LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A one-step binomial tree says the share moves to one of two prices over a single time step, and a call is scored at both endings. The lesson prices the call by holding some number of shares, called delta, that makes the combined portfolio worth the same amount on both branches. That locked-in amount is discounted at the riskless rate, and the call is the remainder. In the worked arithmetic, delta is one third, the locked-in value is 30, the discounted value is 28.57, and the call comes out near 4.70.
Why it matters
This is the Cox, Ross, and Rubinstein tree in its smallest form, one step. It is the same expiration scoring as a call payoff, now paired with a hedge. Three ideas are doing the work, and the lesson lists them on purpose. The price is a no-arbitrage price: once it is set, the share and the call cannot be combined into a riskless gain on either branch. Delta is the share count that creates that riskless portfolio. One step is a sketch. More steps, like more simulated paths, are how the lesson says the price gets finer. No multi-step figure is computed here.
The idea in plain words
The share starts at 100. Over the step, which the lesson takes as one year, the up price used in the payoff is 105 and the down price used in the hedge equation is 90. A spoken 95 for the down move does not fit the arithmetic that follows, so the calculation uses 90. The call's strike in that arithmetic is 100. Up, the positive part of 105 minus 100 is 5. Down, the positive part of 90 minus 100 is 0. The multiplier of 100 shares is left out to keep the units small.
The portfolio is long delta shares and short one call. Up, it is worth 105 times delta, minus 5. Down, it is worth 90 times delta, minus 0. Set them equal and solve. Delta is one third. Either branch is then worth 30. A portfolio that is worth 30 no matter which way the share goes is riskless over the step. Its value today is the discount of 30 at the riskless rate, which the lesson reports as 28.57. The rate itself is not named in the arithmetic. Today the shares in the hedge are worth 100 times one third, about 33.33. The lesson subtracts to a call value of about 4.70.
How to use it
Write the two ending share prices and the two call payoffs. Write the hedged portfolio on each branch as delta times the ending share price, minus the call payoff, because the call is short. Set the branches equal and solve for delta. Plug delta back in to see the common future value. Discount that common value to today at the riskless rate. Today's share piece is the starting price times delta. The call price is the share piece minus the discounted portfolio value, rearranged from the identity that today's hedged portfolio equals the discounted locked-in value.
Check the lesson's figures in that order. One third times 105, minus 5, is 30. One third times 90 is 30. The discounted 30 is 28.57. The starting share piece is about 33.33. The call the lesson states is about 4.70. Small differences are the rounding they flag when 100 divided by 3 repeats.
The no-arbitrage reading is that 4.70 is the call price at which this hedge does not throw off a riskless gain above the financing. Delta is what made the two branches match. Adding steps means splitting the year into smaller up and down moves and repeating the same match on each node. The lesson compares that to adding paths in a Monte Carlo price, and it saves the general delta formula for later.
What usually goes wrong
A down price of 95, said once, does not fit the equation that produces delta of one third. That equation is 105 times delta, minus 5, equals 90 times delta. The payoffs 5 and 0 belong to a strike of 100.
The call is short, so its payoff is subtracted. Discounting 30 and calling that number the call skips the shares. 28.57 is the present value of the hedged package. The call is what remains after the share piece of about 33.33. One step is a teaching sketch with these inputs, not a listed market price.
Questions people ask
What delta makes the two branches equal?
One third of a share. Both the up branch and the down branch are then worth 30.
What call price does the lesson report?
About 4.70, from a share piece of about 33.33 minus a discounted hedged value of 28.57.
What are the three ideas to keep?
The price admits no riskless gain from mixing the call and the share, delta is the hedge that enforces that, and more time steps refine the one-step sketch.