In this workout, we're told that you have sold 50 put options with a $55 strike to a client this morning when the underlying stock price traded at $55 as well.

You hedged your delta at the time and we're asked using the delta table below, answer the following questions.

Well, the first question says, how many shares did you transact in your initial Delta hedge? Well, we were told up in the question that you sold puts, and that's a short put.

A short put leads to a positive delta, so we have to sell shares.

If we have a look at the table, we can see the stock price of $55 gives us a put delta of 0.46.

As we sold 50 put options, we multiply that by the 0.46 puts delta.

That tells us that 23 shares must be sold to be Delta neutral.

In question two, it says, assuming the stock has rallied to $58, now, what do you need to do now to be Delta neutral again, R Well, as the underlying stock price has increased, the put option has moved out of the money, which means the delta has declined From the delta table, we can see at the $58 stock price that the put delta has now changed to 36.

Consequently, the combination of short puts and the initial delta hedge that we had in question one.

It's no longer going to be Delta neutral, but we are still overall short.

The new put delta of 0.36 suggests that for the 50 put options that we sold, this means that 18 shares should have been sold rather than the 23 that we had before.

So in order to correct for this, we'll have to take the 23 minus, the 18 and five shares must be bought back.

Lastly, in question three, it says, assume that after you've executed the trade, described to number two, the underlying price falls back to $55.

What's your profit and loss? Assuming all other price factors remain constant, well, as nothing else has changed, IE, it's still the same date, still got the same price.

We can assume that the profit and loss on the option position is zero because the put premium will be back to its original level.

However, it's on the Delta hedge position where we've got a profit and loss impact as we sold 23 shares in the morning at $55 and bought back five of them later on at $58.

This leads to a loss.

The difference in the price that we bought and sold at was 58 minus the 55. That's three per share multiplied by the five that we had to buy back.

This leads to a realized loss of $15 in total on the five shares, but the remaining 18 shares have been sold at $55 and we were told in the question they're currently trading at $55.

So there's no p and l impact on those 18.