Abstract
Our work is aimed at modeling the American option price by combining the dynamic programming and the optimal stopping time under two asset price models. In doing so, we attempt to control the theoretical error and illustrate the asymptotic characteristics of each model; thus, using a numerical illustration of the convergence of the option price to an equilibrium price, we can notice its behavior when the number of paths tends to be a large number; therefore, we construct a simple estimator on each slice of the number of paths according to an upper and lower bound to control our error. Finally, to highlight our approach, we test it on different asset pricing models, in particular, the exponential Lévy model compared to the simple Black and Scholes model, and we will show how the latter outperforms the former in the real market (Microsoft “MSFT” put option as an example).
Highlights
The first academic work on pricing options, in particular that of Black and Scholes [1], has allowed the development of derivatives markets
The first alternative models to the Gaussian model were the stable model proposed by [4, 5], and within the framework of option valuation, Merton [6] is the first author to have developed a non-Gaussian model; let us note that for a long time, evaluating American option has been considered inadequate with the traditional forward Monte Carlo simulations; the binomial tree model [7] and finite difference [8] methods were used as the only numerical method before the mid-1990s when new methods appeared
The most popular method was proposed by Carriere [9] using optimal stopping times and developed, improved, and popularized by Longstaff and Schwartz [10] by combining between least square regression and Monte Carlo simulation; for a more comprehensive review of the literature, the reader may profitably consult the works of Tankov et al [11]
Summary
The first academic work on pricing options, in particular that of Black and Scholes [1], has allowed the development of derivatives markets. We compare between the models of Black and Scholes in 1973 [1] and an exponential diffusion-jump model, but when using a modeling process using discontinuous trajectories, a delicate problem arises, which is the incompleteness of markets, and the riskneutral measure is not unique. Numerous works such as those of Fujiwara and Miyahara [12] or Tankov et al [11] have been devoted to this subject. We adopt a more direct point of considering view that the dynamic followed by the price of financial assets is given in a risk-neutral universe
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