One Period Binomial Option Pricing Model

  • The following model can be used for options on stocks, currencies, and commodities; points on interest rate option pricing will be made at the end of this section.
  • A critical component for option pricing with the one period binomial model is the notion of constructing a hedged portfolio.

H = nS - c

  • H = The value of the hedged portfolio

  • n = Hedge ratio

  • S = Price of the underlying asset

  • c = Call price

  • A hedged portfolio will be "riskless" in that there is a perfect balance between a long position in the underlying and a short position in the call, so the gain in one offsets the loss to the other.

n = (c+ - c-)/(S+ - S-)

  • c+ = The price of the call when the underlying rises to S+
  • c- = The price of the call with the underlying falls to S- Note that these values are "intrinsic" values (difference between the underlying's future price and the option's strike price). For example, if the strike price is $25/share and the upside is $30/share, then c+ is $5.
  • S+ = The upside price of the underlying asset
  • S- = The downside price of the underlying asset
  • Upside and downside for the underlying must be converted to 1 + the % price move

  • u = (S+/S)

  • d = (S-/S)

  • S = Starting price for the underlying

  • Because the hedged portfolio is a risk free portfolio, the rate of return on the hedged portfolio should equal the risk free rate.

  • The one period model can be expanded to multiple periods.

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