# 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?

• There is no arbitrage here, as every rational agent would immediately exercise a long position in the call option. If they would miss to exercise optimally, then indeed you would earn the risk-free interest rate on the strike over the next period. – LocalVolatility May 15 '18 at 6:46
• @LocalVolatility Thank you for comment. I think you meant the put option.. Could you elaborate on how other investors exercising their put options erases the arbitrage opportunity? Is it because the underlying stock's price will increase? – P E John May 15 '18 at 7:26
• You are right - I meant the “put” option. Any rational agent would exercise the put due to the argument you presented. Thus, you could not carry out the arbitrage in the first place. – LocalVolatility May 15 '18 at 7:33
• @LocalVolatility In my argument the arbitrage is achieved by buying one put option and one share of stock. – P E John May 15 '18 at 7:55
• Please think about it a bit. If you sell the put, then your rational counterpart would exercise it immediately thus your short position in the put would vanish. I.e. you could not hold it over the next period in the first place. I am just repeating myself here. – LocalVolatility May 15 '18 at 9:23