- 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 = 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.
- 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.

**H = nS – c**

**n = (c**

^{+}– c^{–})/(S^{+}– S^{–})
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