5_Binomial_trees.pdf

# L these are not the real world probabilities l the

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l These are not the real world probabilities! l The value of a derivative is then its expected payoff discounted at the risk-free rate. Fin330 13 S 0 u c u S 0 d c d S 0 c (1 p )

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Risk-neutral probabilities l To price a derivative, we assume that investors are risk-neutral (i.e. don’t require higher expected return to compensate for higher risk). l Risk-neutral word has the following features: 1. The expected return on any asset is the risk-free rate. 2. The discount rate used for the expected payoff on any investment is the risk-free rate. l But remember that we compute the expectations using not real world probabilities. Fin330 14
Fin330 Original Example: risk neutral rather than no-arbitrage pricing 6523 . 0 9 . 0 1 . 1 9 . 0 0.25 0.12 = = = × e d u d e p rT S 0 u = 22 c u = 1 S 0 d = 18 c d = 0 S 0 c (1 p ) 15 The value of the option is: e –0.12x0.25 [0.6523x1 + 0.3477x0] = \$ 0.633

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Overview 1. One-step binomial model: 1. Option pricing using no-arbitrage argument 2. Option pricing using risk-neutral probabilities 2. Two-step binomial model 3. Towards the Black-Scholes-Merton formula Fin330 16
Fin330 A Two-Step example l Each time step is 3 months l K =21, r =12% 20 22 18 24.2 19.8 16.2 17 c=?

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Fin330 Valuing a Call Option l Option is out of money at nodes E and F ( è c=0). l Therefore, c=0 also at node C . l Find risk-neutral probabilities using equation for p above with u=1.1, d=0.9, r=0.12, T=0.25. è p=0.65 l Value at node B : e –0.12 × 0.25 (0.65 × 3.2+0.35 × 0) = 2.03 l Value at node A : e –0.12 × 0.25 (0.65 × 2.03+0.35 × 0) = 1.28 20 1.28 22 18 24.2 3.2 19.8 0.0 16.2 0.0 2.03 0.0 A B C D E F 18
Overview 1. One-step binomial model: 1. Option pricing using no-arbitrage argument 2.

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• Spring '16
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