Case 079Option pricing and arbitrage checksHard

Build a two-step binomial tree for a stock at 500 with up 1.1, down 0.9 and 2% a step, and price a 520 put as European and as American. Where is early exercise optimal?

The situation

Mulshi Agro trades at Rs 500. Each step the stock either rises by a factor of 1.1 or falls by a factor of 0.9, and the interest rate is 2% per step. The stock pays no dividend. You are asked to price a put struck at Rs 520 expiring after two steps, first as a European option, then as an American one that can be exercised at any node.

The interviewer wants the risk-neutral probability, the value at every node, the exact node where early exercise is worth it, and the difference between the two prices.

Your task

What are the European and American put prices, where is early exercise optimal, and why does the difference arise?

Quick check

Before building the tree: will the American put be worth more than the European, and why?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The European put is worth about Rs 29.22 and the American put about Rs 33.22; the only early-exercise node is the down node at 450. The risk-neutral probability of an up move is 0.60. Final payoffs are 0, 25 and 115. Discounting gives 9.80 at the up node and 59.80 at the down node, where exercising for 70 beats holding. Replacing 59.80 with 70 and discounting once more lifts today's value by Rs 4.00, the early exercise premium.

Step 1How do you set up the tree and the probability?

Start with the one number the tree lives on. The risk-neutral probabilityThe probability of an up move that makes the stock grow at the interest rate on average. It is a pricing device, not a forecast of what will happen. is the up-probability that makes the stock's expected growth equal the rate: p = (1 + r - d) / (u - d) = (1.02 - 0.9) / (1.1 - 0.9) = 0.60. It is not a forecast; it is the weight that makes a hedged position earn the interest rate, which is the only weight that rules out free money. The stock then goes 500 to 550 or 450, and on to 605, 495 or 405: the tree recombines because up then down equals down then up.

From 500 the stock reaches 550 or 450, then 605, 495 or 405, where the 520 put pays 0, 25 and 115; the European value is 29.22 today, while the American holder exercises at the 450 node for 70 rather than holding for 59.80, which lifts today's value to 33.22.

Step 2How does the European value roll back?

Work from the right. At the up node the put is worth the discounted probability-weighted payoff of its two children: (0.6 times 0 plus 0.4 times 25) divided by 1.02, which is 9.80. At the down node, (0.6 times 25 plus 0.4 times 115) over 1.02 is 59.80. Today's value is one more step of the same rule, (0.6 times 9.80 plus 0.4 times 59.80) over 1.02, which is 29.22. Each step is the same move: average the children with p, then discount one period.

The relationship

V = \frac{p\,V_u + (1-p)\,V_d}{1+r} \qquad p = \frac{1.02 - 0.9}{1.1 - 0.9} = 0.60

V_u, V_d	the option values one step ahead after an up and a down move
p	risk-neutral probability of the up move
1 + r	one period of discounting at 2%

What it says in wordsEvery node's value is the probability-weighted average of its two children, discounted one period; the American version adds a comparison with immediate exercise at each node.

Step 3Where does the American holder do something different?

At every node ask one extra question: is the option worth more dead than alive? At the up node the put is out of the money, so exercise gives nothing and the holder keeps 9.80. At the down node the stock is 450, exercising pays 520 less 450, which is 70, and that beats the hold value of 59.80 by 10.20. The reason is that waiting risks an up move to 495 where the payoff shrinks to 25, and the strike collected today earns interest. Today, immediate exercise would give only 20 against a hold value of 33.22, so the holder waits. A landlord who can take a deposit now or gamble on a bigger one next year takes it now when the odds of a smaller one are high enough.

At the 450 node exercising for 70 beats holding for 59.80, and feeding that choice back to today lifts the put from 29.22 European to 33.22 American, an early exercise premium of Rs 4.00 a share.

Node	Stock	Exercise now	Hold	American value
Up, step 1	550	0	9.80	9.80
Down, step 1	450	70.00	59.80	70.00
Today	500	20.00	33.22	33.22

At each node the American value is the larger of exercising now and holding; only the down node prefers exercise, and today's hold value of 33.22 already includes that choice.

Close with the limit. Two steps is a cartoon; a desk would use hundreds, and the exercise boundary becomes a curve of stock prices below which the put is exercised. The principle survives: an American put on a non-dividend stock carries a premium for the right to collect the strike early, and that premium grows with the rate and with how deep in the money the stock can go. The symmetric claim for calls is the reverse: with no dividends, a call is never exercised early, because selling it always beats exercising it.

Where candidates lose it

The common loss is applying the never-early rule to the put. It belongs to calls on non-dividend stocks. A put holder gives up interest on the strike by waiting, so deep in the money the American put is exercised.

The second is comparing exercise against the European value at the root, 20 against 29, and concluding the American premium is zero. The comparison must be made at every node, and the down node is where it bites.

What the interviewer asks next

Raise the rate to 5% a step. Does the early exercise premium rise or fall, and why?
Price the 520 call on the same tree. Is it ever exercised early?
How would a dividend of Rs 30 paid after the first step change the call's answer?
What happens to the American premium as the number of steps goes to a thousand?

Company names and figures are illustrative.