2013 Q9 September Exam

[Darragh Kelly]

Hi,

I'm a little confused for part (iv) of the 2013 Q9 September exam paper.
I follow the solution regarding finding the investors holding's of the put on stock, based on the criteria that the portfolio needs to be delta and gamma hedged.

What I can't get my head around is the value of the portfolio - am I correct in saying it's equal to the 100,000 call options*price of calls + cash from short selling 100,000 units of stock + cash from short selling the put options?

OR

Does the value of the derivative you've hedged not always have to equal the value of the hedging portfolio you've setup? ie 100,000*2.401(ct) = -100,000*31.45(pt) - 100,000*117.89(St) (which don't equal). I calculated the price of the call option of 2.401 using the put-call-parity relationship. In other past exam paper questions, which were just required to be delta hedged, the second equation we always used to find the holdings was that the value of derivative we are hedging is equal to say the combination of cash and stocks (or puts and stocks in this case).

Little confused on above, and I know we weren't asked here for value just the new holdings of the put and stock.

Many thanks,

Darragh

 

[Steve Hales]

Hi
It is usually the case that when hedging a position we are required to match the value and the delta. However, in this case there are only two variables (the amount of cash and the number of put options) with which we need to set the delta and the gamma to zero. There aren't enough degrees of freedom to match the actual values as well.
The subtlety here is that if we were asked to replicate the behaviour of the call options (or indeed just to delta-hedge it) then we'd match the value and the delta. A delta- and gamma-hedge is concerned with the changes in the portfolio values rather than the absolute values themselves.
Specifically in this question we're concerned about the behaviour of 100,000 call options. The hedging portfolio needs to compensate the investor by whatever amount the call option position changes in value.

 

[Darragh Kelly]

Ok I get you regarding the degrees of freedom, it makes sense. If you have another variable such as cash, then you could match the value too. Is that correct?

I did a calculation to understand what you said regarding the changes being hedged - if you increase the stock by 1 unit to 118.89 the total change in value of the put and shares = -18,673 and the total change in the call is +18,673, therefore we have hedged the call options (as they move in the opposite direction). And if the stock goes down by -1 unit to 116.89 the call options decrease in value by -18,673 and the put and shares increase to a total value of +18,673. Have I done this correctly?

Why is it here in this question, unlike questions where we replicated the behaviour of the call options (ie found the value and, ensured it was delta hedged), that the portfolio automatically moves in the opposite direction of the call option? For example we always in other questions where we replicated the value and delta hedged it, stated the end of the question 'in order to hedge the options we need to take opposite position in this portfolio to the position taken in the options'. Which is correct cause if you don't you just have a portfolio that replicates exactly the value of option but in order to hedge you take opposite position (as its the safety net). .
Many thanks,

 

[Steve Hales]

If you have another variable such as cash, then you could match the value too. Is that correct?

Yes, you could do that.

I did a calculation to understand what you said regarding the changes being hedged - if you increase the stock by 1 unit to 118.89 the total change in value of the put and shares = -18,673 and the total change in the call is +18,673, therefore we have hedged the call options (as they move in the opposite direction). And if the stock goes down by -1 unit to 116.89 the call options decrease in value by -18,673 and the put and shares increase to a total value of +18,673. Have I done this correctly?

Yes, that's how it works - nice numerical example!

Why is it here in this question, unlike questions where we replicated the behaviour of the call options (ie found the value and, ensured it was delta hedged), that the portfolio automatically moves in the opposite direction of the call option? For example we always in other questions where we replicated the value and delta hedged it, stated the end of the question 'in order to hedge the options we need to take opposite position in this portfolio to the position taken in the options'. Which is correct cause if you don't you just have a portfolio that replicates exactly the value of option but in order to hedge you take opposite position (as its the safety net). .

The difference between replicating and hedging was covered in a recent post here. Let me know if that doesn't do it.

 

[Darragh Kelly]

Ok I think I understand what you mean on hedging and replicating so I'm going to use a few examples to check:

2020 Sept Q6

In this question we are hedging a liability (we sold the put options so liability to us), and we want our portfolio to replicate the values in the same way ie if the put option goes up in value, so does our portfolio, if the liability goes down in value so does our portfolio. So we don't take the opposite position, unless we were in the long position ie the puts were an asset to us and we wanted to hedge this asset?

Please if you could confirm my logic.

Q17.5 (iii) from Acted notes

All parts in Q17.5 I fully understand its just the last part (iii).

So we know that the exotic forward is an asset to the investor, therefore we know when we replicate the portfolio we need to take the opposite position to hedge the asset?

My equation for calculating the vega are different to the notes. I use the fact that as we want the portfolio to be vega hedged then the sum of the vegs = 0 ie (+1)(1000)(-0.048522) + x*0.375240 + y*0 = 0. However I get x = +129.31. I used (+1) in from of the vega to denote that we are in the long position ie own the asset. When I used positive 129.31 I cannot get the same answer as the notes. Please can you clarify where I've gone wrong?

Generally for find the replicating portfolio I put a +/-1 in front of the delta/gamma/vega when summing to 0 to denote if we are in the long or short (ie if the derivative is a asset or liability to use). +1 for long position (own asset), -1 for short position (sold asset).

When I applied the above approach to the 2020 Sept Q6 (ii) it works for me ie (-1)(1000)(-0.446)+x*(delta,share)=0, x = -446 shares (short sell). And the rest of the question works out for me in terms of calculating the value of the replicating portfolio.

Can you please see where I'm going wrong?

2013 Sept Q9

So we didn't need to take the opposite position to hedge the asset as we worked out the holdings of the puts and stock based on the criteria of the portfolio being delta-gamma hedged? If the investor in this question didn't hold these calls, but instead sold them (so now calls are liabilities), in calculating the delta-gamma hedged portfolio, would I be correct in saying that the 2 equations would look like this:

delta,porfolio = (-1)*100,000*delta,call +y*delta,put +x*delta,share = 0
gamma,porfolio = (-1)*100,000*gamma,call + y*gamma,put + x*gamma,share = 0

(-1) in front of the delta,call and gamma,call to denote we are now in the short position (ie calls are liabilities).

Then the holdings would be worked out and these are the holdings required to hedge the liability of call options.

Please confirm if this logic is correct.

Apologise on the length of the question, it was hard to explain my issue without comparing a couple of examples.

Many thanks for your help.

Darragh

 

[Steve Hales]

Ok I think I understand what you mean on hedging and replicating so I'm going to use a few examples to check:

2020 Sept Q6

In this question we are hedging a liability (we sold the put options so liability to us), and we want our portfolio to replicate the values in the same way ie if the put option goes up in value, so does our portfolio, if the liability goes down in value so does our portfolio. So we don't take the opposite position, unless we were in the long position ie the puts were an asset to us and we wanted to hedge this asset?

Please if you could confirm my logic.

Yes. Liability (sold puts) and assets (portfolio) move together ensuring stability in total.

Q17.5 (iii) from Acted notes

All parts in Q17.5 I fully understand its just the last part (iii).

So we know that the exotic forward is an asset to the investor, therefore we know when we replicate the portfolio we need to take the opposite position to hedge the asset?

Yes, that's it.

2013 Sept Q9

So we didn't need to take the opposite position to hedge the asset as we worked out the holdings of the puts and stock based on the criteria of the portfolio being delta-gamma hedged? If the investor in this question didn't hold these calls, but instead sold them (so now calls are liabilities), in calculating the delta-gamma hedged portfolio, would I be correct in saying that the 2 equations would look like this:

delta,porfolio = (-1)*100,000*delta,call +y*delta,put +x*delta,share = 0
gamma,porfolio = (-1)*100,000*gamma,call + y*gamma,put + x*gamma,share = 0

(-1) in front of the delta,call and gamma,call to denote we are now in the short position (ie calls are liabilities).

Then the holdings would be worked out and these are the holdings required to hedge the liability of call options.

Please confirm if this logic is correct.

Darragh

That would do it. So long as the totals of the delta and gamma are both zero it's a hedge.

 

[Darragh Kelly]

thanks steve for your help