- 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–)