Amel Bentata, Marc Yor
These notes are the second half of the contents of the course given by the second author at the Bachelier Seminar (8-15-22 February 2008) at IHP. They also correspond to topics studied by the first author for her Ph.D.thesis.
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These notes are the first half of the contents of the course given by the second author at the Bachelier Seminar (February 8-15-22 2008) at IHP. They also correspond to topics studied by the first author for her Ph.D.thesis.
In this paper we study dynamic backward problems, with the computation of conditional expectations as a main objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of Dupire \…
We show that, under certain smoothness conditions, a Brownian martingale at a fixed time can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times the Malliavin derivative. This result can also be seen as a generalization of the semi-group theory of parabolic partial differential equations to the parabolic path-dependent partial differential equations introduced by Dupire (2009) and Cont and Founié (2011). The exponential operator can be calculated explicitly in a series expansion, which resembles the…
When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, Black-Scholes PDE associated with an European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among of them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing …
Multi-Dimensional Martingales from Mutual Information
本文提出了一种基于martingale熵最优传输的多维martingale构造方法，并展示了局部相关性模型在外汇市场中的局限性。
SDEs with uniform distributions: Peacocks, Conic martingales and ergodic uniform diffusions
本文研究均匀分布peacock过程的扩散鞅构造问题，通过逆Fokker-Planck方程推导对应SDE，证明锥条件下的强解唯一性，给出均值回复均匀扩散与遍历性结果，可用于金融相关量建模。
A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we derive existence and uniqueness results for the Black--Scholes equation, and we provide convexity theory for option pricing and derive related ordering results with respect to volatility. We show that American options are convexity preserving, whereas European options preserve concavity for general …
Generalized delayed Black and Scholes Formula
本文针对漂移项含一般延迟的线性微分方程描述的延迟标的股票价格，用基于Girsanov性质的等价鞅测度法推导欧式期权价格显式公式，扩展部分无套利完备模型，指出另一模型因潜在套利无法给出唯一期权价格。
We study the risk premium impact in the Perturbative Black Scholes model. The Perturbative Black Scholes model, developed by Scotti, is a subjective volatility model based on the classical Black Scholes one, where the volatility used by the trader is an estimation of the market one and contains measurement errors. In this article we analyze the correction to the pricing formulas due to the presence of an underlying drift different from the risk free return. We prove that, under some hypothesis on the parameters, if the asset price is a sub-martingale under historical probability, then the impl…
We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf \cite{KR} as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict loca…
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.
Stochastic Knothe-Rosenblatt: Light-speed Calibration of Stochastic Local Volatility Models
本文提出一种基于随机Knothe-Rosenblatt的模块化校准方法，通过递归构造鞅精确匹配边际分布并接近参考动力学，适用于随机和路径依赖波动率模型，支持高效Sinkhorn算法及多资产扩展。
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 exi…
We derive the explicit price of the perpetual American put option cancelled at the last passage time of the underlying above some fixed level. We assume the asset process is governed by a geometric spectrally negative Lévy process. We show that the optimal exercise time is the first epoch when asset price process drops below an optimal threshold. We perform numerical analysis as well considering classical Black-Scholes models and the model where logarithm of the asset price has additional exponential downward shocks. The proof is based on some martingale arguments and fluctuation theory of Lév…
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