Tutorial: Delta Hedging in the Binomial Model

Tutorial: Delta Hedging in the Binomial Model

Let us delta hedge one put option. Select the Binomial Delta Hedging subject of Option Tutor, select to 2 periods, and make sure you have clicked on the initial node. Your display should be:

Note the similarity between this screen and the binomial replication screen. The essential difference is that you cannot trade the bond (since the idea behind hedging is to replicate the bond) and you are told the delta of the option.

Your initial position is one put option and zero stocks in a two-period world. You can see the value of your position at each node (which, since you currently only hold the put, is simply the value of the put option at each node).

You can see that the option is risky. It's value ranges from 0 to 37.50 at the end of two periods. Our aim is to eliminate the risk from holding the option. This means that we want to trade stocks so that all four numbers at the terminal node are the same.

You can see that the delta is -0.3155:

The hedging formula is: 1 option - delta stocks hedges the risk, so we need to sell -(-0.3155) stocks, which is the same as buying 0.3155 stocks. Let's do this and see what happens.

To buy the stocks, you can either click on the >> and > buttons or enter the quantity you want to add or subtract into the text area:

In our case, it is easiest to simply enter 0.3155 and click OK (or press the enter key). The first part of the display now is:

You can see that we are basically hedged in the second period, the portfolio value being about 10.69 (to do it exactly, you should have bought 0.31551 stocks).

We are not hedged at the end of period 2, since our portfolio value ranges from 16.23 to 5.06 to 22.18. To hedge this, click on the top node in the middle (the one that says 10.69), so you are focusing on:

Our position is summarized by:

The value under "Start" shows that we came into this node with 0.3155 stocks. The delta is -0.1667, which means that we must leave this node with 0.1667 stocks. This implies that we have to sell

0.1488= 0.3155 - 0.1667 stocks.

Hint: For simple calculations like this, bring up the calculator from the Calculators menu.

To sell the stocks, type in -0.1488 in the text area and click OK. Your overall display now should be:

You have hedged the top two terminal nodes. To hedge the bottom two terminal nodes, click on the bottom node in the middle (the one that says 10.70). The delta here is -1, and we came into this node with 0.3155 stocks:

The delta tells us that we must leave this node with 1 stock, so we need to buy 1.0 - 0.3155 = 0.6845 more stocks. After you do this, your display is:

which shows you that you have successfully hedged the put.

Try this out for a three period call to see if you understand the principles behind delta hedging. You should pay particular attention to the fact that the delta typically changes as you move from node to node, and therefore you keep having to adjust your stock position. We had to do the same thing when we were replicating an option.