Paper examines floating exercise boundaries for American options in time-inhomogeneous models.
problem Floating exercise boundaries in time-inhomogeneous models with negative interest rates or yields.
method Semi-analytical approach for pricing American options.
result Specialized pricing methodologies are required for models with floating exercise boundaries.
Closed-form solution found for American put option boundary.
problem Finding the optimal exercise boundary for American put options.
method Three models of stock price dynamics with time-dependent parameters, leading to a closed-form solution for the exercise boundary.
result Explicit closed-form solution for the optimal exercise boundary of American put options.
This paper evaluates how XVA affects option exercise decisions.
problem Ignoring XVA in exercise decisions can lead to incorrect valuation.
method Regression techniques to evaluate XVA's impact on exercise decisions.
result XVA significantly impacts the exercise decision at the boundary.
Optimal exercise boundary for put options with delivery lags identified.
problem Analyzing the optimal exercise time for American put options with delivery lags.
method Decomposing the option into a European put and a new American-style derivative, using free boundary techniques.
result The optimal exercise boundary exists and is a strictly increasing and smooth curve.
Pricing Bermudan swaptions with few exercise dates using analytic methods.
problem Pricing Bermudan swaptions with few exercise dates
method Analytic decomposition and backward induction under rolling forward measures
result Pricing formulas with decomposition and boundary linearity
The paper extends VIX option pricing models and finds optimal exercise boundaries.
problem Pricing American VIX options under generalized volatility models.
method Extended 3/2 and 1/2 models, used free-boundary approach and local time-space calculus.
result Found optimal exercise boundaries for VIX factor process.
The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility depending on the option price. We review a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. R…
Paper defines when early exercise of American options is optimal under negative rates.
problem Determining optimal exercise times for American options with negative interest rates.
method Developed a new integral equation to price options and find exercise boundaries under negative rates, using modified fixed point method.
result Successfully developed and validated a new algorithm for pricing American options under negative rates.
In this paper we generalize and analyze the model for pricing American-style Asian options due to (Hansen and Jorgensen 2000) by including a continuous dividend rate q and a general method of averaging of the floating strike. We focus on the qualitative and quantitative analysis of the early exercise boundary. The fi…
The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear vola…
The paper analyzes American options with time-varying caps, finding complex exercise regions and deriving option pricing formulas.
problem Valuation of American capped call options with time-varying caps, especially when the cap grows or decreases over time.
method Probabilistic arguments and local time, characterizing exercise boundaries through recursive integral equations and piecewise constant segments.
result General representation formulas for option prices, derived from exercise boundaries and local time of the underlying process.
Deep learning method solves American options with free boundary using Landau transformation.
problem Solving American options with a free boundary using deep learning.
method Landau transformation, dual solution framework, auxiliary function, feed forward deep neural network (DNN).
result Deep learning method efficiently prices options with early exercise features.
Researchers find the optimal exercise time for American options using a specific type of diffusion process.
problem Finding the optimal time to exercise American options with a time-dependent Ornstein-Uhlenbeck process.
method Optimal stopping problem, probabilistic arguments, non-linear Volterra-type integral equation, Picard iteration algorithm.
result They derive a non-linear Volterra-type integral equation and prove the exercise boundary's Lipschitz continuity and differentiability almost everywhere.
The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the…
In this paper, we present a new method for calculating the limit of early exercise boundary at expiry. We price American style of general derivative using a formula expressed as a sum of the value of European style of derivative and so called American premium. We use the latter expression to calculate an analytic formu…
In this paper we present qualitative and quantitative comparison of various analytical and numerical approximation methods for calculating a position of the early exercise boundary of the American put option paying zero dividends. First we analyze their asymptotic behavior close to expiration. In the second part of the…
Study pricing of American put options with stochastic interest rate and finite maturity.
problem Pricing American put options with stochastic interest rate and finite maturity.
method Applied stochastic calculus and Ito's lemma to derive the option value's formula and optimal exercise boundary.
result Existence and parametrisation of the optimal exercise boundary for the Vasicek model.
Investors often miss out on early exercise of American options with dividends, volatility, and jumps.
problem Investors suboptimal exercise of American call options on dividend-paying stocks.
method Used a fast numerical technique to analyze a large database of investor decisions and incorporated stochastic volatility and jumps in pricing models.
result Pricing models with stochastic volatility and jumps reduce the loss from suboptimal exercise by a quarter.
We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time td during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics and the dividend amount is a deterministic fu…
Study perpetual put options using nonlinear Black-Scholes equations.
problem Analyzing early exercise boundaries for perpetual put options.
method Transformed into a nonlinear stationary Black-Scholes equation and solved numerically.
result Numerical results of early exercise boundary, option price and their parameters.
We derive explicit formulas for time decay, for the European call and put options at expiry, and use them to calculate analytical approximations to the price of the American put and early exercise boundary near expiry. We show that for many families of non-Gaussian processes used in empirical studies of financial marke…
Study on pricing American Exchange options using Lévy processes.
problem Pricing American Exchange options driven by Lévy processes.
method Represented American Exchange options as European options plus early exercise premium; studied properties of free boundary and provided an approximative formula.
result Developed an approximative formula for American Exchange options.
Paper improves American option valuation in complex models.
problem Valuation of American options in time-dependent jump-diffusion models.
method Integral equations and characteristic functions for explicit exercise boundary determination.
result Efficient and accurate pricing method for American options in various models.
Optimal exercise timing of stock options analyzed with varying information on drift change.
problem Analyzing optimal exercise timing of stock options with varying information on drift change.
method Rigorous mathematical analysis and numerical methods to solve optimal stopping problems.
result Characterization of optimal exercise boundaries and smooth pasting properties in both information scenarios.
A new method solves American put options with high accuracy and speed.
problem Solving American put options with high accuracy and speed.
method Adaptive fourth-order Runge-Kutta-Fehlberg method coupled with a fourth-order compact scheme.
result The method provides a more accurate solution and better performance in terms of computational speed.
This paper studies the parabolic free boundary problem arising from pricing American-style put options on an asset whose index follows a geometric Brownian motion process. The contribution is to propose a condition for that the early exercise boundary is a convex function.
In this paper we analyze American style of floating strike Asian call options belonging to the class of financial derivatives whose payoff diagram depends not only on the underlying asset price but also on the path average of underlying asset prices over some predetermined time interval. The mathematical model for the …
New method solves complex financial option pricing with varying time steps.
problem Pricing American options with varying time steps and regime switching.
method Explicit Runge-Kutta-Fehlberg scheme with fourth-order compact finite difference in space and high order analytical approximation.
result The method provides better performance in terms of computational speed and accuracy.
The paper analyzes Variable Annuities with surrender charges, providing a pricing formula and optimal exercise boundary.
problem Analyzing Variable Annuities with surrender charges and early termination rights.
method Formulated as an optimal stopping problem with a discontinuous payoff, non-monotonic optimal stopping boundaries are proven continuous and regular.
result A rigorous pricing formula and optimal exercise boundary for surrender options are derived.
This paper analyzes model risk in American put options using Heston volatility model.
problem Model risk in optimal exercise of American put options.
method Benchmark methodology of Hull and Suo [2002], Heston stochastic volatility model, numerical finite difference methods.
result Optimal exercise behavior is influenced by stochastic volatility dynamics and return-volatility correlation, creating model risk.
New method uses Hermite polynomials for American option valuation.
problem Valuation of American options with complex jump-diffusion dynamics.
method Hermite polynomial expansions of transition density and early exercise premium.
result Converging approximations to true option prices and exercise boundaries.
Two sweeps of the Brennan-Schwartz algorithm solve American options under negative rates.
problem Inability of the Brennan-Schwartz algorithm to solve American options under negative interest rates.
method Two sweeps of the Brennan-Schwartz algorithm in two directions.
result Recovery of the exact solution for American options under negative rates.
In practical work with American put options, it is important to be able to know when to exercise the option, and when not to do so. In computer simulation based on the standard theory of geometric Brownian motion for simulating stock price movements, this problem is fairly easy to handle for options with a short lifesp…
A fast, accurate method for pricing American options with free boundaries.
problem Pricing American options with free boundaries efficiently and accurately.
method A sixth-order compact finite difference scheme with a dynamic staggered boundary scheme and 3(2) R-K Bogacki-Shampine time stepping.
result An efficient sixth-order compact scheme for pricing American options with free boundaries.
A new method for pricing exchange options under stochastic volatility and jumps.
problem Pricing European and American exchange options with stochastic volatility and jumps.
method Equivalent martingale measure, numeraire choice, integral transforms, Kolmogorov backward equation, integral equations.
result Reduced exchange option pricing to a one-dimensional problem of a call option.
In this paper, we price American-style Parisian down-and-in call options under the Black-Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exer…
This paper uses deep learning to price American options under stochastic volatility.
problem Pricing American options with a time-varying exercise boundary under the Heston model.
method Coupled PINNs with curriculum learning and adaptive resampling.
result Demonstrates the effectiveness of the proposed deep learning framework for American option pricing.
Improved fourth-order compact scheme for option valuation with Robin boundary condition.
problem Lower convergence rates in numerical methods for American options.
method High-order compact scheme, Robin boundary condition, coupled nonlinear PDEs.
result Fourth-order convergence rate achieved without mesh refinement.
American options are financial instruments that can be exercised at any time before expiration. In this paper we study the problem of pricing this kind of derivatives within a framework in which some of the properties --volatility and dividend policy-- of the underlaying stock can change at a random instant of time, bu…
We use probabilistic methods to characterise time dependent optimal stopping boundaries in a problem of multiple optimal stopping on a finite time horizon. Motivated by financial applications we consider a payoff of immediate stopping of "put" type and the underlying dynamics follows a geometric Brownian motion. The op…
Model detects exercise fatigue with high accuracy.
problem Detecting exercise fatigue in real-time.
method Feature extraction using AHP, machine learning.
result 98.65% accuracy in detecting exercise fatigue.
This work proposes a method to price American basket options using a Markovian projection.
problem Pricing American basket options in high dimensions is computationally expensive.
method Use a stopping rule based on a low-dimensional Markovian projection of the basket's dynamics.
result Approximate the optimal early-exercise boundary in a lower-dimensional space, providing bounds for the option price.
A new method for pricing American options using exercise rate optimization.
problem Pricing American options efficiently and accurately.
method Monte Carlo simulation and optimization of exercise rates.
result The method provides the correct option price and is efficient for various models.
Continuous-time optimal stopping solved with deep reinforcement learning
problem Optimal stopping problems in continuous time
method CARLOS (Continuous-time Adaptive Reinforcement Learning for Optimal Stopping)
result Higher prices than existing Bermudan solvers, approaching American upper bound
New method for pricing SOFR futures options, solving both American and Asian exercise styles.
problem Lack of pricing models for SOFR futures options post-LIBOR transition.
method Developed a new version of the GIT method to solve semi-analytically.
result Obtained option prices, exercise boundaries, and Greeks for American and Asian options.
Numerical method for pricing exchange options with stochastic volatility and jumps.
problem Pricing exchange options under stochastic volatility and jump-diffusion dynamics.
method Method of lines (MOL) approach to simplify and solve the PDEs.
result Characterization of near-maturity American exchange option boundary and impact of model parameters.
Pen-and-paper exercises cover various machine learning topics.
problem None explicitly stated, focuses on learning through exercises.
method Pen-and-paper exercises on machine learning topics.
result Comprehensive coverage of machine learning concepts through exercises.
Game options study gradual exercise and cancellation with transaction costs.
problem Analyzing game options with gradual exercise and cancellation under proportional transaction costs.
method Developed algorithmic constructions for bid and ask prices, superhedging strategies, and optimal mixed stopping times.
result Increased flexibility in hedging leads to tighter bounds on option price.