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# Notes04 - MA3245 Financial Mathematics I Notes 4 Binomial...

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MA3245: Financial Mathematics I Notes 4: Binomial Tree Model Dr. Oliver Chen [email protected]ath.nus.edu.sg Department of Mathematics, National University of Singapore

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1 Introduction 2 One-period binomial model 3 Risk-neutral world 4 Multi-step binomial model 5 Summary MA3245 Notes 4: Binomial tree 2 / 66
Introduction Introduction Introduction Binomial trees are a common numerical method used to price options. They are introduced at this time as a means of explaining the concept of risk-neutral valuation . The ‘branches’ of a binomial tree represent the different possible paths that the asset price might take from the present time when the asset price is known exactly, to the maturity of the derivative contract in question when the asset price is uncertain. MA3245 Notes 4: Binomial tree 4 / 66

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Introduction Probability preliminaries Probability preliminaries Expected value: The expected value is the weighted average of possible future values. The weighting is by the probability of the future value occurring. We explore the idea of an expected value using several examples with an unbiased coin. MA3245 Notes 4: Binomial tree 5 / 66
Introduction Probability preliminaries Expected value example 1 Suppose that a casino is offering a game where a coin is flipped. If the outcome is heads, then you win \$1. If the outcome is tails, then nothing happens. What is the fair price of playing this game? Each time the game is played, there is a 1 2 probability that you will win \$1, and a 1 2 probability that you will not win (and not lose) anything. MA3245 Notes 4: Binomial tree 6 / 66

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Introduction Probability preliminaries Example 1 (cont.) Put another way, suppose you play the game a large number times, say 1000 times. Then you will expect that heads will come up about 500 times and that tails will come up about 500 times. That is, you will expect to win about \$500. If the cost to play 1000 games is less than \$500, then the casino will expect to lose money. If the cost is greater than \$500, you will expect to lose money. MA3245 Notes 4: Binomial tree 7 / 66
Introduction Probability preliminaries Example 1 (cont.) If the cost to play 1000 games is \$500, then neither you nor the casino can reasonably expect to make (or lose) money. Thus, the fair price of playing 1000 games is \$500. Dividing by 1000, each game should cost \$0 . 50. Remember that the expected value is the weighted average of possible values. So, the expected value is: 1 2 × \$1 + 1 2 × \$0 = \$0 . 50 MA3245 Notes 4: Binomial tree 8 / 66

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Introduction Probability preliminaries Expected value example 2 What if you get \$1 if the coin lands head side up, but you lose \$3 if the coin lands tail side up? The expected value is: 1 2 × \$1 + 1 2 × ( - \$3) = - \$1 So the expected value tells us that the fair price should be - \$1. In other words, the casino needs to pay us \$1. Reasoning in the other way confirms this result: After 1000 games, you can expect to win 500 times (winning \$1 each time) and lose 500 times (losing \$3 each time).
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Notes04 - MA3245 Financial Mathematics I Notes 4 Binomial...

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