Approximating Functional of Local Martingale Under the Lack of Uniqueness of Black-Scholes PDE

Linear vs. Nonlinear Diffusion and Martingale Option Pricing

First, classes of Markov processes that scale exactly with a Hurst exponent H are derived in closed form. A special case of one class is the Tsallis density, advertised elsewhere as nonlinear diffusion or diffusion with nonlinear feedback. But the Tsallis model is only one of a very large class of linear diffusion with a student-t like density. Second, we show by stochastic calculus that our generalization of the Black-Scholes partial differential equation (pde) for variable diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, this was pr…

Option Pricing for Symmetric Lévy Returns with Applications

This paper considers options pricing when the assumption of normality is replaced with that of the symmetry of the underlying distribution. Such a market affords many equivalent martingale measures (EMM). However we argue (as in the discrete-time setting of Klebaner and Landsman, 2007) that an EMM that keeps distributions within the same family is a "natural" choice. We obtain Black-Scholes type option pricing formulae for symmetric Variance-Gamma and symmetric Normal Inverse Gaussian models.

Pricing the European call option in the model with stochastic volatility driven by Ornstein--Uhlenbeck process. Exact formulas

We consider the Black--Scholes model of financial market modified to capture the stochastic nature of volatility observed at real financial markets. For volatility driven by the Ornstein--Uhlenbeck process, we establish the existence of equivalent martingale measure in the market model. The option is priced with respect to the minimal martingale measure for the case of uncorrelated processes of volatility and asset price, and an analytic expression for the price of European call option is derived. We use the inverse Fourier transform of a characteristic function and the Gaussian property of th…

A Theory of Non_Gaussian Option Pricing

Option pricing formulas are derived from a non-Gaussian model of stock returns. Fluctuations are assumed to evolve according to a nonlinear Fokker-Planck equation which maximizes the Tsallis nonextensive entropy of index $q$. A generalized form of the Black-Scholes differential equation is found, and we derive a martingale measure which leads to closed form solutions for European call options. The standard Black-Scholes pricing equations are recovered as a special case ($q = 1$). The distribution of stock returns is well-modelled with $q$ circa 1.5. Using that value of $q$ in the option pricin…

基于埃尔米特基函数的期权价格非参数估计

Nonparametric estimates of option prices via Hermite basis functions

本文提出用重标度埃尔米特多项式线性组合近似对数收益率密度，构造Black-Scholes模型的非参数扰动以定价欧式期权，并通过样本外误差检验表明该方法仅在执行价格接近观测样本时表现较好。

A decomposition formula for fractional Heston jump diffusion models

We present an option pricing formula for European options in a stochastic volatility model. In particular, the volatility process is defined using a fractional integral of a diffusion process and both the stock price and the volatility processes have jumps in order to capture the market effect known as leverage effect. We show how to compute a martingale representation for the volatility process. Finally, using Itô calculus for processes with discontinuous trajectories, we develop a first order approximation formula for option prices. There are two main advantages in the usage of such approxim…

Quantized Interest Rate at the Money for American Options

In this work, we expand the idea of Samuelson[3] and Shepp[2,5,6] for stock optimization using the Bachelier model [4] as our models for the stock price at the money (X[stock price]= K[strike price]) for the American call and put options [1]. At the money (X= K) for American options, the expected payoff of both the call and put options is zero. Shepp investigated several stochastic optimization problems using martingale and stopping time theories [2,5,6]. One of the problems he investigated was how to optimize the stock price using both the Black-Scholes (multiplicative) and the Bachelier (add…

跳跃扩散期权定价的熵动力学

Entropic Dynamics of Jump-Diffusion Option Pricing

研究基于熵推理框架从信息约束推导股票价格动态和期权估值过程，核心是利用市场对回报的偏好确定对数价格为变量，通过连续与跳跃通道表示价格变化，推导出相关方程及结果，不同模型因信息不同而有差异。

Equal Risk Pricing and Hedging of Financial Derivatives with Convex Risk Measures

In this paper, we consider the problem of equal risk pricing and hedging in which the fair price of an option is the price that exposes both sides of the contract to the same level of risk. Focusing for the first time on the context where risk is measured according to convex risk measures, we establish that the problem reduces to solving independently the writer and the buyer's hedging problem with zero initial capital. By further imposing that the risk measures decompose in a way that satisfies a Markovian property, we provide dynamic programming equations that can be used to solve the hedgin…