5_Binomial_trees.pdf

# Option pricing using risk neutral probabilities 2 two

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Option pricing using risk-neutral probabilities 2. Two-step binomial model 3. Towards the Black-Scholes-Merton formula Fin330 19

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Fin330 Delta l Delta ( Δ ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock. l The construction of a riskless portfolio is called delta hedging . l The value of Δ varies from node to node: l 1 st time step: (2.0257-0)/(22-18)=0.5064 l 2 nd time step: l (3.2-0)/(24.2-19.8)=0.7273 l (0-0)/(19.8-16.2)=0 20
Delta l If Δ = 0.65, this means that if the price of the underlying stock increases by \$1, the option price rises by \$0.65, all else equal. l Ranges for Δ : l Calls: have a delta between 0 Δ≤ 1; l Puts: -1 Δ≤ 0. Fin330 21

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Delta l Δ depends on whether the option is in-the- money, at-the-money, or out-of-the- money. l Call options: l In-the-money: Δ à 1 as t à T . l At-the-money: Δ 0.5 . l Out-of-the-money: Δ à 0 as t à T . Fin330 22
Fin330 Choosing u and d l So far, we have set u and d arbitrary. l In practice, u and d are determined from the stock price volatility σ . l One way of matching the volatility is to set: where Δ t is the length of the time step. t t e u d e u Δ σ Δ σ = = = 1 23

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The Black-Scholes-Merton Model l The BSM model can be derived from the binomial model above by reducing the time step Δ t to zero. Fin330 24
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