American-type Options Research Articles

Optimal stopping with signatures

We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process X. We consider classic and randomized stopping times represented by linear and nonlinear functionals of the rough path signature X<∞ associated to X, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature E[X0,T≤N]. By applying a deep neural network approach to approximate the nonlinear signature functionals, we can efficiently solve the optimal stopping problem numerically. The only assumption on the process X is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semimartingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, for example, on financial or electricity markets.

Variance reduction for risk measures with importance sampling in nested simulation

Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are two standard risk measures that are widely adopted in both financial and insurance industries. Simulation-based approaches including nested simulation and least-squares Monte Carlo are effective strategies to yield reliable estimates of these risk measures, but there remain open questions on how importance sampling can be incorporated to improve estimation efficiency. In this paper, we extend the scope of importance sampling from simple Monte Carlo to nested simulation settings and its adaptations for American-type options; we also establish the asymptotic consistency of importance sampling. Numerical results consistent with our theoretical analysis are provided to verify its effectiveness.

A Neural Network Approach to Value R&D Compound American Exchange Option

In this paper we show as the neural network methodology, coupled with the Least Squares Monte Carlo approach, can be very helpful in valuing R&D investment opportunities. As it is well known, R&D projects are made in a phased manner, with the commencement of subsequent phase being dependent on the successful completion of the preceding phase. This is known as a sequential investment and therefore R&D projects can be considered as compound options. In addition, R&D investments often involve considerable cost uncertainty so that they can be viewed as an exchange option, i.e. a swap of an uncertain investment cost for an uncertain gross project value. Finally, the production investment can be realized at any time before the maturity date, after that the effects of R&D disappear. Consequently, an R&D project can be considered as a compound American exchange option. In this context, the Least Squares Monte Carlo method is a powerful and flexible tool for capital budgeting decisions and for valuing American-type options. But, using the simulated values as “targets”, the implementation of a neural network allows to extend the results for any R&D valuation and to abate the waiting time of Least Squares Monte Carlo simulation.

Convertible Bond Valuation with Regime Switching

We present a valuation formula for convertible bonds with regime-switching market conditions by decomposing the convertible bond into a coupon-bearing bond and the American-type exchange option. A coupon-bearing bond component is modeled with a four-factor model: a two-factor affine model for the risk-free rate and a two-factor affine model with stochastic volatility for the credit spreads on the coupon-bearing bond component. We also derive a new valuation formula for the American-type exchange option component.

On extensions of the Barone-Adesi and Whaley method to price American-type options

This paper provides an efficient and accurate hybrid method to price American standard options in certain jump-diffusion models and American barrier-type options under the Black–Scholes framework. Our method generalizes the quadratic approximation scheme of Barone-Adesi and Whaley and several of its extensions. Using perturbative arguments, we decompose the early exercise pricing problem into subproblems of different orders and solve these subproblems successively. The solutions obtained are combined to recover approximations to the original pricing problem of multiple orders, with the zeroth-order version matching the general Barone-Adesi–Whaley ansatz. We test the accuracy and efficiency of the approximations via numerical simulations. The results show a clear dominance of higher-order approximations over their respective zeroth-order versions and reveal that significantly more pricing accuracy can be obtained by relying on approximations of the first few orders. In addition, they suggest that increasing the order of any approximation by one generally refines the pricing precision; however, this happens at the expense of greater computational costs.

SIMPLIFIED OPTION PRICING TECHNIQUES

In this paper we provide alternative methods for pricing European and American call and put options. Our contribution lies in the simplification attempted in the models developed. Such simplification is feasible due to our observation that the value of the option can be derived as a function of the underlying stock price, strike price and time to maturity. This route is supported by the fact that both the risk-free rate and the volatility of the stock are captured by the move of the underlying stock price. Moreover, looking at the properties of the Brownian motion, widely used to map the move of the stock price, we realize that volatility is well depicted by time. Last but not the least, the value of an option is an increasing function of both time and volatility. We find simplified option pricing formulas depending on the underlying asset (price and strike price) and the time to maturity only. We test our formulas against the S&amp;P 500 index options; the advantage of the approach is that less simplifying assumptions are needed and much simpler methods are produced. We provide alternative formulas for pricing European- and American-type options.

Forward–backward SDEs with distributional coefficients

Forward–backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of solution and show its existence and in certain cases its uniqueness. Moreover we establish a link with PDE theory via a non-linear Feynman–Kac formula. The associated semi-linear parabolic PDE is the same for both FBSDEs, also involves distributional coefficients and has not previously been investigated.

On Extensions of the Barone-Adesi &amp;amp; Whaley Method to Price American-Type Options

The present article provides an efficient and accurate hybrid method to price American standard options in certain jump-diffusion models as well as American barrier-type options under the Black & Scholes framework. Our method generalizes the quadratic approximation scheme of Barone-Adesi & Whaley (1987) and several of its extensions. Using perturbative arguments, we decompose the early exercise pricing problem into sub-problems of different orders and solve these sub-problems successively. The obtained solutions are combined to recover approximations to the original pricing problem of multiple orders, with the 0-th order version matching the general Barone-Adesi & Whaley ansatz. We test the accuracy and efficiency of the approximations via numerical simulations. The results show a clear dominance of higher order approximations over their respective 0-th order version and reveal that significantly more pricing accuracy can be obtained by relying on approximations of the first few orders. Additionally, they suggest that increasing the order of any approximation by one generally refines the pricing precision, however that this happens at the expense of greater computational costs.

ЭВОЛЮЦИЯ ФУНКЦИИ ЦЕНЫ АМЕРИКАНСКОГО ОПЦИОНА НА АКЦИИ С ВЫПЛАТОЙ ДИВИДЕНДОВ В МОДЕЛИ ДИФФУЗИИ СО СКАЧКАМИ

This work is devoted to the analysis and evolution of the value function of American type options on a dividend paying stock under jump diffusion processes. An equivalent form of the value function is obtained and analyzed. Moreover, variational inequalities satisfied by this function are investigated. These results can be used to investigate the optimal hedging strategies and optimal exercise boundaries of the corresponding options.

ЭВОЛЮЦИЯ ФУНКЦИИ ЦЕНЫ АМЕРИКАНСКОГО ОПЦИОНА НА АКЦИИ С ВЫПЛАТОЙ ДИВИДЕНДОВ В МОДЕЛИ ДИФФУЗИИ СО СКАЧКАМИ

This work is devoted to the analysis and evolution of the value function of American type options on a dividend paying stock under jump diffusion processes. An equivalent form of the value function is obtained and analyzed. Moreover, variational inequalities satisfied by this function are investigated. These results can be used to investigate the optimal hedging strategies and optimal exercise boundaries of the corresponding options.

A RBSDES APPROACH FOR THE NUMERICAL EVALUATION OF THE AMERICAN OPTION PRICING PROBLEM

The problem of pricing American type options is a typical example of a non lin- ear problem characterized by the absence of closed expressions for its evaluation. Therefore, during recent years, such an issue has been approached , both deterministically and randomly, from the algorithmic point of view, trying to derive suitable numerical approximations. In this paper, starting from the aforementioned solutions, we review some computational, stochastic inspired, methods, mainly based on the the existing link between the above recalled pric- ing task, and the theory of Reflected Backward Stochastic Differential Equations (BSDEs). In particular we show how suitable numerical schemes can be developed within the SBDEs framework by mean of quantization techniques as well as considering Monte Carlo methods.

Robust Monte Carlo Method for R&amp;D Real Options Valuation

This paper is devoted to developing a robust numerical analysis of least squares Monte Carlo (LSM) in valuing R&D investment opportunities. As it is well known, R&D projects are characterized by sequential investments and therefore they can be considered as compound options involving a set of interacting American-type options. The basic Monte Carlo simulation takes a long time and it is computationally intensive and inefficient. In this context, LSM method is a powerful and flexible tool for capital budgeting decisions and for valuing R&D investments. In particular way, numerical tests are performed to examine the optimal choice of basis function and polynomial degree in terms of reduction of the execution time, accuracy and improvement in the simulation.

Stochastic approximation methods for American type options

ABSTRACTStochastic approximation methods for rewards of American type options are studied. Pay-off functions are non random possibly discontinuous functions or random càdlàg functions. General conditions of convergence for binomial, trinomial, and skeleton reward approximations are formulated. Underlying log-price processes are assumed to be random walks. These processes are approximated by log-price processes given by random walks with discrete distributions of jumps. Backward recurrence algorithms for computing of reward functions for approximating log-price processes are given. These approximation algorithms and their rates of convergence are numerically tested for log-price processes represented by Gaussian and compound Gaussian random walks. Comparison of the above approximation methods is made.

A generalization of Yaari’s result on annuitization with optimal retirement

In this paper we generalize the following result of Yaari (1965) on annuitization with an agent’s optimal retirement: it is optimal for individuals to annuitize all of their wealth in the absence of bequest motive. We have other results that refine or extend the result of Yaari (1965). Full annuitization can be viewed as an American-type option that allows an economic agent to exchange the value of labor income with the extra leisure that is brought about by annuitizing all of her wealth. When the agent’s wealth is above a certain wealth threshold, annuitization is triggered, otherwise it is not. Accordingly, wealth plays a key role in controlling the distance to full annuitization.

A Generalization of Yaari's Result on Annuitization with Optimal Retirement

In this paper we generalize the following result of Yaari (1965) on annuitization with an agent's optimal retirement: it is optimal for individuals to annuitize all of their wealth in the absence of bequest motive. We have other results that refine or extend the result of Yaari (1965). Full annuitization can be viewed as an American-type option that allows an economic agent to exchange the value of labor income with the extra leisure that is brought about by annuitizing all of her wealth. When the agent's wealth is above a certain wealth threshold, annuitization is triggered, otherwise it is not. Accordingly, wealth plays a key role in controlling the distance to full annuitization.

THE BRITISH KNOCK-OUT PUT OPTION

Following the economic rationale introduced by Peskir &amp; Samee (2011, 2013) we present a new class of barrier options within the British payoff mechanism where the holder enjoys the early exercise feature of American type options whereupon his payoff (deliverable immediately) is the best prediction of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Should the option holder believe the true drift of the stock price to be unfavorable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. In this paper, we focus on the knock-out put option with an up barrier. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results, we perform a financial analysis of the British knock-out put option. We spot some of the trends previously seen in Peskir &amp; Samee (2011) but observe some behavior unique to the knock-out case. Finally, we derive the British put-call and up-down symmetry relations which express the arbitrage-free price and the rational exercise boundary of the British down-and-out call option in terms of the arbitrage-free price and the rational exercise boundary of the British up-and-out put option.

Convergence of option rewards for multivariate price processes

American type options with general payoff functions possessing polynomial rate of growth are considered for multivariate Markov price processes.Convergence results areobtained for optimal reward fu ...

A high-order compact method for nonlinear Black–Scholes option pricing equations of American options

Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black–Scholes model become unrealistic and the model results in nonlinear, possibly degenerate, parabolic diffusion–convection equations. Since in general, a closed-form solution to the nonlinear Black–Scholes equation for American options does not exist (even in the linear case), these problems have to be solved numerically. We present from the literature different compact finite difference schemes to solve nonlinear Black–Scholes equations for American options with a nonlinear volatility function. As compact schemes cannot be directly applied to American type options, we use a fixed domain transformation proposed by Ševčovič and show how the accuracy of the method can be increased to order four in space and time.

On a constant related to American type options

On a constant related to American type options

Convergence of option rewards for Markov type price processes modulated by stochastic indices. II

Conditions of convergence for optimal expected rewards of American type options are given for perturbed Markov type price processes controlled by stochstic indices