Options on US stocks and ETFs are American style: the owner can exercise on any trading day. The formula behind this module prices an option that can only be exercised at expiry, and a formula cannot handle a choice you can make at every moment. A tree of possible prices, worked backward one step at a time, can, and it puts a dollar price on the right to exercise early.
The last lesson showed why a hedged option earns the interest rate, so a model can grow the stock at the rate and discount the average payoff. A tree applies that rule to one short step at a time.
One step
Take a $100 stock and one step of three months. The stock either rises 10% to $110 or falls to $90.91, the drop that exactly undoes a 10% rise. With a 4% rate, $100 left in cash grows to $101.01 over the step. The tree uses the chance of an up move that makes the stock's average outcome equal that $101.01, the forward price from Module 2. Solving p × 110 + (1 - p) × 90.91 = 101.01 gives p = 0.529. This is the risk-neutral probability: not a forecast, just the chance that prices the stock fairly against cash.
Now price a $100 call. It pays $10 at the top and nothing at the bottom. The average, using the risk-neutral chance, is 0.529 × $10 = $5.29, and discounting that back three months at 4% gives $5.24. The $100 put pays $9.09 at the bottom and nothing at the top, so it is worth 0.471 × $9.09, discounted, or $4.24. The two differ by 99.5 cents, the interest on the $100 strike for three months, exactly as parity says they must.
The one-step example: $100 today, $110 or $90.91 in three months, and a $100 call worth $5.24. Switch to a put for the $4.24. Then add steps and watch the value in the note zigzag around the Black-Scholes figure and close in on it.
Rolling back from expiry
A real tree cuts the option's life into many short steps. Each node splits in two, and because up then down lands back where it started, the branches rejoin: after 100 steps there are only 101 possible prices. The size of each move comes from the volatility, so a more volatile stock spreads its tree wider.
Pricing runs backward. At expiry every node's value is just the payoff. One step earlier, each node is worth the risk-neutral average of the two nodes it can move to, discounted for one step. Repeat, column by column, until one number is left at today's node. That is rolling back, the whole method. A few steps give a rough answer that zigzags as you add more; a few hundred land within a cent of the Black-Scholes value for an option exercisable only at expiry. The formula is what the tree becomes as the steps shrink to nothing.
American options: one extra comparison
A European option can only be exercised at expiry, so rolling back is all there is. An American option adds one check at every node: compare the value of holding, the rolled-back number, with the value of exercising right now, the intrinsic value, and keep the larger. That single comparison is the only difference between the two styles, and it means an American option is never worth less than the European one: it carries every right the European one does, plus one.
Take the same tree with two steps of three months and a $110 put. At expiry the stock is at $121.00, $100.00 or $82.64, and the put pays $0, $10.00 or $27.36. Roll back to the lower node after one step, where the stock is $90.91. Holding is worth 0.990 × (0.529 × $10.00 + 0.471 × $27.36) = $18.00. Exercising is worth $110 minus $90.91, or $19.09. The American put takes the $19.09. Carry that higher number back to today and the American put is worth $11.35, against $10.84 for the European one.
The two-step $110 put from the text. The red node is where exercising beats holding. Switch to European and the red goes, taking 51 cents with it. Drag the rate to zero and it goes too. Add steps and the premium settles around 60 cents.
The 51 cents between the two is the early-exercise premium: the price of the right to exercise before expiry. It is largest for puts deep in the money with a long time left and a high rate, where the cash from exercising has the most interest to earn. For a one-month put 5% in the money it is small, around 5 cents on a $100 stock.
Calls are different. On a stock with no dividend the tree never finds a node where exercising a call wins, so the American and European calls are worth the same. For a put, once the rate is above zero some price is low enough that taking the strike in cash today wins. Module 10 explains when exercising early makes sense in practice, and what a dividend changes.
Why US stock options are American
Listed options on US single stocks and ETFs are American style under exchange rules and settle in shares. Most broad index options, SPX among them, are European and settle in cash; Module 13 covers settlement. So every price on a stock or ETF chain is an American price, and a European model would price its deep in-the-money puts too low.
The maths: the tree's three formulas and the hedge behind them
Cut the option's life of T years into n steps of Δt = T / n.
u = e^(v √Δt)andd = 1 / uare the up and down multipliers, withvthe volatility as a decimal.p = (e^((r - q) Δt) - d) / (u - d)is the risk-neutral chance of an up move, withrthe rate andqthe dividend yield.- At each node,
V = e^(-r Δt) × [p × V_up + (1 - p) × V_down]. For an American option,V = max(that, intrinsic value).
The example above used v = 19.06% and Δt = 0.25, so u = e^(0.1906 × 0.5) = e^0.0953 = 1.100 and d = 1 / 1.100 = 0.909. With r = 4% and no dividend, p = (e^0.01 - 0.909) / (1.100 - 0.909) = (1.0101 - 0.9091) / 0.1909 = 0.529.
A hedge produces the same answer. Buy (10 - 0) / (110 - 90.91) = 0.5238 shares for $52.381 and borrow $47.145, the value today of $47.619 repaid in three months. At $110 the shares are worth $57.619, and after repaying $47.619 you have $10. At $90.91 they are worth $47.619, which repays the loan and leaves $0. That copies the call in both states, so the call costs what the copy costs: $52.381 - $47.145 = $5.236.
Your turn
Open AMZN on the Options Chain and find the put about 5% in the money. Work out its intrinsic value, read its mid and take the difference. On an American put it should never be negative, and a small slice of it is the early-exercise premium you just priced.