Paper explores asset pricing dynamics in Bachelier model.
problem Understanding risky asset price dynamics in Bachelier model.
method Analyzes Bachelier market model to represent risky asset price dynamics.
result Defines riskless assets within the Bachelier model.
Study reviews Bachelier model for negative oil prices post-COVID.
problem Negative oil futures prices due to COVID-19.
method Literature review and model comparison.
result Connects Black-Scholes and Bachelier models for option pricing.
We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe -- theoretically as well as for Bachelier's original data -- that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we ex…
Integrates ESG factors into Bachelier's model for asset pricing.
problem Incorporating ESG factors into classical finance models.
method Defines ESG price process and integrates into Bachelier's model.
result Enables option pricing valuation with ESG factors.
This paper examines Bachelier implied volatility at extreme strikes.
problem Investigates appropriate implied volatility extrapolation at extreme strikes.
method Compares Bachelier and Black-Scholes models, focusing on normal distribution and vanilla options.
result Bachelier implied variance grows at most linearly in log-moneyness, similar to Black-Scholes.
Matrix approximation method for Bachelier option pricing and Greeks under stochastic volatility models
problem Computing option prices and Greeks for stochastic volatility models
method Matrix approximation using elementary linear algebra
result Option prices and Greeks computed for infinitely many strikes with a finite number of expectations
Study examines implied volatility behavior in Bachelier model.
problem Characterizing implied volatility in Bachelier model for large strikes.
method Exploiting regular variation theory, derived explicit expressions for Bachelier implied volatility.
result Established a rigorous connection between characteristic function analyticity and volatility smile asymptotic slope.
Unified model integrates Bachelier and Black-Scholes-Merton for asset pricing.
problem Study of asset pricing in a natural world with negative prices or riskless rates.
method Unified framework combining Bachelier and Black-Scholes-Merton models.
result Unified model shows different option pricing depending on riskless instruments used.
Survey revisits Bachelier and Dupire, highlighting optimal transport's role.
problem Finding arbitrage-free models calibrated to volatility surfaces.
method Revisits mathematical finance principles, uses optimal transport results.
result Optimal transport provides rigorous foundations for Dupire's model.
Study scaling limits of utility indifference prices in discretized Bachelier model.
problem Analyzing utility indifference prices for path-dependent European options in a discretized Bachelier model.
method Purely probabilistic approach, including duality argument, optimal drift control problem, martingale techniques, and strong invariance principles.
result Obtained a scaling limit for utility indifference prices as the number of trading times increases.
Study utility indifference pricing in a Bachelier model with small linear price impact.
problem Utility indifference pricing in a model with linear price impact.
method Analyzes the Bachelier model with exponential utility indifference prices for vanilla European options.
result Computes the scaling limit of utility indifference prices for a vanishing price impact inversely proportional to risk aversion.
The paper calculates Bachelier option prices using Taylor expansions and applies it as a variance reduction technique.
problem Calculating Bachelier option prices and variance reduction in correlated cases.
method Taylor expansions and classical Itô calculus to derive option prices, uses negative powers of future mean volatility.
result The paper provides a new method to calculate Bachelier option prices and applies it to reduce variance in Monte Carlo simulations.
The paper analyzes implied volatility for European and Asian options under stochastic volatility Bachelier model.
problem Analyzing implied volatility for European and Asian options under stochastic volatility.
method Using Malliavin calculus and anticipating Ito's formula, the paper computes and finds asymptotic formulas for implied volatility and skew.
result The paper provides a short maturity asymptotic formula for the skew of implied volatility that depends on the roughness of the volatility model.
A new Bachelier model explains oil option volatility during the pandemic.
problem Describing and predicting the volatility surface of oil options during the pandemic.
method Additive Bachelier model with three parameters: volatility term structure, vol-of-vol, and skew.
result The model accurately describes the volatility surface and supports efficient pricing of exotic options.
Study utility indifference pricing with delayed investment information in a Bachelier model.
problem Investment decisions based on delayed information in a Bachelier model.
method Developed discrete-time duality and used techniques from [7] to compute scaling limits.
result Utility indifference prices scaling limit for vanishing delay with quadratic penalty.
Extends BBSM model to incorporate ESG ratings and path dynamics.
problem Price stock options considering historical market index dynamics and ESG ratings.
method Develops discrete, binary tree option pricing model under BBSM with ESG valuation.
result Model accurately fits stock price changes and European call option prices.
Monotonicity of normalized implied-volatility coordinates under no-arbitrage
problem Monotonicity of normalized implied-volatility coordinates under no-arbitrage
method Elementary discrete no-arbitrage proof
result Monotonicity principle extended to Bachelier implied volatility
Two new rational formulae for normal implied volatility are presented.
problem Calculating normal implied volatility using iterative methods.
method Two explicit rational formulae that avoid iteration and logarithms.
result Accurate and fast formulae for normal implied volatility.
This paper surveys options pricing under arithmetic Brownian motion and derives formulas for various types of options.
problem The use of arithmetic Brownian motion in finance is not widely adopted.
method Risk-neutral valuation and derivation of formulas for European options under three types of underlying assets.
result Derivation of formulas for European options and partial differential equations 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 cal…
Study on short-term behavior of ATM-IV for jump-diffusion model.
problem Analyzing the short-time behavior of ATM-IV for a specific stochastic volatility model.
method Used Malliavin Calculus techniques to derive expressions for ATM-IV level and skew.
result Short-time behavior of ATM-IV level is consistent for all pure-jump Lévy processes.
In this paper we compare two classical one-factor diffusion models which are used to model the term structure of interest rates. One of them is based on the Wiener-Bachelier process while the second one is based on the Ornstein-Uhlenbeck process. We show essential differences between the prices of European call options…
We study optimal trading in an Almgren-Chriss model with running and terminal inventory costs and general predictive signals about price changes. As a special case, this allows to treat optimal liquidation in "target zone models": asset prices with a reflecting boundary enforced by regulatory interventions. In this cas…
Innovative extensions to option pricing models using asymmetric Brownian motion and random walk approaches.
problem Capturing empirical phenomena like return skewness, heavy tails, and volatility asymmetry in option pricing models.
method Developing the Geometric Asymmetric Brownian Motion (GABM) within the Bachelier--Black--Scholes--Merton framework.
result Deriving closed-form option pricing formulas and a discrete-time binomial tree algorithm that converges to the GABM limit.
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.
Scaling properties of the BUX index are similar to those observed in other parts of the world. The main difference is that the traditional quantities like volatility, growth and autocorrelation of returns follows more closely the assumptions of the traditional stock market theory developed by Bachelier and by Black and…
Model calculates optimal trading time for derivatives orders.
problem Balancing execution costs and market risks in large order execution.
method Time Is Money model using Bachelier model and central limit order book.
result Demonstrates a continuous-time Arrival Price framework.
The paper solves a utility-based hedging problem with quadratic costs.
problem Optimal trading strategy for hedging European contingent claims with quadratic transaction costs.
method Duality theory applied to exponential utility maximization problem.
result Explicit computation of optimal trading strategy for quadratic payoffs.
We study the class of Azéma-Yor processes defined from a general semimartingale with a continuous running maximum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the drawdown equation. Solutions of the latter have the drawdown…
We provide approximations for VIX futures and options in forward variance models.
problem Modeling VIX futures and options in forward variance models.
method Weak approximations and explicit formula derivation for VIX futures and options.
result Explicit combinations of Black-Scholes prices and greeks for option price approximations.
Modeling stock price fluctuations using Brownian motion and stochastic differential equations.
problem Capturing the stochastic behavior of stock prices.
method Developed a stochastic differential equation to model stock price fluctuations, incorporating Itô integration.
result Backtesting showed a strong correlation coefficient between the model and actual stock price movements.
Study optimal liquidation under high risk aversion and small price impact.
problem Optimal liquidation of options under high risk aversion and linear price impact.
method Analyzes Bachelier model with linear price impact, computes utility indifference prices, and finds asymptotically optimal portfolios.
result Establishes a scaling limit for vanishing price impact and computes corresponding utility indifference prices.
The paper extends a variance gamma model to quadratic functions, reducing arbitrage and computational costs.
problem Creating an arbitrage-free interpolation for option pricing models.
method Generalizing the local variance gamma model to a piecewise quadratic local variance function.
result The quadratic model results in an arbitrage-free interpolation of class C3, reducing knots and computational cost.
The paper studies scaling limits of hedging prices in financial models.
problem Scaling limits of exponential utility indifference prices in financial models.
method Formulated dual problem as stochastic control, solved HJB equation for upper bound, used duality result for lower bound.
result Represented scaling limit in terms of specific relative entropy and constructed asymptotic optimal hedging strategies.
Using agent-based modelling, empirical evidence and physical ideas, such as the energy function and the fact that the phase space must have twice the dimension of the configuration space, we argue that the stochastic differential equations which describe the motion of financial prices with respect to real world probabi…
Paper provides an explicit formula for local volatility in Cheyette models.
problem Approximating local volatility in Cheyette interest rate models.
method Extended Dupire framework, perturbation methods, probabilistic techniques.
result Explicit analytical formula for local volatility in Cheyette models.
The thesis examines stochastic calculus in option pricing with logistic models and numerical methods.
problem Exploring the application of stochastic calculus in option pricing.
method Monte-Carlo Simulation and machine learning algorithms.
result Insights from Peter Carr and Lorenzo Torricelli's convex duality in continuous models.
Modeling the evolution of a financial index as a stochastic process is a problem awaiting a full, satisfactory solution since it was first formulated by Bachelier in 1900. Here it is shown that the scaling with time of the return probability density function sampled from the historical series suggests a successful mode…
Investor optimizes investment timing with future knowledge, overcoming transaction costs.
problem Optimal investment timing with future peeking, constrained by transaction costs.
method Solves control problem with infinite-dimensional memory using Gaussian Volterra integral equations.
result Explicit solution to optimal investment problem in Bachelier setting.
Constructs supermartingale couplings with full marginals constraints.
problem Optimal transport for supermartingale couplings with multiple marginals.
method Markovian iteration of one-period optimal supermartingale couplings.
result Explicit construction of supermartingale processes solving optimal transport problem.
This study compares deep generative models to traditional methods for generating financial time series.
problem Generating realistic multivariate financial time series for risk management and portfolio optimization.
method Systematic comparison of deep generative models (DGMs) against state-of-the-art parametric models on synthetic and empirical data.
result Deep generative models outperform traditional parametric models in generating financial time series.
The financial market is nonpredictable, as according to the Bachelier, the mathematical expectation of the speculator is zero. Nevertheless, we observe in the price fluctuations the two distinct scales, short and long time. Behaviour of a market in long terms, such as year intervals, is different from that in short ter…
This paper demonstrates the efficiency of using Edgeworth and Gram-Charlier expansions in the calibration of the Libor Market Model with Stochastic Volatility and Displaced Diffusion (DD-SV-LMM). Our approach brings together two research areas; first, the results regarding the SV-LMM since the work of Wu and Zhang (200…
The price of financial assets are, since Bachelier, considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works. In this letter we investigate the question of sca…
The paper introduces a new price model based on entropy that better fits high-frequency market data.
problem Understanding fair prices in high-frequency markets with bid-ask imbalance.
method A parametrized family of prices derived from the Maximum Entropy Principle, minimizing bias given volume imbalance.
result The model can generate higher kurtosis and heavy-tailed distributions compared to standard models.
We consider the problem of hedging a European contingent claim in a Bachelier model with transient price impact as proposed by Almgren and Chriss. Following the approach of Rogers and Singh and Naujokat and Westray, the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strat…
The paper models Gasoil options using Brent benchmarks, improving volatility estimation.
problem Inability to directly model illiquid Gasoil options market.
method Jointly models Brent and Gasoil futures prices with a correlated Bachelier model, estimating volatility spread.
result The proposed framework accurately maps Brent implied volatilities to Gasoil implied volatilities.
Implied volatilities form a well-known structure of smile or surface which accommodates the Bachelier model and observed market prices of interest rate options. For the swaptions that we study, three parameters are taken into account for indexing the implied volatilities and form a "volatility cube": strike (or moneyne…