Dynamic Programming Algorithms for the Ask and Bid Prices

12. American Put Option 12.3. The American Put Option 0 T K π∗ We will now prove the second property for any Markovian risk neutral dynamics, in particular our GBM risk neutral dynamics. Suppose that it is optimal to exercise in state (S,t). This means that the cashflow from exercising, equal to K− S is at least the expected discounted to risk management, from option pricing to model calibration can be solved e ciently using modern optimization techniques. This course discusses sev-eral classes of optimization problems (including linear, quadratic, integer, dynamic, stochastic, conic, and robust programming) encountered in nan-cial models. to determine the price of European and American options. Monte Carlo simulation is a numerical method for pricing options. It assumes that in order to value an option, we need to find the expected value of the price of the underlying asset on the expiration date. Since the price is a random variable, one Proposition 1. The optimal exercise policy for the owner of an American call option is to hold the option until expiration, that is, ˝ = T. Proof. Let ˝ T be any stopping time. If the American option were exercised at time ˝, the payo would be (S˝ K)+, and so the value at time zero to a holder of the option planning to exercise Reference: Longstaff-Schwartz paper on Pricing American Options (industry-standard approach) Reference: A paper on RL for Optimal Exercise of American Options; Implement standard binary tree/grid-based numerical algorithm for American Option Pricing and ensure it validates against Black-Scholes formula for Europeans

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