# Optimization Methods in Finance

- Optimization Methods in Finance
- Numerical Methods for Pricing Exotic Options
- CME 241: Reinforcement Learning for Stochastic Control
- Dynamic Programming Algorithms for the Ask and Bid Prices
- Dynamic Programming Algorithms for the Ask and Bid Prices

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 The bid and ask prices of an American option are represented in terms of the expectation of the option payoff evaluated at an optimal stopping time with respect to an optimal martingale probability measure. As a by-product a similar dynamic programming algorithm is obtained for pricing and hedging European contingent claims in the same setting. 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. Dynamic programming algorithms are developed for computing the ask and bid prices of American contingent claims in a binary tree setting in the presence of small proportional transaction costs

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