CT8 September 2013 - Q9(iv)

[JuppWeber]

Hi,
Is someone able to explain why the new holdings of the put and the stock are indeed 100,000? I appreciate the mark scheme notes as much and it must also be the case - but can't quite seem to get my head around it ...
Thanks in advance.

 

[John Potter]

Start with put-call parity. Differentiate twice. Delta(call) = Delta(put) + 1 and Gammas are equal.
Go from there and get back to me if this doesn't sort it
John

 

[NateJ95]

Start with put-call parity. Differentiate twice. Delta(call) = Delta(put) + 1 and Gammas are equal.
Go from there and get back to me if this doesn't sort it
John

Hi John, I am just going through this question now myself, still a little lost but I will explain where I am at the moment:

From the put call parity without any differentiation:

Ct + 150exp(-0.02) = 31.4517 + 117.98, fair enough I can calculate the price of the call here which is 2.401899 rounded.

From the usually delta formulas I can see that the delta of the put would be -0.81327 i.e. we would need to be short here. Given we are told that they are hedging the 100,000 calls with 18673 shares then we need to neutralize the delta so could we not say N (no of puts) * delta of put = 18673 or is that incorrect?

Just a little lost as to how they end up with the answer of 100,000 for both put options (can understand this somewhat) and 100,000 shares from this?

 

[John Potter]

Differentiate put-call parity twice:

c + Ke^-rT = p + S
DeltaC + 0 = DeltaP + 1
GammaC = GammaP

The portfolio is gamma hedged. The only way this could happen is if the investor is hedging their 100,000 calls by shorting 100,000 puts

This means that

Delta Portfolio = number of calls * DeltaC - number of puts * DeltaP + number of shares * 1 = 0

= 100,000 * DeltaC - 100,000 * (DeltaC - 1) + number of shares = 0

So, number of shares = -100,000