금융 모델링: Binomial Option Pricing Model

* One-Step Model
  1) Assume there are three assets: Stock, a riskless bond, an option

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   2) Consider a portfolio consisting of one long option and h units short of the stock

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   3) In order for ,

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• h is the "delta"

   4) Choosing h this way, it leads us to the risk free rate return, then

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   5) Risk neutral probabilities p and q are drawn:

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* Multi-Step

   1) Two-step or multiple-step models can be calculated the same way

• However, hedging parameter h is not constant, but changing over time
• , , 
• Therefore, 
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• This is generalized in the Black-Scholes Model

   2) Options values can be calculated by going "backward"

• Because American options can be exercised at any nodes, "backward" method should be used
• European options do not need not be "backward", but the value can be calculated by below generalized equation
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• Where, 
•  is a payoff function for state x

   3) Dividends

• Assume the dividend yield is 
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   4) American options

• Backward method is used
• When going backward (two-step for example = 3 nodes), compare 1) the payoff in "immediate exercise" case at t=1, and 2) the value at t=1 by discounting payoffs at t=2, and choose the larger one
• [i.e. 1)=value of exercising early, 2)=value of holding the option]
• Using the same way described above, choose the right value for the American option at t=0
• Therefore, when valuing American option by binomial tree, each node should be examined