布朗运动和泊松过程共同驱动下的欧式期权定价

针对布朗运动和泊松过程共同驱动下股票价格的随机微分方程,利用It0公式和随机积分的方法,得到了该形式下欧式期权定价的模型,并给出了模型的求解.     

On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing

This paper considers the estimation of an unknown function h that can be characterized as a solution to a nonlinear operator equation mapping between two infinite dimensional Hilbert spaces. The nonlinear operator is unknown but can be consistently estimated, and its inverse is discontinuous, rendering the problem ill-posed. We establish the consistency for the class of estimators that are regularized using general lower semicompact penalty functions. We derive the optimal convergence rates of the estimators under the Hilbert scale norms. We apply our results to two important problems in economics and finance: (1) estimating the parameters of the pricing kernel of defaultable bonds; (2) recovering the volatility surface implied by option prices allowing for measurement error in the option prices and numerical error in the computation of the operator. The first anther was supported by US National Science Foundation (Grant No. SES-0631613) and the Cowles Foundation for Research in Economics     

A non random walk theory of exchange rate dynamics with applications to option pricing

A two dimensional stochastic process is developed to model exchange rate dynamics. We incorporate the non random walk influence of pur–chasing power parity, to synthesise the theories of international trade and foreign currency options. Our results, which include a closed form expression for the transition density function of the exchange rate and an exact formula to price currency options, offer a theoretical framework for further study of foreign exchange markets     

Asymptotic growth of trajectories of multifractional Brownian motion,with statistical applications to drift parameter estimation

We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein–Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian process and some functionals of it, in terms of the covariance function of its increments.     

分数布朗运动驱动下带比例交易成本的期权定价

在标的资产价格服从几何分数布朗运动模型条件下,利用分数布朗运动随机分析理论和偏微分方程方法,建立了几何分数布朗运动驱动下的金融市场模型,讨论了带比例交易成本的欧式期权,并且得到了相应的期权定价公式.     

The maximal and minimal ranks of a quaternion matrix expression with applications

In this paper, we establish the formulas of the extermal ranks of the quaternion matrix expression f(X₁, X₂) = C₇ ? A₄X₁B₄ ? A₅X₂B₅ where X₁, X₂ are variant quaternion matrices subject to quaternion matrix equations A₁X₁ = C₁, A₂X₁ = C₂, A₃X₁ = C₃, X₂B₁ = C₄, X₂B₂ = C₅, X₂B₃ = C₆. As applications, we give a new necessary and sufficient condition for the existence of solutions to some systems of quaternion matrix equations. Some results can be viewed as special cases of the results of this paper.     

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X$X$ in a bounded κ$\kappa$-fat open set; if u$u$ is a positive harmonic function with respect to X$X$ in a bounded κ$\kappa$-fat open set D$D$ and h$h$ is a positive harmonic function in D$D$ vanishing on D^c$D^c$, then the non-tangential limit of u/h$u/h$ exists almost everywhere with respect to the Martin-representing measure of h$h$.     

An extension of the Euler Laplace transform inversion algorithm with applications in option pricing

We show that the Euler algorithm for Laplace transform inversion can be extended to functions defined on the entire real line, if they have specific decay features. Our objective is to apply the method to option pricing problems, specifically when inverting Laplace transforms of the option price in the logarithm of the strike.     

Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values

We construct simultaneous rational approximations to q-series L₁(x₁; q) and L₁(x₂; q) and, if x = x₁ = x₂, to series L₁(x; q) and L₂(x; q), where

Fractional Brownian motion with complex variance via random walk in the complex plane and applications

A model of complex-valued fractional Brownian motion has been built up recently as the limit of a random walk in the complex plane, but this model involves radial steps only. It is shown that, by using non-radial steps, this model can be easily extended to define a fractional Brownian motion with complex-valued variance. The relations between complex-valued Brownian motion and the heat equation of order n is clarified and mainly one obtains the general expression of the probability density functions for these processes. One shows that the maximum entropy principle (MPE) provides the probability density of the complex-valued fractional Brownian motion, exactly like for the standard Brownian motion. And lastly, one shows that the heat equation of order 2n (which is the Fokker–Planck equation (FPE) of the complex-valued Brownian motion) has a solution which is similar to that of the FPE of fractional order introduced before by the author, therefore, to some extent, an identification between the complex-valued model via random walk in the complex plane and the model involving a derivative of fractional order.     

Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.     

Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift — Their characteristics and applications

In ?ochowski (2008) [9] we defined truncated variation of Brownian motion with drift, W_t=B_t+μt,t≥0, where (B_t) is a standard Brownian motion. Truncated variation differs from regular variation in neglecting jumps smaller than some fixed c>0. We prove that truncated variation is a random variable with finite moment-generating function for any complex argument.We also define two closely related quantities — upward truncated variation and downward truncated variation.The defined quantities may have interpretations in financial mathematics. The exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process.We calculate the Laplace transform with respect to the time parameter of the moment-generating functions of the upward and downward truncated variations.As an application of the formula obtained we give an exact formula for the expected values of upward and downward truncated variations. We also give exact (up to universal constants) estimates of the expected values of the quantities mentioned.     

Brownian fluctuations in space-time with applications to vibrations of rods

In this paper Brownian fluctuations in space-time are considered. Time is assumed to run alternately forward and backward, the alternance being marked by a Poisson process with rate λ. It is shown that the law of this motion is a solution of a fourth-order partial differential equation. Furthermore the law of this movement in the presence of an absorbing barrier is derived. The equation ruling the movement analysed, when λ = 0 and is submitted to the change t' = −it, reduces to the equation of vibrations of rods. This fact is exploited to obtain the solution of boundary value problems concerning the equation of vibrating beams by means of Brownian motion techniques.     

A system of non-local parabolic PDE and application to option pricing

This article includes a proof of well posedness of an initial-boundary value problem involving a system of non-local parabolic partial differential equation (PDE), which naturally arises in the study of derivative pricing in a generalized market model, which is known as a semi-Markov modulated geometric Brownian motion (GBM) model We study the well posedness of the problem via a Volterra integral equation of second kind. A probabilistic approach, in particular the method of conditioning on stopping times is used for showing the uniqueness.     

A fast numerical approach to option pricing with stochastic interest rate,stochastic volatility and double jumps

This study proposes a pricing model through allowing for stochastic interest rate and stochastic volatility in the double exponential jump-diffusion setting. The characteristic function of the proposed model is then derived. Fast numerical solutions for European call and put options pricing based on characteristic function and fast Fourier transform (FFT) technique are developed. Simulations show that our numerical technique is accurate, fast and easy to implement, the proposed model is suitable for modeling long-time real-market changes. The model and the proposed option pricing method are useful for empirical analysis of asset returns and risk management in firms.