How Monte Carlo approach can be used in option pricing?
American and Bermudan option pricing in a Monte-Carlo framework is, in theory, impossible. Because it requires the value of the option at intermediary dates. In order to decide whether to exercise or to keep it an information that is usually not provided.
Actually, rather than the value of the option, one needs an optimal exercise strategy. That is, for each trajectory, an “optimal” date when to exercise the option.
A strategy is acceptable if the decision to exercise or not only depends on the available information. And not on the future of the trajectory. For instance, it is not acceptable to exercise the option at the maximum of its exercise value along the trajectory. Because at a given date t, one doesn’t know whether the trajectory will end at its maximum.
We describe a general technique to compute optimal exercise criterions. This technique requires a multi-dimensional nonlinear optimizer.
Theoretically, there exists for each exercise date and for each date in the case of an American option an optimal exercise boundary. One should keep the option, when beneath the boundary. And beyond the boundary, exercised. By no mean should the decision involve any type of randomness.
Professor Douady told Rebellion about his paper:
“I wrote it to show a couple of Monte-Carlo algorithms to solve free boundary problems, which are the essence of control theory (HJB is a free boundary problem). This is a well-known challenge, as we are caught in the dilemma of ‘available information.’ I describe in the part on Longstaff-Schwartz method.”
Research Professor at University of Paris I: Pantheon-Sorbonne
Raphael is a mathematician and economist, specialised in quantitative finance and chaos theory. With more than fifteen years experience in the banking industry (risk management, option models, trading strategies) and thirty years research in pure and applied mathematics, Raphael developed highly sophisticated quantitative solutions and statistical analysis. A former fellow of Ecole Normale Supérieure in Paris, earned his Ph.D. in 1982 in Hamiltonian dynamics and became strongly involved in Finance in 1993.
Currently affiliated with University of Paris 1-Sorbonne Economic Center (CES) and the French National Center for Scientific Research (CNRS). Raphael also received the appointment of International Associate Professor at New York University Polytechnic Institute. Raphael led and organized numerous academic, as well as practitioner conferences around the world, including the New York University seminar of Mathematical Finance and Paris Europlace conferences. Raphael’s most recent research topics are Hedge Funds risks, for which Raphael developed especially suited powerful nonlinear statistical models, and systemic risk.
Raphael is one of the founders and the research director of Riskdata, a market-leading provider of risk management tools for investors, asset managers, hedge funds, fund of funds, and pension funds. Furthermore, Raphael became appointed as academic director of a French “Laboratory of Excellence” devoted to financial regulation (LabEx ReFi). Raphael is also a member of the Praxis Club, a New York based think tank advising the French government on its economic policy and other related topics and on the “risk committee” of Finance Innovation, a French official entity supporting innovation in financial software.