American Put Exercise and a Proposed Binomial-Model Arbitrage

Summary

The document poses an arbitrage question about an American put in a binomial stock-price tree. At an intermediate node, it assumes early exercise is optimal and that every terminal descendant remains below the strike. The proposed portfolio combines one put with one share: its initial cost appears to equal the strike, while its terminal value also appears to equal the strike in every outcome.

The question compares that apparent payoff with a risk-free investment that would cost less than the strike at the intermediate node, suggesting a possible arbitrage from shorting the portfolio and lending the proceeds. It does not include an answer or establish whether the assumptions can hold together in an arbitrage-free model. The example is therefore useful as a prompt to examine early-exercise valuation, discounted payoffs, and consistency between option prices and the underlying tree, rather than as a validated strategy.

Key ideas

• The proposed portfolio combines an American put with the underlying share at an intermediate node.
• The question assumes early exercise is optimal and all subsequent terminal stock prices are below the strike.
• The apparent guaranteed strike payoff is compared with the lower present value of a risk-free bond.
• The document raises the arbitrage puzzle but provides no resolution or proof that its assumptions are consistent.

Tags

# American put option in binomial model - arbitrage opportunity?


# American put option in binomial model - arbitrage opportunity?












I'm sorry this must be an elementary question. I spent a good deal of time searching through webs including this site for the problem but I got none.

Here's the problem:

Say we have a binomial tree that is originally(that is, before the put option is introduced) arbitrage-free for sure, and an American put option with strike price K.

Let's assume there's a node (that is not a terminal node) s.t.

(1) on that node, the intrinsic value of the put option is greater than its discounted expected value, hence the option is priced at the intrinsic value.

(2) on every terminal node(the nodes at the end) that comes out from that node, the option is in the money. And again, at those nodes the price of the option equals the intrinsic value, which is the payoff.

e.g.

Now, if someone makes a portfolio consists of one option and the underlying stock at the node described above, it costs K to set up and its final value is always K, i.e. it is worth K at every terminal node coming from the node.

I'm having trouble understanding how this is not an arbitrage opportunity. At and after the node the portfolio is equivalent to zero coupon bond with payoff K and they should be worth the same at the node. However, the bond is worth $K*(1+r)^-t$ whereas the set-up cost of the portfolio is K. An arbitrageur would short the portfolio and lend it to the money market taking risk-free profit.

What did I get wrong?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.