In this workouts, you are a market maker that has sold 1000 put options to a client.

The time to expire is a month, and the strike is set at $55.

Let's remember that number.

The underlying asset also currently trades at $55 in the spot market as well.

Oh, great. We're told using the delta table below, answer the following questions.

So we have the initial information from the sentence above repeated in this table, but then we've got some more information underneath the questions.

So let's have a read through the questions. Number one, how many shares do you have to buy or sell to be Delta neutral? Well, a short put position means we have positive delta.

This means we have to sell shares to achieve a Delta neutral position.

So we are going to be short, but buy how much we need to find the option delta.

We are going to take the option delta from the stock price of $55.

I'll then take that 0.46.

I'll then multiply that by the 1000 options.

This means we need to sell 460 shares to be Delta neutral.

Question two says, after the initial Delta hedge has been executed, the stock price rallies to $57.

You adjust the delta hedge to remain Delta neutral on aggregates.

But how many shares are bought and sold when we do that? So number two, adjusting the Delta hedge, we've now got a new option delta.

If the price has gone up from 55 to 57, the new put delta is 0.39.

This means the new number of shares for Delta Hedge is going to be that 0.39 multiplied by the thousand up at the top.

Ah, instead of selling 460 shares, we should have sold 390.

Right? So the number of shares to trade means we need to now buy 70 shares. Instead of selling four 60, we really want to sell three 90.

That means we need to buy 70.

So to achieve an overall Delta neutral position, we have to buy back 70 shares.

Great. And that's the final item here.

Direction of adjustment is buy for question three.

A few moments later, the underlying asset drops in price back to $55.

Oh, you adjust your delta hedge again to bring the aggregate delta to zero.

What is your p and l profit and loss at this time? And for simplicity, volatility has remained constant.

So the trade required to be Delta neutral again means we need to unwind the 70 shares that we just bought.

So we will have to sell those 70 shares again.

Now, the gain or loss on the options.

Well, nothing's really happened here.

The options have returned to their initial price with none being traded.

There's no profit and loss there. Great.

However, on the shares, we have had an impact.

Shares have returned to the original price, but 70 were bought at $57 and have just been sold at $55.

Oh dear. We've made a bit of a loss there.

So gain and loss on the options was zero, but now we've had a 55 minus the 57. We've made a loss on the 70 shares.

We've got a loss there of 140.

Our overall position is a loss of 140.

Lastly, question four, how would the p and l profit loss have looked if you decided not to adjust the Delta hedge after the initial move up to $57? What would've been the risk in not doing so? Ah, well, initially, with the benefit of hindsight, we can say it would've been better not to adjust the Delta hedge after the initial move to 57.

We wouldn't have had this overall loss of 140.

So it's very tempting for us to think, ah, maybe we should not have done that hedge.

The risk in not doing so would've been that this move was the beginning of a new trend and prices of the underlying would've continued to go up past 57 to 59 to 61 to 63, while we had a growing short position.