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Binomial Pricing
The binomial pricing model for options was first proposed by Cox, Ross and Rubinstein in 1979. This is the simplest technique used for option pricing, and because of the flexible nature of its design, binomial pricing can be used to price a wider range of option types. This model assumes that price of the security follows a random walk.
Suppose the price of an underlying is S0 and that over a small time interval Δt it may have only two potential future values S0u or S0d. Probability p is assigned to the likelihood that the price will rise and a probability of 1-p is assigned to that of a fall in the stock price.
where,
So = stock price
p = probability of an increase in price
u = specific factor by which the price rises (u >= 0)
d = specific factor by which the price falls (0 < d <= 1)
σ = standard deviation
If the price moved up to Su in the first period then the price may move next to either Suu or Sud. However if the price moved down in the first period to Sd then in the second period it may move to either Sdu or Sdd.
In general, the time period between today and the day of expiry of the option is sliced into many small time periods. Each point in the tree is referred to as a node. The tree contains potential future asset prices for each time period from today through to expiry.
Payoffs can be calculated at each node corresponding to the time of expiry. The payoffs of simple put and call options are given by the formulae:
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Payoff for put option = max(X - SN,0) Payoff for call option = max(SN - X,0) where, SN = Price of the underlying asset at each node X = Strike price
Discounted payoffs of the option at expiry node is achieved by the process called backwards induction. This involves stepping backwards through time calculating the option value at each node. The backward induction starts from N i.e expiry time and decreases down to 0 i.e today. For american put option = max (X-Sn,e-rΔt(pVu+(1-p)Vd)) For american call option = max (Sn-X,e-rΔt(pVu+(1-p)Vd)) For european call/put option = e-rΔt(pVu+(1-p)Vd) |
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where, X = strike price. Sn = price of the underlying asset. p = probability of the price rise. Vu = option value from upper node at n+1. Vd = option value from the lower node at n+1. r = risk-free rate.
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