# DELTA for put option and subsequent arbitrage opportunity

We know that, 1. Short call and long stock as per hedge ratio will generate arbitrage profit if call option is over priced. 2. For put option, the direction of long/short of put option and stock is the same way unlike the earlier mentioned call option. My question is? 1. Is arbitrage opportunity really exists when call price is underpriced? Then we have to take long in call option and short stock? 2. Is really arbitrage opportunity exist for put option when it is both underpriced or overpriced? If put option is underpriced, then what will be our strategy? Both long stock and long put option? NB. Pricing of option derived from one period or two period pricing model.

Need to know this ans badly.

Ok, I assume that you know to calculate the delta hedge ratio. It is (payoff diff between the nodes/ Price difference between the nodes). Once to get that you should be able to calculate the number of shares required to hedge. For all arbitrage opportunity the mantra "buy low sell highâ€™.

Say, the call price according to the model is 5 and the actual price in the market is 4, then buy call options in the market and sell shares. Similarly if the price according to the binomial model is \$6 and price in the market is \$8, then you need to sell calls and buy shares. This is in node 0 and this will give your investment amount.

In Node 1, you net off the shares and the calls and make an arbitrage profit. Say in node 0, you bought shares, in Node 1 you will sell those shares. Also consider the share price. If it is greater than strike price, you will pay out the difference. If share price is low, then the call expires worthless and you will be left with only the (number of shares * Share price). Trick - in Node 1, the net amount under both the possibilities will have to equal to the amount when the option is worthless.

For Put, you should do the same thing but the direction is the same not opposite. Arrive at the decision to buy or sell put based on the price and then buy/sell shares in the same direction.

Also remember that the delta hedge ratio is different for each node, so for a 2 period model you will have 3 delta hedge ratios.

How can I be able to give such clear and cognitive explanation like you? Very well reply indeedâ€¦