above ("An asset with a known future price"), and as above, for no arbitrage to be possible, the present value of the position must be its expected future value discounted at the risk free rate, r. The value of a call is then found by equating the two. Solve for Δ such that: value of position in one period = Δ × S up - (S up – strike price, 0) = Δ × S down - (S down – strike price, 0) Solve for the value of the call, using Δ, where: value of position today = value of position in one period ÷ (1 + r) = Δ × S current – value of call The replicating portfolio It is possible to create a position consisting of Δ shares and $B borrowed at the risk free rate, which will produce identical cash flows to one option on the underlying share. The position created is known as a "replicating portfolio" since its cash flows replicate those of the option. As shown above ("Assets with identical cash flows"), in the absence of arbitrage opportunities, since the cash flows produced are identical, the price of the option today must be the same as the value of the position today. Solve simultaneously for Δ and B such that: Δ × S up - B × (1 + r) = (0, S up – strike price) Δ × S down - B × (1 + r) = (0, S down – strike price) Solve for the value of the call, using Δ and B, where: call = Δ × S current - B Note that there is no discounting here the interest rate appears only as part of the construction. This approach is therefore used in preference to others where it is not clear whether the risk free rate may be applied as the discount rate at each decision point, or whether, instead, a premium over risk free, differing by state, would be required. The best example of this would be under real options analysis where managements' actions actually change the risk characteristics of the project in question, and hence the Required rate of return could differ in the up- and down-states. Here, in the above formulae, we then have: "Δ × S up - B × (1 + r up)..." and "Δ × S down - B × (1 + r down)...".