges at a geometric rate in any Hölder space of an arbitrary order $γ<1/2$; this expression is obtained through the expansion of the random function $s\mapsto Φ(X(s))$ on the Haar basis. Finally, thanks to it, we give an efficient iterative simulation method for ${Z(t)}_{t\in[0,1]}$.

math.PR↗