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 I. Basic math.
 1 Conditional probability.
 A. Definition of conditional probability.
 B. A bomb on a plane.
 C. Dealing a pair in the "hold' em" poker.
 D. Monty-Hall problem.
 E. Two headed coin drawn from a bin of fair coins.
 F. Randomly unfair coin.
 G. Recursive Bayesian calculation.
 H. Birthday problem.
 I. Backward induction.
 J. Conditional expectation. Filtration. Flow of information. Stopping time.
 2 Normal distribution.
 3 Brownian motion.
 4 Poisson process.
 5 Ito integral.
 6 Ito calculus.
 7 Change of measure.
 8 Girsanov's theorem.
 9 Forward Kolmogorov's equation.
 10 Backward Kolmogorov's equation.
 11 Optimal control, Bellman equation, Dynamic programming.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Backward induction.

die is rolled three times. After the first and the second roll a player has an option to take gain equal to result of the rolling and end the game or discard the result and continue the game to the next rolling. At the last rolling the player's gain is the result of the last rolling. What is the optimal stopping strategy and what is the fair price for participation in the game?

We introduce the function Clearly, Also, where the is the mathematical expectation applied to the variable . We have Therefore, after the second rolling the player should reject 3, . Similarly, where Therefore, after the first rolling the player should reject 4, . The fair price for participation in the game is

 Notation. Index. Contents.