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.